APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN

MOTION

HARVEY BARNHARD

Abstract This expository paper shows how waiting times of certain queuingsystems can be approximated by Brownian motion In particular when cus-tomers exit a queue at a slightly faster rate than they enter the waiting timeof the nth customer can be approximated by the supremum of reflected Brow-nian motion with negative drift Along the way we introduce fundamentalconcepts of queuing theory and Brownian motion This paper assumes somefamiliarity of stochastic processes

Contents

1 Introduction 22 Queuing Theory 221 Continuous-Time Markov Chains 322 Birth-Death Processes 523 Queuing Models 624 Properties of Queuing Models 73 Brownian Motion 831 Random Walks 932 Brownian Motion Definition 1033 Brownian Motion Construction 11331 Scaled Random Walks 11332 Skorokhod Embedding 12333 Strong Approximation 1434 Brownian Motion Variants 154 Heavy Traffic Approximation 17Acknowledgements 19References 20

Date November 21 2018

1

2 HARVEY BARNHARD

1 Introduction

This paper discusses a Brownian Motion approximation of waiting times in sto-chastic systems known as queues The paper begins by covering the fundamentalconcepts of queuing theory the mathematical study of waiting lines After waitingtimes and heavy traffic are introduced we define and construct Brownian motionThe construction of Brownian motion in this paper will be less thorough than othertexts on the subject We instead emphasize the components of the constructionmost relevant to the final resultmdashthe waiting time of customers in a simple queuecan be approximated with Brownian motion

Throughout the paper we use the terminology of a convenience store customerspopulate a queue and the amount of time it takes for a customer to exit the systemonce at the front of the line is called a service time This paper focuses on one of thesimplest queuing models and seeks to answer one simple question when customersare being serviced at a rate only slightly faster than customers are arriving howlong can a customer expect to wait in line before being serviced

2 Queuing Theory

Consider a queue in which inter-arrival times and service times are exponentiallydistributed Note that a process with exponential inter-arrival times may also beregarded as a Poisson process where the number of customers arriving in a giventime period follows the Poisson distribution LetNt denote the number of customersin line at time t Let λN and microN be the rates at which customers arrive and areserviced given that there are N customers in line For example if λN = 1 and theunit of time is seconds then we expect one arrival per second If λN = 12 then weexpect two arrivals per second Denote a(t) as the probability that the next arrivalwill occur t units of time from now and s(t) as the probability that the next servicewill be complete t units of time from now The probability distributions are thusgiven by

aN (t) = λNeminusλN t sN (t) = microNeminusmicroN t

So if A sim aN (t) and S sim sN (t) are random variables then

E [A] =1

λNand E [S] =

1

microN

Figure 1 depicts such a queue Rectangles represent customers waiting in line andthe circle represents customers currently being serviced

λN microN

Figure 1

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 3

We can also consider the possible number of customers in a queue as states sothat the state space consists of the non-negative integers where λN and microN are theprobabilistic rate of transition between states

0 1 2 3 4

λ0 λ1 λ2 λ3

micro4micro3micro2micro1

Figure 2

21 Continuous-Time Markov ChainsA system with a countable state-space and probabilistic transitions between statessuggests the use of Markov chains However standard Markov chains of elemen-tary probability theory operate in a discrete-time setting In this paper we con-sider exponential service and inter-arrival times As a result transitions betweenstates can occur anytime in the continuous timeline of non-negative reals Wetherefore aim to construct a countable-space continuous-time version of a discrete-time Markov chain We first introduce two properties that define continuous-timeMarkov chainsmdashthe Markov property and time-homogeneity

Definition 21 Consider a stochastic process Xt taking values in a state spaceS Xt is said to exhibit the Markov property if for y isin S and t ge s

PXt = y | Xr 0 le r le s = PXt = y | Xs

Definition 22 A time-homogenous Markov chain is a stochastic process Xttge0

such that

PXt = y | Xs = x = PXtminuss = y | X0

If a process exhibits time-homogeneity transition probabilities depend only onthe state of the process not the absolute time

Definition 23 A continuous-time Markov chain is a stochastic process Xttge0

taking values in a state space S and satisfying

PXt+∆t = i | Xt = i = 1minus qii∆t+ o(∆t)

PXt+∆t = j | Xt = i = qij∆t+ o(∆t)

where qij represents the transition rate from i isin S to j isin S and

qii equiv983131

j ∕=i

qij

The use of qij as a rate of transition between states requires further explanationEach qij is the derivative of pij(t) the probability of starting in state i and ending

4 HARVEY BARNHARD

in state j after ∆t units of time have elapsed (here we have assumed differentiabilityof pij(t) at ∆t = 0)

qij = lim∆trarr0

=PXt+∆t = j | Xt = i

∆t

The matrix P(t) is the transition matrix and its (i j)th element is pij(t)

P(t) =

983093

983097983095p11(t) p12(t) middot middot middotp21(t) p22(t) middot middot middot

983094

983098983096

The matrix Q is called the infinitesimal generator and its (i j)th element is qij Note that if zero time has elapsed then there is zero probability of transitioningout of the original state so pii(0) = 1 and P(0) = I We can therefore write

Q = Pprime(0)

and

P(t) = I+Q∆t

Example 25 will illustrate a deeper connection between qij and pij(t) but we mustfirst define what it means for two states to be communicable

Definition 24 Let i and j be two states of a Markov chain We say that i and jcommunicate if there is a positive probability that one state will lead to the otherstate given a certain amount of time In notation i and j communicate if thereexists time durations r gt 0 and s gt 0 such that

PXt+r = i | Xt = j PXt+s = j | Xt = i

A communication class is the collection of all such states that communicate withone another The set of communication classes partition the state space into disjointsets If there exists only one communication class then the Markov chain is said tobe irreducible In this paper the state-space of queues comprises the non-negativeintegers and is irreducible

Example 25 Let Xt be an irreducible continuous-time Markov chain Showthat for each i j and every t gt 0

P Xt = j | X0 = i gt 0

ProofFix i and j Since Xt is irreducible there exists some time t such that

P Xt = j|X0 = i gt 0

We want to show that this holds for all t Consider the sequence of events thatoccur between i and j and denote them as k1 kn where k1 occurs immediatelyafter i and kn occurs immediately before j It is clear that

qklkl+1gt 0 for all l

Or equivalently for some times s lt t

P Xt = kl+1 | Xs = kl gt 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 5

Now pick any time t gt 0 Note that there are potentially many paths from i to jand k1 kn represents one such path Therefore there is a non-zero probabilitythat i will go to j by this transition path In notation

P Xt = j | X0 = i genminus1983132

l=1

P983153X (l+1)t

n= kl+1|X lt

n= kl

983154gt 0

We now extend the result of the example to show how each continuous-timeMarkov chain induces a discrete-time Markov chain Let Xttge0 be an irreduciblecontinuous-time Markov chain Let S be a countable state-space and i j be statesin S Let t0 t1 t2 be distinct times with t0 lt t1 lt t2 Define

pij(t0 t2) equiv P Xt2 = j | Xt0 = ito be the probability of Xt being in state j at time t2 given that the process wasin state i at time t0 Then for each transition probability we can write

pij(t0 t2) =983131

jisinS

pij(t0 t1)pjk(t1 t2)

Enumerate the sequence of states in order of occurrence

n0 = 0

n1 = inft Xt ∕= X0n2 = inft ge n1 Xt ∕= Xn1

nm = inft ge nmminus1 Xt ∕= Xnmminus1then Xn equiv Xn n isin n0 n1 forms a discrete-time Markov chain out of acontinuous-time Markov chain This process is known as embedding and Xn iscalled an embedded Markov chain A similar procedure will be used later in Section33 when we construct a random walk a discrete-time process out of a Brownianmotion a continuous-time process Note that the embedding process causes us tolose information about the holding times of each state the amount of time spent ina state before transitioning to another state We therefore do not lose any furtherinformation by normalizing the time between states to some constant c gt 0

nm minus nmminus1 = c for all m ge 1

So after normalization the transition-rate matrix for Xn simply becomes P(c)as defined above

22 Birth-Death ProcessesA birth-death process is a non-negative integer valued continuous-time Markovchain That is a stochastic process Nt fulfilling the following two conditions

(1) Nt isin 0 1 2 3 (2) as s rarr 0 (Nt+s minusNt) isin 0minus1 1

The second condition states that for an infinitesimal change in time the popula-tion of the system will either increase by one decrease by one or remain the sameA birth-death process is a system where the rate of exit out of and entrance intothe system are known and the collection of such processes forms a large family of

6 HARVEY BARNHARD

stochastic processes For instance a Poisson process is one of the simplest birth-death processes where the ldquobirthrdquo rate is constant and the ldquodeathrdquo rate is zero Inparticular a queuing process is a birth-death process where customer arrivals areldquobirthsrdquo and service completions are ldquodeathsrdquo

23 Queuing Models

Example 26 (MMc) In this model inter-arrival times and service times areexponentially distributed with rates λ and micro respectively In this queue λ and microare constant and independent of the current state That is λN = λ and microN = micro forall N Service is received on a ldquofirst come first servedrdquo basis and c customers maybe serviced at a time The simplest case is when c = 1 where only one customeris serviced at a time In an MMc queue where customers are serviced as soon asthey arrive regardless of the population of the system we set c = infin The MMcmodel assumes that no customers leave the system between arrival and completionof service

This paper focuses on the MM1 queue but we introduce a few more modelsas examples of the breadth of situations that can be analyzed through the lens ofqueuing theory

Example 27 (GGc) This model is the same as the MMc model exceptinter-arrival times and service times follow some general (represented by G) notnecessarily exponential distribution

Example 28 (GGcK) This model is the same as the GGc model but withan upper bound K on the number of customers that can occupy the system at agiven time This type of queue is called a truncated queue

Example 29 (ABCDE) The above queuing models are identified usingKendall Notation a combination of letters and slashes Each position in the nota-tion corresponds to a certain queuing characteristic (eg inter-arrival time distri-bution) and the letter specifies the characteristic (eg exponentially distributedinter-arrival times) The following table covers a wide selection of queuing modelsthat are easily be described using Kendall notation

Characteristic Symbol Explanation(A) Inter-arrival time distribution M Exponential(B) Service time distribution D Deterministic

Ek Erlang type kG General

(C) of parallel servers 1 2 infin(D) Max system capacity 1 2 infin(E) Order of service FCFS First come first served

LCFS Last come first servedRSS Random selectionPR PriorityGD General

The table above is by no means a definitive list of queuing characteristics Forinstance inter-arrival and service times may be state-dependent as introduced inthe beginning of this section with rates λN and microN Queues can also be cyclic in

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 7

the sense that once a customer is serviced the customer immediately returns to aposition in line to await service once again

24 Properties of Queuing ModelsThe first two properties one may ask of a queue are the expected number of cus-tomers in line and the duration of time a customer must wait before being ser-viced Let Nt be the number of customers in a GGc queue at time t If we letpn = PNt = n the expected population of the system is

E[Nt] =

infin983131

n=0

npn

and the expected population of those in the queue (and not being serviced) is

E[Ntq] =

infin983131

n=0

(nminus c)pn

Definition 210 The nth waiting time is the amount of time the nth customerspends waiting in line prior to entering service We denote waiting time as the

random variable W(n)q and the total time a customer is in the system including

service time as W (n) The nth customer is the nth overall customer to enter thesystem not the customer in the nth position in line

We now consider the waiting time of the nth customer to enter a queuing systemwhere the initial population is zero that is N0 = 0 Let S(n) be the nth servicetime and let I(n) be the nth inter-arrival time ie the time between the (nminus 1)stcustomer and the nth customer arriving in the system Define

U (n) equiv S(n) minus I(n)

to be the time between the nth inter-arrival time and the nth service time

Theorem 211 (Lindleyrsquos Equation) In a single-server queue where customersare serviced on a first-come first-served basis the waiting time of the (n + 1)thcustomer is recursively given by

W (n+1)q = max

9831530W (n)

q + S(n) minus I(n)983154

ProofSince the initial population of the system is zero the first customer to arrive will

be serviced immediately so W(1)q = 0 The waiting time for the second customer

will be the time it takes for the first customer to finish being serviced If the firstcustomer has already been serviced by the time the second customer arrives thenthe waiting time for the second customer will be zero

W (2)q = max

9831530 S(1) minus I(1)

983154= max

9831530W (1)

q + S(1) minus I(1)983154

For n ge 3 the waiting time of the nth customer is simply the waiting time of the(nminus 1)th customer and the amount of time it takes for the (nminus 1)th customer tobe serviced or zero if the (nminus 1)th customer has already been serviced by the timethe nth customer enters the system

Lemma 212 Under the same conditions as the preceding theorem W(n)q may be

rewritten as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

8 HARVEY BARNHARD

Proof

We prove the lemma by iterating over W(n)q

W (n)q = max

983153W (nminus1)

q + U (nminus1) 0983154

= max983153max

983153W (nminus2)

q + U (nminus2) 0983154+ U (nminus1) 0

983154

= max983153max

983153W (nminus2)

q + U (nminus2) + U (nminus1) U (nminus1)983154 0983154

= max983153W (nminus2)

q + U (nminus2) + U (nminus1) U (nminus1) 0983154

Continue this process in a recursive manner until we have

W (n)q = max

983153W (1)

q + U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1)983154

The result of the lemma is now immediate since we have assumed thatW(1)q = 0

The goal of this paper is to approximate W(n)q the waiting time of the nth

customer From the preceding lemma it is clear that the waiting time depends onthe waiting times of all preceding customers in line More precisely the waitingtime of the nth customer is the maximum of the partial sums of the decreasingsequence

983051U (nminusi)

983052n

i=1 an iid sequence of random variables Section 3 will show

that this decreasing sequence is a random walk and when n is sufficiently large thissequence can be approximated by Brownian motion For such an approximation tobe accurate however each U (nminusi) must be sufficiently small small This situationarises in queues that exhibit heavy traffic

Definition 213 The traffic intensity ρ of a given queue is defined as the ratioof arrival times to service times So for an MMc queue the traffic intensity is

ρ =λ

cmicro

where λ and micro are the arrival and service rate respectively of customers in thequeue For a single server MM1 queue the traffic intensity is

ρ =λ

micro

Remark 214 When ρ is close to zero waiting times approach zero as queue lengthshortens and new customers are serviced immediately upon arrival When ρ isgreater than one waiting times successively increase as the queue population ldquoex-plodesrdquo to infinity The interesting case occurs when ρ is close tomdashbut does notexceedmdashone A system is said to be in heavy traffic when ρ isin (1minus ε 1) and ε gt 0is small

3 Brownian Motion

This section introduces the stochastic process of Brownian motion viewed as thelimit of random walks In this paper we confine our study of Brownian motion tothe one-dimensional case

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 9

31 Random Walks

Definition 31 A one-dimensional random walk is a stochastic process constructedas the sum of iid random variables That is if Xi is a sequence of iid randomvariables with Xi isin R (regarded as steps) then the sum of the first n such randomvariables

Rn =

n983131

i=1

Xi

is the n-th value of a random walk A random walk is said to be symmetric ifXi isin R is symmetrically distributed about zero A random walk is called simple ifPXi = 1+ PXi = minus1 = 1

Example 32 Figure 3 depicts a simple random walk simulated using the statisti-cal computing software R with PXi = minus1 = PXi = 1 = 1

2 and 15 total stepsTaking R0 = X0 = 0 the values of Xi and Ri are

Xi15i=1 = 1minus1 1minus1minus1minus1minus1 1minus1minus1 1 1minus1 1minus1Ri15i=1 = 0 1 0 1 0minus1minus2minus3minus2minus3minus4minus3minus2minus3minus2minus3

0 5 10 15

minus4minus3

minus2minus1

01

n

S n

Figure 3

An important property of random walks is that they exhibit independent incre-ments meaning that for any selection of positive integers t1 lt t2 lt middot middot middot lt tn therandom variables

Rt2 minusRt1 Rtn minusRtnminus1

are independent This property immediately follow from the fact that each incre-ment is the sum of iid random variables

10 HARVEY BARNHARD

32 Brownian Motion DefinitionWe now introduce the continuous-time analog to a random walkmdashBrownian motionA one-dimensional standard Brownian motion with variance parameter σ2 is a real-valued process Bttge0 defined in the probability space (ΩF P) that has thefollowing properties

(I) If t0 lt t1 lt middot middot middot lt tn then Bt0 Bt1 minusBt0 Btn minusBtnminus1are independent

(II) If s t ge 0 then

P (Bs+t minusBs isin A) =

983133

A

1radic2πσ2t

exp(minusx22σ2t)

(III) With probability 1 t 983041rarr Bt is continuous

The first two conditions may be concisely reworded as

(I) Independent increments(II) Increment lengths are normally distributed with mean 0 and variance σ2t

Bs+t minusBs sim N(0σ2t)

Remark 33Brownian motion is a collection of stochastic processes When we say a Brownianmotion or the Brownian motion we are talking about a particular realization ofBrownian motion in the space of continuous functions C[0 T ] where T is a stoppingtime

The above definition establishes the necessary conditions for a stochastic process tobe Brownian but the existence of such a process requires further work Throughan application of the Kolmogorov extension theorem existence is proved in [4]However such a proof is bereft of intuition for queueing theory and approximatingwaiting times Instead we take the existence of Brownian motion as given andshow that Brownian motion can be constructed as a limit of random walksmdashtheresult of Donskerrsquos Theorem Brownian motion constructed in such a manner iscalled the standard Wiener process

Before constructing Brownian motion we must first establish what type of con-vergence we are discussing

Definition 34 A sequence of random variables Xiinfini=1 converges almost surelyto a random variable X in the underlying probability space (ΩF P) (denoted

Xnasminusminusrarr X) if

P983153ω isin Ω lim

nrarrinfinXn(ω) = X(ω)

983154= 1

We aim to show almost sure convergence of a scaled random walk to a Brownianmotion in (C[0 T ] d) the metric space of continuous functions on the interval [0 T ]with d as the supremum norm

d(f g) = suptisin[0T ]

|f(t)minus g(t)|

Note that random walks and Brownian motions are stochastic processes while dmeasures the distance between two continuous functions In order to measure thedistance between random walks and Brownian motion we instead consider the pathstaken by these stochastic processes

t 983041rarr Rt and t 983041rarr Bt

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 11

33 Brownian Motion Construction In this paper the construction of Brow-nian motion comprises the following steps

(1) Construct a simple random walk out of a Brownian motion path usingSkorokhod embedding

(2) Linearly interpolate and scale the embedded random walk creating a con-tinuous function from [0 n] 983041rarr R

(3) Take the uniform limit of this scaled random walk to derive the originalBrownian motion path

The paper presents an alternative formulation to the canonical construction ofBrownian motion which involves defining Brownian motion on the dyadic rationalsand using continuity to extend the definition to the reals To motivate this particu-lar construction we will proceed out of order we will begin with step (2) backtrackto step (1) and conclude by stringing together steps (1) through (3) in subsection333 (Strong Approximation)

331 Scaled Random WalksLet Rm be a random walk where E[Xi] = 0 VarXi = σ2 and

Rm =

m983131

i=1

Xi

Definition 35 A scaled random walk with scaling parameter n is a real valuedrandom process

983051R(n)

983052 defined on t ge 0 For t such that nt is an integer

R(n)t =

1radicnσ2

Rnt

For t such that nt is not an integer S(n)(t) is defined by linear interpolation

R(n)t =

1radicnσ2

983063Rlfloorntrfloor + (ntminus lfloorntrfloor)

983043Rlceilntrceil minusRlfloorntrfloor

983044 983064

This method of scaling operates on two aspects of random walks as n increasesthe time between state transitions decreases and increment lengths decrease Lett0 = 0 and tk = inftkt isin Z nt gt nt(kminus1) then as n rarr infin

tk minus t(kminus1) rarr 0 and R(n)tk

minusR(n)t(kminus1)

rarr 0

As a result R(n)(t) is continuous with probability 1 on t ge 0 for all n The scalingfactor 1radic

nσ2ensures constant variance for a given t

Var

9830691radicnσ2

Rnt

983070= Var

9830831radicnσ2

nt983131

i=1

Xi

983084=

9830611radicnσ2

9830622

ntVarXi = tVar Xi

σ2= t

By observing the limiting behavior of R(n)t (ie as n rarr infin) we establish a con-

struction of Brownian motion

12 HARVEY BARNHARD

332 Skorokhod EmbeddingTo construct Brownian motion out of random walks we must first construct randomwalks out of Brownian motion Structuring the proof in this matter is not withoutreason there are uncountably many realizations of Brownian motion so we want toensure that the random walks to which we are taking the limit do in fact convergeto the Brownian motion of interest Such a procedure of deriving a random walkfrom a Brownian motion is called Skorokhod embedding The procedure is given inthe following pseudocode

Set t0 = 0 and n = 0 Let T be a stopping time of Brownian motion Btso that Bt is defined on [0 T ] Let R0 = 0 be the initial value

of the Skorokhod embedding

While tn + t le TInitialize t = 0

While |Bt minusBtn | le 1 Continue Bt until |Bt minusBtn | = 1 for the first time

Set t(n+1) = inft |Bt minusBtn | = 1If Bt(n+1)

minusBtn = 1 then set X(n+1) = 1

Otherwise Bt(n+1)minusBtn = minus1 so set X(n+1) = minus1

Let the (n+ 1)th value of the Skorokhod embedding be

R(n+1) = Rn +Xn+1

Increment to the next step of the inner while loop n = n+ 1

The resulting sequence Rnn=12 is the Skorokhod embedding of Bttisin[0T ]

Theorem 36 Let Bt be a standard Brownian motion Set t0 = 0 and let tn bethe stopping time where |Bt| = 1 for the nth time

tn equiv inft ge 0 |Bt minusBtn | = 1

If we define Rn equiv Btn then Rn is a simple random walk

ProofTo prove that B0 Bt1 Bt2 is a simple random walk we merely need to showthat

(1) PBtn+1minusBtn = 1 = PBtn+1

minusBtn = minus1 = 12 for all n isin N

(2) tn+1 minus tn are iid random variables

The first item follows from the symmetric property of Brownian increments prop-erty (II) in the definition of Brownian motion Bs+t minusBs sim N(0σ2t)

The second item follows from the strong Markov property of Brownian motionwhich states that if τ is a stopping time with respect to Bt then Bt minus Bτ is

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 13

also a Brownian motion We do not include a rigorous treatment of stopping timesor a proof of the strong Markov property

We now state a theorem that places a probabilistic bound on the differencebetween a Brownian motion and its Skorokhod embedding

Theorem 37 Define Θ to be the maximum distance between a Brownian motionand its Skorokhod embedding

Θ(BRT ) = max0letleT

|Bt minusRt|

Note that Θ is equivalent to the distance between Bt and Rt in the space ofcontinuous functions on the time interval [0 T ] equipped with the supremum normThere exist c a isin [0infin) such that for all r le n14 and all integers n ge 3

P983153Θ(BRT ) ge rn14

983155log n

983154le ceminusar

ProofIt will suffice to prove the theorem for r ge 9c2 where c is the constant such that

P983153osc(B δ T ) gt r

983155δ log(1δ)

983154le cT δ(rc)

2

andosc(B δ T ) = sup |Bt minusBs| s t isin D s t isin [0 T ] |sminus t| le δ

is the oscillation of Bt restricted to t isin D the dyadic rationals The proof thatsuch a c exists can be found in [3] page 68 where it is also shown that for n isin N

Θ(BSn) le 1 + osc(B 1 n) + max|Bj minusBτj | j = 1 nNow suppose 9c2 le r le n14 If |Bn minusBτn | is large then either |nminus τn| is large

or the oscillation of B is large Consider the three events

(1)983051Θ(BRn) ge rn14

radiclog n

983052

(2)983051osc(B r

radicn 2n) ge (r3)n14

radiclog n

983052

(3) max1lejlen |τj minus j| ge rradicn

From above we know that event (1) is contained in the union of events (2) and (3)As a consequence we need only prove the result of the theorem for events (2) and(3) We first tackle event (2)

P983153osc(B r

radicn 2n) gt (r3)n(14)

983155log n

983154le 3P

983153osc(B r

radicn n) gt (r3)n(14)

983155log n

983154

= 3P983153osc(B rnminus12) gt (r3)nminus(14)

983155log n

983154

le 3P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log

983043n12r

983044983070

Ifradicr3 ge c and r le n14 we can conclude that there exist c and a such that

P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log(n12)

983070le ceminusar logn

For event (3) we refer to a proof on page 266 in the appendix of [3] to show thatthere exist c a such that

P983069

max1lejlen

|τj minus j| gt rradicn

983070le ceminusar2

14 HARVEY BARNHARD

333 Strong ApproximationWith preliminaries in order we now show that Brownian motion can be thought ofas a limit of simple random walks a construction known as the strong approximationof Brownian motion Let Bt be a standard Brownian motion with variance

parameter 1 as defined in the probability space (ΩF P) and let B(n)t be the

scaled Brownian motion

B(n)t =

1radicnBnt

From B(n) we use the Skorokhod embedding to derive the simple random walkS(n) And from S(n) we derive the scaled simple random walk R(n)

R(n)t =

1radicnS(n)nt

From Theorem 37 we know that there exists c a isin [0infin) such that for all positiveintegers T

P983069

max0letleTn

983055983055983055S(n)t minusB

(n)t

983055983055983055 ge cr(Tn)14983155log (Tn)

983070le ceminusar

multiplying by 1radicn this becomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge crT 14nminus14983155log (Tn)

983070le ceminusar

Letting r = c log n where c is sufficiently large for the given T the inequalitybecomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

983070le c

n2

To proceed we require the use of the Borel-Cantelli lemma the result of which isprovided below

Lemma 38 (Borel-Cantelli)If An is a sequence of events in Ω and

983123infini=1 PAn lt infin then

PAn io = 0

where An io = lim supnrarrinfin An is the event that infinitely many An occur

If we set

An = max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

we can then apply the Borel-Cantelli lemma to the above inequality to concludethat

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 le cnminus14 log32 n

with probability one for all n sufficiently large Since

limnrarrinfin

cnminus14 log32 n = 0

we conclude that

R(n)t

asminusminusrarr Bt in C[0 T ]The following theorem condenses the results of this section up to this point

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 15

Theorem 39 (Donskerrsquos Theorem) Let Xi be a sequence of iid random vari-

ables with mean 0 and variance σ2 If R(n)t is a scaled random walk as defined

above then R(n)t rArr Bt where Bt is a standard Brownian motion This result is

also known as the functional central limit theorem

34 Brownian Motion VariantsThe preceding construction of Brownian motion will prove particularly useful in ourapproximation of heavy traffic waiting times However this process requires furthermodification Recall that when traffic intensity ρ is slightly less than 1 customersare being serviced at a slightly faster rate then they are entering the queue Thusgiven enough time traffic will eventually ldquoclear outrdquo due to faster service timesHowever this particular construction of Brownian motion has increments of meanzero

E[Bs+t minusBs] = 0

meaning that the approximate number of customers in line wouldmdashon averagemdashremain unchanged regardless of how long the line is observed Such a propertyis called the martingale property Instead we want a process that exhibits thesupermartingale property

E[Bs+t minusBs] lt 0

Definition 310 Let Fn be an increasing sequence of σ-fields (Fn is called afiltration) Let Xn be a sequence of random variables with finite mean andXn isin Fn for all n Martingales supermartingales and submartingales are thendefined as follows for all n

Martingale E [Xn+1 | Fn] = Xn

Supermartingale E [Xn+1 | Fn] le Xn

Submartingale E [Xn+1 | Fn] ge Xn

In words a sequence that we expect to remain the same over time is a martingalea sequence we expect to increase is a submartingale and a sequence we expect todecrease is a supermartingale To approximate heavy traffic we want a Brownianmotion-like process that exhibits the supermartingale property

Definition 311 A Brownian motion with drift parameter α and variance param-eter σ2 Bα

t is defined by the following properties

(1) Independent increments(2) Increment lengths are normally distributed with mean αt and variance σ2t(3) the path t 983041rarr Bα

t is continuous

Note that these properties are the same as standard Brownian motion exceptincrement lengths now have non-zero means Since we have already shown theexistence of standard Brownian motion the construction of Brownian motion withdrift is now easy

Lemma 312 Let Bt be a standard Brownian motion with variance parameterσ2 then Bα

t = Bt + αt is Brownian motion with drift parameter α and varianceparameter σ2 Moreover Bα

t fulfills the supermartingale property when α lt 0

16 HARVEY BARNHARD

ProofIndependence of increments follows directly from independence of increments forBt and the fact that αt is a constant for a given t To prove that Bα

t has stationaryincrements we have to show that the distribution of increments Bα

s+tminusBαs depends

only the time interval t and not the absolute time Recall from the definition of Bt

that Bs+t minusBt are normal random variables with mean 0 and variance t for s gt 0

E983045Bα

s+t minusBαs

983046= E

983045Bα

s+t

983046minus E [Bα

s ]

= E983045Bα

s+t + α(s+ t)983046minus E [Bs + αs]

= α(s+ t)minus αs

= αt

Var983051Bα

s+t minusBαs

983052= Var Bs+t minusBs + αt= Var Bs+t minusBs= σ2t

And since for fixed t αt is just a constant we use the property of normal randomvariables to conclude that

Bαs+t minusBα

s sim N(αtσ2t) for all s gt 0

Continuity of Bαt directly follows from continuity of Bt We have shown that

Bαt constructed as the sum of standard Brownian motion and drift αt is indeed

Brownian motion with drift

Brownian motion with negative drift provides a better approximation of thewaiting-time of customers in a queue than does standard Brownian motion How-ever the unboundedness of Brownian motion suggests that waiting times couldperhaps be negative In order to circumvent this issue we instead use reflectedBrownian motion with drift Reflected Brownian motion acts much the same asBrownian motion except on a given boundary off of which the Brownian motionismdashunsurprisinglymdashreflected In the one-dimensional case the boundaries may beany interval on the reals that provide a lower and upper bound to the Brownianmotion For our purposes we only use a single boundary zero that gives a lowerbound to Brownian motion That is for all t ge 0 we want Bα

t ge 0

Definition 313 Let Bαt be a standard Brownian motion with drift parameter α

and let Mαt be the running maximum of Bα

t

Mαt = sup

0lesletBα

s

Then reflected Brownian motion with drift parameter minusα and boundary 0 definedon R+ can be constructed as follows

Rα0t equiv max0Mα

t minusBαt

Note that max0 Sαt ge 0 and Mα

t ge Bαt B

α0t ge 0 for all t Most importantly

the local behavior of Bα0t is exactly like Brownian motion with drift parameter minusα

since max0 Sαt is simply a constant when Bα

t ∕= 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 17

We have now provided sufficient modifications of Brownian motion and we nextmove onmdashwhat we will ultimately use to approximate heavy traffic waiting timesThe next example provides us with the distribution of the supremum of a Brownianmotion with negative drift

Example 314Let Bt be a Brownian motion with drift α lt 0 and variance parameter σ2 Define

Mt = sup0leslet

Bs

to be the running maximum then

Minfin sim exp

9830612|α|σ2

983062

that is

PW ge w = eminusminus2|α|σ2 w w ge 0

E [Minfin] =σ2

2|α|

Prooffor constants a b gt 0 Let T (minusa b) be the first time that B minus t hits minusa or b

T (minusa b) = inft Bt = minusa or Bt = bIn [7] it is proved that

P983051BT (minusab)=b

983052=

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)

Since α lt 0 exp(2αaσ2) rarr 0 we have

limararrinfin

PBT (minusab) = b = limararrinfin

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)= exp(2αbσ2)

The left-hand side becomes the probability that the process will reach b somewherealong its path (ie that the maximum of the process exceeds b somewhere alongits path) Therefore we have

PW ge b = exp(2αbσ2) = expminus2|α|bσ2

This example shows us that the supremum of a Brownian motion observed ldquofor-everrdquo (ie on the interval [0infin)) is exponentially distributed with rate 2|α|σ2

4 Heavy Traffic Approximation

We now have all we need to provide a heuristic argument for the approximationof single-servers queue in heavy traffic using Brownian Motion For a rigorousalbeit opaque treatment of this approximation see [8] the seminal work on thissubject by Kingman

Theorem 41 The waiting time of the nth customer in an MM1 queue canbe approximated by the supremum of reflected Brownian motion with negative driftand boundary at 0

18 HARVEY BARNHARD

ProofRecall that an MM1 queue is a process where the number of customer arrivalsfollows a poisson process with rate λ and service times are exponentially distributedwith rate micro From Lindleyrsquos equation in Section 2 we can write the waiting timefor the nth customer as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

where U (n) = S(n)minus I(n) is the difference between the nth service time and the nthinter-arrival time Define the partial sum P

(n)k to be zero when k = 0 and

P(n)k equiv

k983131

i=1

U (nminusi) for k ge 1

We can then rewrite the waiting time of the nth customer as

W (n)q = max

0lekle(nminus1)P

(n)k

The fact that P(n)k is the kth value of a random walk follows immediately from

U (1) U (nminus1) being iid random variables Let α and σ2 be the expectationand variance of U (i) We calculate α and σ2 by first noting that S(i) and I(i) areexponentially distributed with rates micro and λ Recall that ρ = λmicro is the trafficintensity as defined in the first section of this paper

α = E983147U (i)

983148= E

983147S(i)

983148minus E

983147I(i)

983148=

1

microminus 1

λ=

ρminus 1

λ

σ2 = Var983153U (i)

983154= Var

983153S(i)

983154+Var

983153I(i)

983154=

1

micro2+

1

λ2

We next compute the expectation and variance of P(n)k using the fact that U (i) are

iid random variables

E983147P

(n)k

983148=

k983131

i=1

E983147U (nminusi)

983148= kα

Var983153P

(n)k

983154=

k983131

i=1

Var983153U (nminusi)

983154= kσ2

For a finite n W(n)q is therefore the maximum value of a random walk with n steps

We now show that Pnk can be approximated by Brownian motion with drift

parameter α and variance parameter σ2 when n is sufficiently large We restate theproperties of such a Brownian motion Bt below

(1) B0 = 0(2) Bt has stationary and independent increments(3) Bt sim N(αtσ2t)(4) t 983041rarr Bt is continuous with probability one

The first property is clear as P(n)0 = 0 for all n The second property follows from

the fact that U (i) are iid random variables The third property is a result of thecentral limit theorem for iid sequences the result of which is provided below

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

2 HARVEY BARNHARD

1 Introduction

This paper discusses a Brownian Motion approximation of waiting times in sto-chastic systems known as queues The paper begins by covering the fundamentalconcepts of queuing theory the mathematical study of waiting lines After waitingtimes and heavy traffic are introduced we define and construct Brownian motionThe construction of Brownian motion in this paper will be less thorough than othertexts on the subject We instead emphasize the components of the constructionmost relevant to the final resultmdashthe waiting time of customers in a simple queuecan be approximated with Brownian motion

Throughout the paper we use the terminology of a convenience store customerspopulate a queue and the amount of time it takes for a customer to exit the systemonce at the front of the line is called a service time This paper focuses on one of thesimplest queuing models and seeks to answer one simple question when customersare being serviced at a rate only slightly faster than customers are arriving howlong can a customer expect to wait in line before being serviced

2 Queuing Theory

Consider a queue in which inter-arrival times and service times are exponentiallydistributed Note that a process with exponential inter-arrival times may also beregarded as a Poisson process where the number of customers arriving in a giventime period follows the Poisson distribution LetNt denote the number of customersin line at time t Let λN and microN be the rates at which customers arrive and areserviced given that there are N customers in line For example if λN = 1 and theunit of time is seconds then we expect one arrival per second If λN = 12 then weexpect two arrivals per second Denote a(t) as the probability that the next arrivalwill occur t units of time from now and s(t) as the probability that the next servicewill be complete t units of time from now The probability distributions are thusgiven by

aN (t) = λNeminusλN t sN (t) = microNeminusmicroN t

So if A sim aN (t) and S sim sN (t) are random variables then

E [A] =1

λNand E [S] =

1

microN

Figure 1 depicts such a queue Rectangles represent customers waiting in line andthe circle represents customers currently being serviced

λN microN

Figure 1

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 3

We can also consider the possible number of customers in a queue as states sothat the state space consists of the non-negative integers where λN and microN are theprobabilistic rate of transition between states

0 1 2 3 4

λ0 λ1 λ2 λ3

micro4micro3micro2micro1

Figure 2

21 Continuous-Time Markov ChainsA system with a countable state-space and probabilistic transitions between statessuggests the use of Markov chains However standard Markov chains of elemen-tary probability theory operate in a discrete-time setting In this paper we con-sider exponential service and inter-arrival times As a result transitions betweenstates can occur anytime in the continuous timeline of non-negative reals Wetherefore aim to construct a countable-space continuous-time version of a discrete-time Markov chain We first introduce two properties that define continuous-timeMarkov chainsmdashthe Markov property and time-homogeneity

Definition 21 Consider a stochastic process Xt taking values in a state spaceS Xt is said to exhibit the Markov property if for y isin S and t ge s

PXt = y | Xr 0 le r le s = PXt = y | Xs

Definition 22 A time-homogenous Markov chain is a stochastic process Xttge0

such that

PXt = y | Xs = x = PXtminuss = y | X0

If a process exhibits time-homogeneity transition probabilities depend only onthe state of the process not the absolute time

Definition 23 A continuous-time Markov chain is a stochastic process Xttge0

taking values in a state space S and satisfying

PXt+∆t = i | Xt = i = 1minus qii∆t+ o(∆t)

PXt+∆t = j | Xt = i = qij∆t+ o(∆t)

where qij represents the transition rate from i isin S to j isin S and

qii equiv983131

j ∕=i

qij

The use of qij as a rate of transition between states requires further explanationEach qij is the derivative of pij(t) the probability of starting in state i and ending

4 HARVEY BARNHARD

in state j after ∆t units of time have elapsed (here we have assumed differentiabilityof pij(t) at ∆t = 0)

qij = lim∆trarr0

=PXt+∆t = j | Xt = i

∆t

The matrix P(t) is the transition matrix and its (i j)th element is pij(t)

P(t) =

983093

983097983095p11(t) p12(t) middot middot middotp21(t) p22(t) middot middot middot

983094

983098983096

The matrix Q is called the infinitesimal generator and its (i j)th element is qij Note that if zero time has elapsed then there is zero probability of transitioningout of the original state so pii(0) = 1 and P(0) = I We can therefore write

Q = Pprime(0)

and

P(t) = I+Q∆t

Example 25 will illustrate a deeper connection between qij and pij(t) but we mustfirst define what it means for two states to be communicable

Definition 24 Let i and j be two states of a Markov chain We say that i and jcommunicate if there is a positive probability that one state will lead to the otherstate given a certain amount of time In notation i and j communicate if thereexists time durations r gt 0 and s gt 0 such that

PXt+r = i | Xt = j PXt+s = j | Xt = i

A communication class is the collection of all such states that communicate withone another The set of communication classes partition the state space into disjointsets If there exists only one communication class then the Markov chain is said tobe irreducible In this paper the state-space of queues comprises the non-negativeintegers and is irreducible

Example 25 Let Xt be an irreducible continuous-time Markov chain Showthat for each i j and every t gt 0

P Xt = j | X0 = i gt 0

ProofFix i and j Since Xt is irreducible there exists some time t such that

P Xt = j|X0 = i gt 0

We want to show that this holds for all t Consider the sequence of events thatoccur between i and j and denote them as k1 kn where k1 occurs immediatelyafter i and kn occurs immediately before j It is clear that

qklkl+1gt 0 for all l

Or equivalently for some times s lt t

P Xt = kl+1 | Xs = kl gt 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 5

Now pick any time t gt 0 Note that there are potentially many paths from i to jand k1 kn represents one such path Therefore there is a non-zero probabilitythat i will go to j by this transition path In notation

P Xt = j | X0 = i genminus1983132

l=1

P983153X (l+1)t

n= kl+1|X lt

n= kl

983154gt 0

We now extend the result of the example to show how each continuous-timeMarkov chain induces a discrete-time Markov chain Let Xttge0 be an irreduciblecontinuous-time Markov chain Let S be a countable state-space and i j be statesin S Let t0 t1 t2 be distinct times with t0 lt t1 lt t2 Define

pij(t0 t2) equiv P Xt2 = j | Xt0 = ito be the probability of Xt being in state j at time t2 given that the process wasin state i at time t0 Then for each transition probability we can write

pij(t0 t2) =983131

jisinS

pij(t0 t1)pjk(t1 t2)

Enumerate the sequence of states in order of occurrence

n0 = 0

n1 = inft Xt ∕= X0n2 = inft ge n1 Xt ∕= Xn1

nm = inft ge nmminus1 Xt ∕= Xnmminus1then Xn equiv Xn n isin n0 n1 forms a discrete-time Markov chain out of acontinuous-time Markov chain This process is known as embedding and Xn iscalled an embedded Markov chain A similar procedure will be used later in Section33 when we construct a random walk a discrete-time process out of a Brownianmotion a continuous-time process Note that the embedding process causes us tolose information about the holding times of each state the amount of time spent ina state before transitioning to another state We therefore do not lose any furtherinformation by normalizing the time between states to some constant c gt 0

nm minus nmminus1 = c for all m ge 1

So after normalization the transition-rate matrix for Xn simply becomes P(c)as defined above

22 Birth-Death ProcessesA birth-death process is a non-negative integer valued continuous-time Markovchain That is a stochastic process Nt fulfilling the following two conditions

(1) Nt isin 0 1 2 3 (2) as s rarr 0 (Nt+s minusNt) isin 0minus1 1

The second condition states that for an infinitesimal change in time the popula-tion of the system will either increase by one decrease by one or remain the sameA birth-death process is a system where the rate of exit out of and entrance intothe system are known and the collection of such processes forms a large family of

6 HARVEY BARNHARD

stochastic processes For instance a Poisson process is one of the simplest birth-death processes where the ldquobirthrdquo rate is constant and the ldquodeathrdquo rate is zero Inparticular a queuing process is a birth-death process where customer arrivals areldquobirthsrdquo and service completions are ldquodeathsrdquo

23 Queuing Models

Example 26 (MMc) In this model inter-arrival times and service times areexponentially distributed with rates λ and micro respectively In this queue λ and microare constant and independent of the current state That is λN = λ and microN = micro forall N Service is received on a ldquofirst come first servedrdquo basis and c customers maybe serviced at a time The simplest case is when c = 1 where only one customeris serviced at a time In an MMc queue where customers are serviced as soon asthey arrive regardless of the population of the system we set c = infin The MMcmodel assumes that no customers leave the system between arrival and completionof service

This paper focuses on the MM1 queue but we introduce a few more modelsas examples of the breadth of situations that can be analyzed through the lens ofqueuing theory

Example 27 (GGc) This model is the same as the MMc model exceptinter-arrival times and service times follow some general (represented by G) notnecessarily exponential distribution

Example 28 (GGcK) This model is the same as the GGc model but withan upper bound K on the number of customers that can occupy the system at agiven time This type of queue is called a truncated queue

Example 29 (ABCDE) The above queuing models are identified usingKendall Notation a combination of letters and slashes Each position in the nota-tion corresponds to a certain queuing characteristic (eg inter-arrival time distri-bution) and the letter specifies the characteristic (eg exponentially distributedinter-arrival times) The following table covers a wide selection of queuing modelsthat are easily be described using Kendall notation

Characteristic Symbol Explanation(A) Inter-arrival time distribution M Exponential(B) Service time distribution D Deterministic

Ek Erlang type kG General

(C) of parallel servers 1 2 infin(D) Max system capacity 1 2 infin(E) Order of service FCFS First come first served

LCFS Last come first servedRSS Random selectionPR PriorityGD General

The table above is by no means a definitive list of queuing characteristics Forinstance inter-arrival and service times may be state-dependent as introduced inthe beginning of this section with rates λN and microN Queues can also be cyclic in

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 7

the sense that once a customer is serviced the customer immediately returns to aposition in line to await service once again

24 Properties of Queuing ModelsThe first two properties one may ask of a queue are the expected number of cus-tomers in line and the duration of time a customer must wait before being ser-viced Let Nt be the number of customers in a GGc queue at time t If we letpn = PNt = n the expected population of the system is

E[Nt] =

infin983131

n=0

npn

and the expected population of those in the queue (and not being serviced) is

E[Ntq] =

infin983131

n=0

(nminus c)pn

Definition 210 The nth waiting time is the amount of time the nth customerspends waiting in line prior to entering service We denote waiting time as the

random variable W(n)q and the total time a customer is in the system including

service time as W (n) The nth customer is the nth overall customer to enter thesystem not the customer in the nth position in line

We now consider the waiting time of the nth customer to enter a queuing systemwhere the initial population is zero that is N0 = 0 Let S(n) be the nth servicetime and let I(n) be the nth inter-arrival time ie the time between the (nminus 1)stcustomer and the nth customer arriving in the system Define

U (n) equiv S(n) minus I(n)

to be the time between the nth inter-arrival time and the nth service time

Theorem 211 (Lindleyrsquos Equation) In a single-server queue where customersare serviced on a first-come first-served basis the waiting time of the (n + 1)thcustomer is recursively given by

W (n+1)q = max

9831530W (n)

q + S(n) minus I(n)983154

ProofSince the initial population of the system is zero the first customer to arrive will

be serviced immediately so W(1)q = 0 The waiting time for the second customer

will be the time it takes for the first customer to finish being serviced If the firstcustomer has already been serviced by the time the second customer arrives thenthe waiting time for the second customer will be zero

W (2)q = max

9831530 S(1) minus I(1)

983154= max

9831530W (1)

q + S(1) minus I(1)983154

For n ge 3 the waiting time of the nth customer is simply the waiting time of the(nminus 1)th customer and the amount of time it takes for the (nminus 1)th customer tobe serviced or zero if the (nminus 1)th customer has already been serviced by the timethe nth customer enters the system

Lemma 212 Under the same conditions as the preceding theorem W(n)q may be

rewritten as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

8 HARVEY BARNHARD

Proof

We prove the lemma by iterating over W(n)q

W (n)q = max

983153W (nminus1)

q + U (nminus1) 0983154

= max983153max

983153W (nminus2)

q + U (nminus2) 0983154+ U (nminus1) 0

983154

= max983153max

983153W (nminus2)

q + U (nminus2) + U (nminus1) U (nminus1)983154 0983154

= max983153W (nminus2)

q + U (nminus2) + U (nminus1) U (nminus1) 0983154

Continue this process in a recursive manner until we have

W (n)q = max

983153W (1)

q + U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1)983154

The result of the lemma is now immediate since we have assumed thatW(1)q = 0

The goal of this paper is to approximate W(n)q the waiting time of the nth

customer From the preceding lemma it is clear that the waiting time depends onthe waiting times of all preceding customers in line More precisely the waitingtime of the nth customer is the maximum of the partial sums of the decreasingsequence

983051U (nminusi)

983052n

i=1 an iid sequence of random variables Section 3 will show

that this decreasing sequence is a random walk and when n is sufficiently large thissequence can be approximated by Brownian motion For such an approximation tobe accurate however each U (nminusi) must be sufficiently small small This situationarises in queues that exhibit heavy traffic

Definition 213 The traffic intensity ρ of a given queue is defined as the ratioof arrival times to service times So for an MMc queue the traffic intensity is

ρ =λ

cmicro

where λ and micro are the arrival and service rate respectively of customers in thequeue For a single server MM1 queue the traffic intensity is

ρ =λ

micro

Remark 214 When ρ is close to zero waiting times approach zero as queue lengthshortens and new customers are serviced immediately upon arrival When ρ isgreater than one waiting times successively increase as the queue population ldquoex-plodesrdquo to infinity The interesting case occurs when ρ is close tomdashbut does notexceedmdashone A system is said to be in heavy traffic when ρ isin (1minus ε 1) and ε gt 0is small

3 Brownian Motion

This section introduces the stochastic process of Brownian motion viewed as thelimit of random walks In this paper we confine our study of Brownian motion tothe one-dimensional case

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 9

31 Random Walks

Definition 31 A one-dimensional random walk is a stochastic process constructedas the sum of iid random variables That is if Xi is a sequence of iid randomvariables with Xi isin R (regarded as steps) then the sum of the first n such randomvariables

Rn =

n983131

i=1

Xi

is the n-th value of a random walk A random walk is said to be symmetric ifXi isin R is symmetrically distributed about zero A random walk is called simple ifPXi = 1+ PXi = minus1 = 1

Example 32 Figure 3 depicts a simple random walk simulated using the statisti-cal computing software R with PXi = minus1 = PXi = 1 = 1

2 and 15 total stepsTaking R0 = X0 = 0 the values of Xi and Ri are

Xi15i=1 = 1minus1 1minus1minus1minus1minus1 1minus1minus1 1 1minus1 1minus1Ri15i=1 = 0 1 0 1 0minus1minus2minus3minus2minus3minus4minus3minus2minus3minus2minus3

0 5 10 15

minus4minus3

minus2minus1

01

n

S n

Figure 3

An important property of random walks is that they exhibit independent incre-ments meaning that for any selection of positive integers t1 lt t2 lt middot middot middot lt tn therandom variables

Rt2 minusRt1 Rtn minusRtnminus1

are independent This property immediately follow from the fact that each incre-ment is the sum of iid random variables

10 HARVEY BARNHARD

32 Brownian Motion DefinitionWe now introduce the continuous-time analog to a random walkmdashBrownian motionA one-dimensional standard Brownian motion with variance parameter σ2 is a real-valued process Bttge0 defined in the probability space (ΩF P) that has thefollowing properties

(I) If t0 lt t1 lt middot middot middot lt tn then Bt0 Bt1 minusBt0 Btn minusBtnminus1are independent

(II) If s t ge 0 then

P (Bs+t minusBs isin A) =

983133

A

1radic2πσ2t

exp(minusx22σ2t)

(III) With probability 1 t 983041rarr Bt is continuous

The first two conditions may be concisely reworded as

(I) Independent increments(II) Increment lengths are normally distributed with mean 0 and variance σ2t

Bs+t minusBs sim N(0σ2t)

Remark 33Brownian motion is a collection of stochastic processes When we say a Brownianmotion or the Brownian motion we are talking about a particular realization ofBrownian motion in the space of continuous functions C[0 T ] where T is a stoppingtime

The above definition establishes the necessary conditions for a stochastic process tobe Brownian but the existence of such a process requires further work Throughan application of the Kolmogorov extension theorem existence is proved in [4]However such a proof is bereft of intuition for queueing theory and approximatingwaiting times Instead we take the existence of Brownian motion as given andshow that Brownian motion can be constructed as a limit of random walksmdashtheresult of Donskerrsquos Theorem Brownian motion constructed in such a manner iscalled the standard Wiener process

Before constructing Brownian motion we must first establish what type of con-vergence we are discussing

Definition 34 A sequence of random variables Xiinfini=1 converges almost surelyto a random variable X in the underlying probability space (ΩF P) (denoted

Xnasminusminusrarr X) if

P983153ω isin Ω lim

nrarrinfinXn(ω) = X(ω)

983154= 1

We aim to show almost sure convergence of a scaled random walk to a Brownianmotion in (C[0 T ] d) the metric space of continuous functions on the interval [0 T ]with d as the supremum norm

d(f g) = suptisin[0T ]

|f(t)minus g(t)|

Note that random walks and Brownian motions are stochastic processes while dmeasures the distance between two continuous functions In order to measure thedistance between random walks and Brownian motion we instead consider the pathstaken by these stochastic processes

t 983041rarr Rt and t 983041rarr Bt

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 11

33 Brownian Motion Construction In this paper the construction of Brow-nian motion comprises the following steps

(1) Construct a simple random walk out of a Brownian motion path usingSkorokhod embedding

(2) Linearly interpolate and scale the embedded random walk creating a con-tinuous function from [0 n] 983041rarr R

(3) Take the uniform limit of this scaled random walk to derive the originalBrownian motion path

The paper presents an alternative formulation to the canonical construction ofBrownian motion which involves defining Brownian motion on the dyadic rationalsand using continuity to extend the definition to the reals To motivate this particu-lar construction we will proceed out of order we will begin with step (2) backtrackto step (1) and conclude by stringing together steps (1) through (3) in subsection333 (Strong Approximation)

331 Scaled Random WalksLet Rm be a random walk where E[Xi] = 0 VarXi = σ2 and

Rm =

m983131

i=1

Xi

Definition 35 A scaled random walk with scaling parameter n is a real valuedrandom process

983051R(n)

983052 defined on t ge 0 For t such that nt is an integer

R(n)t =

1radicnσ2

Rnt

For t such that nt is not an integer S(n)(t) is defined by linear interpolation

R(n)t =

1radicnσ2

983063Rlfloorntrfloor + (ntminus lfloorntrfloor)

983043Rlceilntrceil minusRlfloorntrfloor

983044 983064

This method of scaling operates on two aspects of random walks as n increasesthe time between state transitions decreases and increment lengths decrease Lett0 = 0 and tk = inftkt isin Z nt gt nt(kminus1) then as n rarr infin

tk minus t(kminus1) rarr 0 and R(n)tk

minusR(n)t(kminus1)

rarr 0

As a result R(n)(t) is continuous with probability 1 on t ge 0 for all n The scalingfactor 1radic

nσ2ensures constant variance for a given t

Var

9830691radicnσ2

Rnt

983070= Var

9830831radicnσ2

nt983131

i=1

Xi

983084=

9830611radicnσ2

9830622

ntVarXi = tVar Xi

σ2= t

By observing the limiting behavior of R(n)t (ie as n rarr infin) we establish a con-

struction of Brownian motion

12 HARVEY BARNHARD

332 Skorokhod EmbeddingTo construct Brownian motion out of random walks we must first construct randomwalks out of Brownian motion Structuring the proof in this matter is not withoutreason there are uncountably many realizations of Brownian motion so we want toensure that the random walks to which we are taking the limit do in fact convergeto the Brownian motion of interest Such a procedure of deriving a random walkfrom a Brownian motion is called Skorokhod embedding The procedure is given inthe following pseudocode

Set t0 = 0 and n = 0 Let T be a stopping time of Brownian motion Btso that Bt is defined on [0 T ] Let R0 = 0 be the initial value

of the Skorokhod embedding

While tn + t le TInitialize t = 0

While |Bt minusBtn | le 1 Continue Bt until |Bt minusBtn | = 1 for the first time

Set t(n+1) = inft |Bt minusBtn | = 1If Bt(n+1)

minusBtn = 1 then set X(n+1) = 1

Otherwise Bt(n+1)minusBtn = minus1 so set X(n+1) = minus1

Let the (n+ 1)th value of the Skorokhod embedding be

R(n+1) = Rn +Xn+1

Increment to the next step of the inner while loop n = n+ 1

The resulting sequence Rnn=12 is the Skorokhod embedding of Bttisin[0T ]

Theorem 36 Let Bt be a standard Brownian motion Set t0 = 0 and let tn bethe stopping time where |Bt| = 1 for the nth time

tn equiv inft ge 0 |Bt minusBtn | = 1

If we define Rn equiv Btn then Rn is a simple random walk

ProofTo prove that B0 Bt1 Bt2 is a simple random walk we merely need to showthat

(1) PBtn+1minusBtn = 1 = PBtn+1

minusBtn = minus1 = 12 for all n isin N

(2) tn+1 minus tn are iid random variables

The first item follows from the symmetric property of Brownian increments prop-erty (II) in the definition of Brownian motion Bs+t minusBs sim N(0σ2t)

The second item follows from the strong Markov property of Brownian motionwhich states that if τ is a stopping time with respect to Bt then Bt minus Bτ is

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 13

also a Brownian motion We do not include a rigorous treatment of stopping timesor a proof of the strong Markov property

We now state a theorem that places a probabilistic bound on the differencebetween a Brownian motion and its Skorokhod embedding

Theorem 37 Define Θ to be the maximum distance between a Brownian motionand its Skorokhod embedding

Θ(BRT ) = max0letleT

|Bt minusRt|

Note that Θ is equivalent to the distance between Bt and Rt in the space ofcontinuous functions on the time interval [0 T ] equipped with the supremum normThere exist c a isin [0infin) such that for all r le n14 and all integers n ge 3

P983153Θ(BRT ) ge rn14

983155log n

983154le ceminusar

ProofIt will suffice to prove the theorem for r ge 9c2 where c is the constant such that

P983153osc(B δ T ) gt r

983155δ log(1δ)

983154le cT δ(rc)

2

andosc(B δ T ) = sup |Bt minusBs| s t isin D s t isin [0 T ] |sminus t| le δ

is the oscillation of Bt restricted to t isin D the dyadic rationals The proof thatsuch a c exists can be found in [3] page 68 where it is also shown that for n isin N

Θ(BSn) le 1 + osc(B 1 n) + max|Bj minusBτj | j = 1 nNow suppose 9c2 le r le n14 If |Bn minusBτn | is large then either |nminus τn| is large

or the oscillation of B is large Consider the three events

(1)983051Θ(BRn) ge rn14

radiclog n

983052

(2)983051osc(B r

radicn 2n) ge (r3)n14

radiclog n

983052

(3) max1lejlen |τj minus j| ge rradicn

From above we know that event (1) is contained in the union of events (2) and (3)As a consequence we need only prove the result of the theorem for events (2) and(3) We first tackle event (2)

P983153osc(B r

radicn 2n) gt (r3)n(14)

983155log n

983154le 3P

983153osc(B r

radicn n) gt (r3)n(14)

983155log n

983154

= 3P983153osc(B rnminus12) gt (r3)nminus(14)

983155log n

983154

le 3P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log

983043n12r

983044983070

Ifradicr3 ge c and r le n14 we can conclude that there exist c and a such that

P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log(n12)

983070le ceminusar logn

For event (3) we refer to a proof on page 266 in the appendix of [3] to show thatthere exist c a such that

P983069

max1lejlen

|τj minus j| gt rradicn

983070le ceminusar2

14 HARVEY BARNHARD

333 Strong ApproximationWith preliminaries in order we now show that Brownian motion can be thought ofas a limit of simple random walks a construction known as the strong approximationof Brownian motion Let Bt be a standard Brownian motion with variance

parameter 1 as defined in the probability space (ΩF P) and let B(n)t be the

scaled Brownian motion

B(n)t =

1radicnBnt

From B(n) we use the Skorokhod embedding to derive the simple random walkS(n) And from S(n) we derive the scaled simple random walk R(n)

R(n)t =

1radicnS(n)nt

From Theorem 37 we know that there exists c a isin [0infin) such that for all positiveintegers T

P983069

max0letleTn

983055983055983055S(n)t minusB

(n)t

983055983055983055 ge cr(Tn)14983155log (Tn)

983070le ceminusar

multiplying by 1radicn this becomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge crT 14nminus14983155log (Tn)

983070le ceminusar

Letting r = c log n where c is sufficiently large for the given T the inequalitybecomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

983070le c

n2

To proceed we require the use of the Borel-Cantelli lemma the result of which isprovided below

Lemma 38 (Borel-Cantelli)If An is a sequence of events in Ω and

983123infini=1 PAn lt infin then

PAn io = 0

where An io = lim supnrarrinfin An is the event that infinitely many An occur

If we set

An = max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

we can then apply the Borel-Cantelli lemma to the above inequality to concludethat

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 le cnminus14 log32 n

with probability one for all n sufficiently large Since

limnrarrinfin

cnminus14 log32 n = 0

we conclude that

R(n)t

asminusminusrarr Bt in C[0 T ]The following theorem condenses the results of this section up to this point

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 15

Theorem 39 (Donskerrsquos Theorem) Let Xi be a sequence of iid random vari-

ables with mean 0 and variance σ2 If R(n)t is a scaled random walk as defined

above then R(n)t rArr Bt where Bt is a standard Brownian motion This result is

also known as the functional central limit theorem

34 Brownian Motion VariantsThe preceding construction of Brownian motion will prove particularly useful in ourapproximation of heavy traffic waiting times However this process requires furthermodification Recall that when traffic intensity ρ is slightly less than 1 customersare being serviced at a slightly faster rate then they are entering the queue Thusgiven enough time traffic will eventually ldquoclear outrdquo due to faster service timesHowever this particular construction of Brownian motion has increments of meanzero

E[Bs+t minusBs] = 0

meaning that the approximate number of customers in line wouldmdashon averagemdashremain unchanged regardless of how long the line is observed Such a propertyis called the martingale property Instead we want a process that exhibits thesupermartingale property

E[Bs+t minusBs] lt 0

Definition 310 Let Fn be an increasing sequence of σ-fields (Fn is called afiltration) Let Xn be a sequence of random variables with finite mean andXn isin Fn for all n Martingales supermartingales and submartingales are thendefined as follows for all n

Martingale E [Xn+1 | Fn] = Xn

Supermartingale E [Xn+1 | Fn] le Xn

Submartingale E [Xn+1 | Fn] ge Xn

In words a sequence that we expect to remain the same over time is a martingalea sequence we expect to increase is a submartingale and a sequence we expect todecrease is a supermartingale To approximate heavy traffic we want a Brownianmotion-like process that exhibits the supermartingale property

Definition 311 A Brownian motion with drift parameter α and variance param-eter σ2 Bα

t is defined by the following properties

(1) Independent increments(2) Increment lengths are normally distributed with mean αt and variance σ2t(3) the path t 983041rarr Bα

t is continuous

Note that these properties are the same as standard Brownian motion exceptincrement lengths now have non-zero means Since we have already shown theexistence of standard Brownian motion the construction of Brownian motion withdrift is now easy

Lemma 312 Let Bt be a standard Brownian motion with variance parameterσ2 then Bα

t = Bt + αt is Brownian motion with drift parameter α and varianceparameter σ2 Moreover Bα

t fulfills the supermartingale property when α lt 0

16 HARVEY BARNHARD

ProofIndependence of increments follows directly from independence of increments forBt and the fact that αt is a constant for a given t To prove that Bα

t has stationaryincrements we have to show that the distribution of increments Bα

s+tminusBαs depends

only the time interval t and not the absolute time Recall from the definition of Bt

that Bs+t minusBt are normal random variables with mean 0 and variance t for s gt 0

E983045Bα

s+t minusBαs

983046= E

983045Bα

s+t

983046minus E [Bα

s ]

= E983045Bα

s+t + α(s+ t)983046minus E [Bs + αs]

= α(s+ t)minus αs

= αt

Var983051Bα

s+t minusBαs

983052= Var Bs+t minusBs + αt= Var Bs+t minusBs= σ2t

And since for fixed t αt is just a constant we use the property of normal randomvariables to conclude that

Bαs+t minusBα

s sim N(αtσ2t) for all s gt 0

Continuity of Bαt directly follows from continuity of Bt We have shown that

Bαt constructed as the sum of standard Brownian motion and drift αt is indeed

Brownian motion with drift

Brownian motion with negative drift provides a better approximation of thewaiting-time of customers in a queue than does standard Brownian motion How-ever the unboundedness of Brownian motion suggests that waiting times couldperhaps be negative In order to circumvent this issue we instead use reflectedBrownian motion with drift Reflected Brownian motion acts much the same asBrownian motion except on a given boundary off of which the Brownian motionismdashunsurprisinglymdashreflected In the one-dimensional case the boundaries may beany interval on the reals that provide a lower and upper bound to the Brownianmotion For our purposes we only use a single boundary zero that gives a lowerbound to Brownian motion That is for all t ge 0 we want Bα

t ge 0

Definition 313 Let Bαt be a standard Brownian motion with drift parameter α

and let Mαt be the running maximum of Bα

t

Mαt = sup

0lesletBα

s

Then reflected Brownian motion with drift parameter minusα and boundary 0 definedon R+ can be constructed as follows

Rα0t equiv max0Mα

t minusBαt

Note that max0 Sαt ge 0 and Mα

t ge Bαt B

α0t ge 0 for all t Most importantly

the local behavior of Bα0t is exactly like Brownian motion with drift parameter minusα

since max0 Sαt is simply a constant when Bα

t ∕= 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 17

We have now provided sufficient modifications of Brownian motion and we nextmove onmdashwhat we will ultimately use to approximate heavy traffic waiting timesThe next example provides us with the distribution of the supremum of a Brownianmotion with negative drift

Example 314Let Bt be a Brownian motion with drift α lt 0 and variance parameter σ2 Define

Mt = sup0leslet

Bs

to be the running maximum then

Minfin sim exp

9830612|α|σ2

983062

that is

PW ge w = eminusminus2|α|σ2 w w ge 0

E [Minfin] =σ2

2|α|

Prooffor constants a b gt 0 Let T (minusa b) be the first time that B minus t hits minusa or b

T (minusa b) = inft Bt = minusa or Bt = bIn [7] it is proved that

P983051BT (minusab)=b

983052=

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)

Since α lt 0 exp(2αaσ2) rarr 0 we have

limararrinfin

PBT (minusab) = b = limararrinfin

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)= exp(2αbσ2)

The left-hand side becomes the probability that the process will reach b somewherealong its path (ie that the maximum of the process exceeds b somewhere alongits path) Therefore we have

PW ge b = exp(2αbσ2) = expminus2|α|bσ2

This example shows us that the supremum of a Brownian motion observed ldquofor-everrdquo (ie on the interval [0infin)) is exponentially distributed with rate 2|α|σ2

4 Heavy Traffic Approximation

We now have all we need to provide a heuristic argument for the approximationof single-servers queue in heavy traffic using Brownian Motion For a rigorousalbeit opaque treatment of this approximation see [8] the seminal work on thissubject by Kingman

Theorem 41 The waiting time of the nth customer in an MM1 queue canbe approximated by the supremum of reflected Brownian motion with negative driftand boundary at 0

18 HARVEY BARNHARD

ProofRecall that an MM1 queue is a process where the number of customer arrivalsfollows a poisson process with rate λ and service times are exponentially distributedwith rate micro From Lindleyrsquos equation in Section 2 we can write the waiting timefor the nth customer as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

where U (n) = S(n)minus I(n) is the difference between the nth service time and the nthinter-arrival time Define the partial sum P

(n)k to be zero when k = 0 and

P(n)k equiv

k983131

i=1

U (nminusi) for k ge 1

We can then rewrite the waiting time of the nth customer as

W (n)q = max

0lekle(nminus1)P

(n)k

The fact that P(n)k is the kth value of a random walk follows immediately from

U (1) U (nminus1) being iid random variables Let α and σ2 be the expectationand variance of U (i) We calculate α and σ2 by first noting that S(i) and I(i) areexponentially distributed with rates micro and λ Recall that ρ = λmicro is the trafficintensity as defined in the first section of this paper

α = E983147U (i)

983148= E

983147S(i)

983148minus E

983147I(i)

983148=

1

microminus 1

λ=

ρminus 1

λ

σ2 = Var983153U (i)

983154= Var

983153S(i)

983154+Var

983153I(i)

983154=

1

micro2+

1

λ2

We next compute the expectation and variance of P(n)k using the fact that U (i) are

iid random variables

E983147P

(n)k

983148=

k983131

i=1

E983147U (nminusi)

983148= kα

Var983153P

(n)k

983154=

k983131

i=1

Var983153U (nminusi)

983154= kσ2

For a finite n W(n)q is therefore the maximum value of a random walk with n steps

We now show that Pnk can be approximated by Brownian motion with drift

parameter α and variance parameter σ2 when n is sufficiently large We restate theproperties of such a Brownian motion Bt below

(1) B0 = 0(2) Bt has stationary and independent increments(3) Bt sim N(αtσ2t)(4) t 983041rarr Bt is continuous with probability one

The first property is clear as P(n)0 = 0 for all n The second property follows from

the fact that U (i) are iid random variables The third property is a result of thecentral limit theorem for iid sequences the result of which is provided below

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 3

We can also consider the possible number of customers in a queue as states sothat the state space consists of the non-negative integers where λN and microN are theprobabilistic rate of transition between states

0 1 2 3 4

λ0 λ1 λ2 λ3

micro4micro3micro2micro1

Figure 2

21 Continuous-Time Markov ChainsA system with a countable state-space and probabilistic transitions between statessuggests the use of Markov chains However standard Markov chains of elemen-tary probability theory operate in a discrete-time setting In this paper we con-sider exponential service and inter-arrival times As a result transitions betweenstates can occur anytime in the continuous timeline of non-negative reals Wetherefore aim to construct a countable-space continuous-time version of a discrete-time Markov chain We first introduce two properties that define continuous-timeMarkov chainsmdashthe Markov property and time-homogeneity

Definition 21 Consider a stochastic process Xt taking values in a state spaceS Xt is said to exhibit the Markov property if for y isin S and t ge s

PXt = y | Xr 0 le r le s = PXt = y | Xs

Definition 22 A time-homogenous Markov chain is a stochastic process Xttge0

such that

PXt = y | Xs = x = PXtminuss = y | X0

If a process exhibits time-homogeneity transition probabilities depend only onthe state of the process not the absolute time

Definition 23 A continuous-time Markov chain is a stochastic process Xttge0

taking values in a state space S and satisfying

PXt+∆t = i | Xt = i = 1minus qii∆t+ o(∆t)

PXt+∆t = j | Xt = i = qij∆t+ o(∆t)

where qij represents the transition rate from i isin S to j isin S and

qii equiv983131

j ∕=i

qij

The use of qij as a rate of transition between states requires further explanationEach qij is the derivative of pij(t) the probability of starting in state i and ending

4 HARVEY BARNHARD

in state j after ∆t units of time have elapsed (here we have assumed differentiabilityof pij(t) at ∆t = 0)

qij = lim∆trarr0

=PXt+∆t = j | Xt = i

∆t

The matrix P(t) is the transition matrix and its (i j)th element is pij(t)

P(t) =

983093

983097983095p11(t) p12(t) middot middot middotp21(t) p22(t) middot middot middot

983094

983098983096

The matrix Q is called the infinitesimal generator and its (i j)th element is qij Note that if zero time has elapsed then there is zero probability of transitioningout of the original state so pii(0) = 1 and P(0) = I We can therefore write

Q = Pprime(0)

and

P(t) = I+Q∆t

Example 25 will illustrate a deeper connection between qij and pij(t) but we mustfirst define what it means for two states to be communicable

Definition 24 Let i and j be two states of a Markov chain We say that i and jcommunicate if there is a positive probability that one state will lead to the otherstate given a certain amount of time In notation i and j communicate if thereexists time durations r gt 0 and s gt 0 such that

PXt+r = i | Xt = j PXt+s = j | Xt = i

A communication class is the collection of all such states that communicate withone another The set of communication classes partition the state space into disjointsets If there exists only one communication class then the Markov chain is said tobe irreducible In this paper the state-space of queues comprises the non-negativeintegers and is irreducible

Example 25 Let Xt be an irreducible continuous-time Markov chain Showthat for each i j and every t gt 0

P Xt = j | X0 = i gt 0

ProofFix i and j Since Xt is irreducible there exists some time t such that

P Xt = j|X0 = i gt 0

We want to show that this holds for all t Consider the sequence of events thatoccur between i and j and denote them as k1 kn where k1 occurs immediatelyafter i and kn occurs immediately before j It is clear that

qklkl+1gt 0 for all l

Or equivalently for some times s lt t

P Xt = kl+1 | Xs = kl gt 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 5

Now pick any time t gt 0 Note that there are potentially many paths from i to jand k1 kn represents one such path Therefore there is a non-zero probabilitythat i will go to j by this transition path In notation

P Xt = j | X0 = i genminus1983132

l=1

P983153X (l+1)t

n= kl+1|X lt

n= kl

983154gt 0

We now extend the result of the example to show how each continuous-timeMarkov chain induces a discrete-time Markov chain Let Xttge0 be an irreduciblecontinuous-time Markov chain Let S be a countable state-space and i j be statesin S Let t0 t1 t2 be distinct times with t0 lt t1 lt t2 Define

pij(t0 t2) equiv P Xt2 = j | Xt0 = ito be the probability of Xt being in state j at time t2 given that the process wasin state i at time t0 Then for each transition probability we can write

pij(t0 t2) =983131

jisinS

pij(t0 t1)pjk(t1 t2)

Enumerate the sequence of states in order of occurrence

n0 = 0

n1 = inft Xt ∕= X0n2 = inft ge n1 Xt ∕= Xn1

nm = inft ge nmminus1 Xt ∕= Xnmminus1then Xn equiv Xn n isin n0 n1 forms a discrete-time Markov chain out of acontinuous-time Markov chain This process is known as embedding and Xn iscalled an embedded Markov chain A similar procedure will be used later in Section33 when we construct a random walk a discrete-time process out of a Brownianmotion a continuous-time process Note that the embedding process causes us tolose information about the holding times of each state the amount of time spent ina state before transitioning to another state We therefore do not lose any furtherinformation by normalizing the time between states to some constant c gt 0

nm minus nmminus1 = c for all m ge 1

So after normalization the transition-rate matrix for Xn simply becomes P(c)as defined above

22 Birth-Death ProcessesA birth-death process is a non-negative integer valued continuous-time Markovchain That is a stochastic process Nt fulfilling the following two conditions

(1) Nt isin 0 1 2 3 (2) as s rarr 0 (Nt+s minusNt) isin 0minus1 1

The second condition states that for an infinitesimal change in time the popula-tion of the system will either increase by one decrease by one or remain the sameA birth-death process is a system where the rate of exit out of and entrance intothe system are known and the collection of such processes forms a large family of

6 HARVEY BARNHARD

stochastic processes For instance a Poisson process is one of the simplest birth-death processes where the ldquobirthrdquo rate is constant and the ldquodeathrdquo rate is zero Inparticular a queuing process is a birth-death process where customer arrivals areldquobirthsrdquo and service completions are ldquodeathsrdquo

23 Queuing Models

Example 26 (MMc) In this model inter-arrival times and service times areexponentially distributed with rates λ and micro respectively In this queue λ and microare constant and independent of the current state That is λN = λ and microN = micro forall N Service is received on a ldquofirst come first servedrdquo basis and c customers maybe serviced at a time The simplest case is when c = 1 where only one customeris serviced at a time In an MMc queue where customers are serviced as soon asthey arrive regardless of the population of the system we set c = infin The MMcmodel assumes that no customers leave the system between arrival and completionof service

This paper focuses on the MM1 queue but we introduce a few more modelsas examples of the breadth of situations that can be analyzed through the lens ofqueuing theory

Example 27 (GGc) This model is the same as the MMc model exceptinter-arrival times and service times follow some general (represented by G) notnecessarily exponential distribution

Example 28 (GGcK) This model is the same as the GGc model but withan upper bound K on the number of customers that can occupy the system at agiven time This type of queue is called a truncated queue

Example 29 (ABCDE) The above queuing models are identified usingKendall Notation a combination of letters and slashes Each position in the nota-tion corresponds to a certain queuing characteristic (eg inter-arrival time distri-bution) and the letter specifies the characteristic (eg exponentially distributedinter-arrival times) The following table covers a wide selection of queuing modelsthat are easily be described using Kendall notation

Characteristic Symbol Explanation(A) Inter-arrival time distribution M Exponential(B) Service time distribution D Deterministic

Ek Erlang type kG General

(C) of parallel servers 1 2 infin(D) Max system capacity 1 2 infin(E) Order of service FCFS First come first served

LCFS Last come first servedRSS Random selectionPR PriorityGD General

The table above is by no means a definitive list of queuing characteristics Forinstance inter-arrival and service times may be state-dependent as introduced inthe beginning of this section with rates λN and microN Queues can also be cyclic in

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 7

the sense that once a customer is serviced the customer immediately returns to aposition in line to await service once again

24 Properties of Queuing ModelsThe first two properties one may ask of a queue are the expected number of cus-tomers in line and the duration of time a customer must wait before being ser-viced Let Nt be the number of customers in a GGc queue at time t If we letpn = PNt = n the expected population of the system is

E[Nt] =

infin983131

n=0

npn

and the expected population of those in the queue (and not being serviced) is

E[Ntq] =

infin983131

n=0

(nminus c)pn

Definition 210 The nth waiting time is the amount of time the nth customerspends waiting in line prior to entering service We denote waiting time as the

random variable W(n)q and the total time a customer is in the system including

service time as W (n) The nth customer is the nth overall customer to enter thesystem not the customer in the nth position in line

We now consider the waiting time of the nth customer to enter a queuing systemwhere the initial population is zero that is N0 = 0 Let S(n) be the nth servicetime and let I(n) be the nth inter-arrival time ie the time between the (nminus 1)stcustomer and the nth customer arriving in the system Define

U (n) equiv S(n) minus I(n)

to be the time between the nth inter-arrival time and the nth service time

Theorem 211 (Lindleyrsquos Equation) In a single-server queue where customersare serviced on a first-come first-served basis the waiting time of the (n + 1)thcustomer is recursively given by

W (n+1)q = max

9831530W (n)

q + S(n) minus I(n)983154

ProofSince the initial population of the system is zero the first customer to arrive will

be serviced immediately so W(1)q = 0 The waiting time for the second customer

will be the time it takes for the first customer to finish being serviced If the firstcustomer has already been serviced by the time the second customer arrives thenthe waiting time for the second customer will be zero

W (2)q = max

9831530 S(1) minus I(1)

983154= max

9831530W (1)

q + S(1) minus I(1)983154

For n ge 3 the waiting time of the nth customer is simply the waiting time of the(nminus 1)th customer and the amount of time it takes for the (nminus 1)th customer tobe serviced or zero if the (nminus 1)th customer has already been serviced by the timethe nth customer enters the system

Lemma 212 Under the same conditions as the preceding theorem W(n)q may be

rewritten as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

8 HARVEY BARNHARD

Proof

We prove the lemma by iterating over W(n)q

W (n)q = max

983153W (nminus1)

q + U (nminus1) 0983154

= max983153max

983153W (nminus2)

q + U (nminus2) 0983154+ U (nminus1) 0

983154

= max983153max

983153W (nminus2)

q + U (nminus2) + U (nminus1) U (nminus1)983154 0983154

= max983153W (nminus2)

q + U (nminus2) + U (nminus1) U (nminus1) 0983154

Continue this process in a recursive manner until we have

W (n)q = max

983153W (1)

q + U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1)983154

The result of the lemma is now immediate since we have assumed thatW(1)q = 0

The goal of this paper is to approximate W(n)q the waiting time of the nth

customer From the preceding lemma it is clear that the waiting time depends onthe waiting times of all preceding customers in line More precisely the waitingtime of the nth customer is the maximum of the partial sums of the decreasingsequence

983051U (nminusi)

983052n

i=1 an iid sequence of random variables Section 3 will show

that this decreasing sequence is a random walk and when n is sufficiently large thissequence can be approximated by Brownian motion For such an approximation tobe accurate however each U (nminusi) must be sufficiently small small This situationarises in queues that exhibit heavy traffic

Definition 213 The traffic intensity ρ of a given queue is defined as the ratioof arrival times to service times So for an MMc queue the traffic intensity is

ρ =λ

cmicro

where λ and micro are the arrival and service rate respectively of customers in thequeue For a single server MM1 queue the traffic intensity is

ρ =λ

micro

Remark 214 When ρ is close to zero waiting times approach zero as queue lengthshortens and new customers are serviced immediately upon arrival When ρ isgreater than one waiting times successively increase as the queue population ldquoex-plodesrdquo to infinity The interesting case occurs when ρ is close tomdashbut does notexceedmdashone A system is said to be in heavy traffic when ρ isin (1minus ε 1) and ε gt 0is small

3 Brownian Motion

This section introduces the stochastic process of Brownian motion viewed as thelimit of random walks In this paper we confine our study of Brownian motion tothe one-dimensional case

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 9

31 Random Walks

Definition 31 A one-dimensional random walk is a stochastic process constructedas the sum of iid random variables That is if Xi is a sequence of iid randomvariables with Xi isin R (regarded as steps) then the sum of the first n such randomvariables

Rn =

n983131

i=1

Xi

is the n-th value of a random walk A random walk is said to be symmetric ifXi isin R is symmetrically distributed about zero A random walk is called simple ifPXi = 1+ PXi = minus1 = 1

Example 32 Figure 3 depicts a simple random walk simulated using the statisti-cal computing software R with PXi = minus1 = PXi = 1 = 1

2 and 15 total stepsTaking R0 = X0 = 0 the values of Xi and Ri are

Xi15i=1 = 1minus1 1minus1minus1minus1minus1 1minus1minus1 1 1minus1 1minus1Ri15i=1 = 0 1 0 1 0minus1minus2minus3minus2minus3minus4minus3minus2minus3minus2minus3

0 5 10 15

minus4minus3

minus2minus1

01

n

S n

Figure 3

An important property of random walks is that they exhibit independent incre-ments meaning that for any selection of positive integers t1 lt t2 lt middot middot middot lt tn therandom variables

Rt2 minusRt1 Rtn minusRtnminus1

are independent This property immediately follow from the fact that each incre-ment is the sum of iid random variables

10 HARVEY BARNHARD

32 Brownian Motion DefinitionWe now introduce the continuous-time analog to a random walkmdashBrownian motionA one-dimensional standard Brownian motion with variance parameter σ2 is a real-valued process Bttge0 defined in the probability space (ΩF P) that has thefollowing properties

(I) If t0 lt t1 lt middot middot middot lt tn then Bt0 Bt1 minusBt0 Btn minusBtnminus1are independent

(II) If s t ge 0 then

P (Bs+t minusBs isin A) =

983133

A

1radic2πσ2t

exp(minusx22σ2t)

(III) With probability 1 t 983041rarr Bt is continuous

The first two conditions may be concisely reworded as

(I) Independent increments(II) Increment lengths are normally distributed with mean 0 and variance σ2t

Bs+t minusBs sim N(0σ2t)

Remark 33Brownian motion is a collection of stochastic processes When we say a Brownianmotion or the Brownian motion we are talking about a particular realization ofBrownian motion in the space of continuous functions C[0 T ] where T is a stoppingtime

The above definition establishes the necessary conditions for a stochastic process tobe Brownian but the existence of such a process requires further work Throughan application of the Kolmogorov extension theorem existence is proved in [4]However such a proof is bereft of intuition for queueing theory and approximatingwaiting times Instead we take the existence of Brownian motion as given andshow that Brownian motion can be constructed as a limit of random walksmdashtheresult of Donskerrsquos Theorem Brownian motion constructed in such a manner iscalled the standard Wiener process

Before constructing Brownian motion we must first establish what type of con-vergence we are discussing

Definition 34 A sequence of random variables Xiinfini=1 converges almost surelyto a random variable X in the underlying probability space (ΩF P) (denoted

Xnasminusminusrarr X) if

P983153ω isin Ω lim

nrarrinfinXn(ω) = X(ω)

983154= 1

We aim to show almost sure convergence of a scaled random walk to a Brownianmotion in (C[0 T ] d) the metric space of continuous functions on the interval [0 T ]with d as the supremum norm

d(f g) = suptisin[0T ]

|f(t)minus g(t)|

Note that random walks and Brownian motions are stochastic processes while dmeasures the distance between two continuous functions In order to measure thedistance between random walks and Brownian motion we instead consider the pathstaken by these stochastic processes

t 983041rarr Rt and t 983041rarr Bt

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 11

33 Brownian Motion Construction In this paper the construction of Brow-nian motion comprises the following steps

(1) Construct a simple random walk out of a Brownian motion path usingSkorokhod embedding

(2) Linearly interpolate and scale the embedded random walk creating a con-tinuous function from [0 n] 983041rarr R

(3) Take the uniform limit of this scaled random walk to derive the originalBrownian motion path

The paper presents an alternative formulation to the canonical construction ofBrownian motion which involves defining Brownian motion on the dyadic rationalsand using continuity to extend the definition to the reals To motivate this particu-lar construction we will proceed out of order we will begin with step (2) backtrackto step (1) and conclude by stringing together steps (1) through (3) in subsection333 (Strong Approximation)

331 Scaled Random WalksLet Rm be a random walk where E[Xi] = 0 VarXi = σ2 and

Rm =

m983131

i=1

Xi

Definition 35 A scaled random walk with scaling parameter n is a real valuedrandom process

983051R(n)

983052 defined on t ge 0 For t such that nt is an integer

R(n)t =

1radicnσ2

Rnt

For t such that nt is not an integer S(n)(t) is defined by linear interpolation

R(n)t =

1radicnσ2

983063Rlfloorntrfloor + (ntminus lfloorntrfloor)

983043Rlceilntrceil minusRlfloorntrfloor

983044 983064

This method of scaling operates on two aspects of random walks as n increasesthe time between state transitions decreases and increment lengths decrease Lett0 = 0 and tk = inftkt isin Z nt gt nt(kminus1) then as n rarr infin

tk minus t(kminus1) rarr 0 and R(n)tk

minusR(n)t(kminus1)

rarr 0

As a result R(n)(t) is continuous with probability 1 on t ge 0 for all n The scalingfactor 1radic

nσ2ensures constant variance for a given t

Var

9830691radicnσ2

Rnt

983070= Var

9830831radicnσ2

nt983131

i=1

Xi

983084=

9830611radicnσ2

9830622

ntVarXi = tVar Xi

σ2= t

By observing the limiting behavior of R(n)t (ie as n rarr infin) we establish a con-

struction of Brownian motion

12 HARVEY BARNHARD

332 Skorokhod EmbeddingTo construct Brownian motion out of random walks we must first construct randomwalks out of Brownian motion Structuring the proof in this matter is not withoutreason there are uncountably many realizations of Brownian motion so we want toensure that the random walks to which we are taking the limit do in fact convergeto the Brownian motion of interest Such a procedure of deriving a random walkfrom a Brownian motion is called Skorokhod embedding The procedure is given inthe following pseudocode

Set t0 = 0 and n = 0 Let T be a stopping time of Brownian motion Btso that Bt is defined on [0 T ] Let R0 = 0 be the initial value

of the Skorokhod embedding

While tn + t le TInitialize t = 0

While |Bt minusBtn | le 1 Continue Bt until |Bt minusBtn | = 1 for the first time

Set t(n+1) = inft |Bt minusBtn | = 1If Bt(n+1)

minusBtn = 1 then set X(n+1) = 1

Otherwise Bt(n+1)minusBtn = minus1 so set X(n+1) = minus1

Let the (n+ 1)th value of the Skorokhod embedding be

R(n+1) = Rn +Xn+1

Increment to the next step of the inner while loop n = n+ 1

The resulting sequence Rnn=12 is the Skorokhod embedding of Bttisin[0T ]

Theorem 36 Let Bt be a standard Brownian motion Set t0 = 0 and let tn bethe stopping time where |Bt| = 1 for the nth time

tn equiv inft ge 0 |Bt minusBtn | = 1

If we define Rn equiv Btn then Rn is a simple random walk

ProofTo prove that B0 Bt1 Bt2 is a simple random walk we merely need to showthat

(1) PBtn+1minusBtn = 1 = PBtn+1

minusBtn = minus1 = 12 for all n isin N

(2) tn+1 minus tn are iid random variables

The first item follows from the symmetric property of Brownian increments prop-erty (II) in the definition of Brownian motion Bs+t minusBs sim N(0σ2t)

The second item follows from the strong Markov property of Brownian motionwhich states that if τ is a stopping time with respect to Bt then Bt minus Bτ is

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 13

also a Brownian motion We do not include a rigorous treatment of stopping timesor a proof of the strong Markov property

We now state a theorem that places a probabilistic bound on the differencebetween a Brownian motion and its Skorokhod embedding

Theorem 37 Define Θ to be the maximum distance between a Brownian motionand its Skorokhod embedding

Θ(BRT ) = max0letleT

|Bt minusRt|

Note that Θ is equivalent to the distance between Bt and Rt in the space ofcontinuous functions on the time interval [0 T ] equipped with the supremum normThere exist c a isin [0infin) such that for all r le n14 and all integers n ge 3

P983153Θ(BRT ) ge rn14

983155log n

983154le ceminusar

ProofIt will suffice to prove the theorem for r ge 9c2 where c is the constant such that

P983153osc(B δ T ) gt r

983155δ log(1δ)

983154le cT δ(rc)

2

andosc(B δ T ) = sup |Bt minusBs| s t isin D s t isin [0 T ] |sminus t| le δ

is the oscillation of Bt restricted to t isin D the dyadic rationals The proof thatsuch a c exists can be found in [3] page 68 where it is also shown that for n isin N

Θ(BSn) le 1 + osc(B 1 n) + max|Bj minusBτj | j = 1 nNow suppose 9c2 le r le n14 If |Bn minusBτn | is large then either |nminus τn| is large

or the oscillation of B is large Consider the three events

(1)983051Θ(BRn) ge rn14

radiclog n

983052

(2)983051osc(B r

radicn 2n) ge (r3)n14

radiclog n

983052

(3) max1lejlen |τj minus j| ge rradicn

From above we know that event (1) is contained in the union of events (2) and (3)As a consequence we need only prove the result of the theorem for events (2) and(3) We first tackle event (2)

P983153osc(B r

radicn 2n) gt (r3)n(14)

983155log n

983154le 3P

983153osc(B r

radicn n) gt (r3)n(14)

983155log n

983154

= 3P983153osc(B rnminus12) gt (r3)nminus(14)

983155log n

983154

le 3P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log

983043n12r

983044983070

Ifradicr3 ge c and r le n14 we can conclude that there exist c and a such that

P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log(n12)

983070le ceminusar logn

For event (3) we refer to a proof on page 266 in the appendix of [3] to show thatthere exist c a such that

P983069

max1lejlen

|τj minus j| gt rradicn

983070le ceminusar2

14 HARVEY BARNHARD

333 Strong ApproximationWith preliminaries in order we now show that Brownian motion can be thought ofas a limit of simple random walks a construction known as the strong approximationof Brownian motion Let Bt be a standard Brownian motion with variance

parameter 1 as defined in the probability space (ΩF P) and let B(n)t be the

scaled Brownian motion

B(n)t =

1radicnBnt

From B(n) we use the Skorokhod embedding to derive the simple random walkS(n) And from S(n) we derive the scaled simple random walk R(n)

R(n)t =

1radicnS(n)nt

From Theorem 37 we know that there exists c a isin [0infin) such that for all positiveintegers T

P983069

max0letleTn

983055983055983055S(n)t minusB

(n)t

983055983055983055 ge cr(Tn)14983155log (Tn)

983070le ceminusar

multiplying by 1radicn this becomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge crT 14nminus14983155log (Tn)

983070le ceminusar

Letting r = c log n where c is sufficiently large for the given T the inequalitybecomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

983070le c

n2

To proceed we require the use of the Borel-Cantelli lemma the result of which isprovided below

Lemma 38 (Borel-Cantelli)If An is a sequence of events in Ω and

983123infini=1 PAn lt infin then

PAn io = 0

where An io = lim supnrarrinfin An is the event that infinitely many An occur

If we set

An = max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

we can then apply the Borel-Cantelli lemma to the above inequality to concludethat

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 le cnminus14 log32 n

with probability one for all n sufficiently large Since

limnrarrinfin

cnminus14 log32 n = 0

we conclude that

R(n)t

asminusminusrarr Bt in C[0 T ]The following theorem condenses the results of this section up to this point

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 15

Theorem 39 (Donskerrsquos Theorem) Let Xi be a sequence of iid random vari-

ables with mean 0 and variance σ2 If R(n)t is a scaled random walk as defined

above then R(n)t rArr Bt where Bt is a standard Brownian motion This result is

also known as the functional central limit theorem

34 Brownian Motion VariantsThe preceding construction of Brownian motion will prove particularly useful in ourapproximation of heavy traffic waiting times However this process requires furthermodification Recall that when traffic intensity ρ is slightly less than 1 customersare being serviced at a slightly faster rate then they are entering the queue Thusgiven enough time traffic will eventually ldquoclear outrdquo due to faster service timesHowever this particular construction of Brownian motion has increments of meanzero

E[Bs+t minusBs] = 0

meaning that the approximate number of customers in line wouldmdashon averagemdashremain unchanged regardless of how long the line is observed Such a propertyis called the martingale property Instead we want a process that exhibits thesupermartingale property

E[Bs+t minusBs] lt 0

Definition 310 Let Fn be an increasing sequence of σ-fields (Fn is called afiltration) Let Xn be a sequence of random variables with finite mean andXn isin Fn for all n Martingales supermartingales and submartingales are thendefined as follows for all n

Martingale E [Xn+1 | Fn] = Xn

Supermartingale E [Xn+1 | Fn] le Xn

Submartingale E [Xn+1 | Fn] ge Xn

In words a sequence that we expect to remain the same over time is a martingalea sequence we expect to increase is a submartingale and a sequence we expect todecrease is a supermartingale To approximate heavy traffic we want a Brownianmotion-like process that exhibits the supermartingale property

Definition 311 A Brownian motion with drift parameter α and variance param-eter σ2 Bα

t is defined by the following properties

(1) Independent increments(2) Increment lengths are normally distributed with mean αt and variance σ2t(3) the path t 983041rarr Bα

t is continuous

Note that these properties are the same as standard Brownian motion exceptincrement lengths now have non-zero means Since we have already shown theexistence of standard Brownian motion the construction of Brownian motion withdrift is now easy

Lemma 312 Let Bt be a standard Brownian motion with variance parameterσ2 then Bα

t = Bt + αt is Brownian motion with drift parameter α and varianceparameter σ2 Moreover Bα

t fulfills the supermartingale property when α lt 0

16 HARVEY BARNHARD

ProofIndependence of increments follows directly from independence of increments forBt and the fact that αt is a constant for a given t To prove that Bα

t has stationaryincrements we have to show that the distribution of increments Bα

s+tminusBαs depends

only the time interval t and not the absolute time Recall from the definition of Bt

that Bs+t minusBt are normal random variables with mean 0 and variance t for s gt 0

E983045Bα

s+t minusBαs

983046= E

983045Bα

s+t

983046minus E [Bα

s ]

= E983045Bα

s+t + α(s+ t)983046minus E [Bs + αs]

= α(s+ t)minus αs

= αt

Var983051Bα

s+t minusBαs

983052= Var Bs+t minusBs + αt= Var Bs+t minusBs= σ2t

And since for fixed t αt is just a constant we use the property of normal randomvariables to conclude that

Bαs+t minusBα

s sim N(αtσ2t) for all s gt 0

Continuity of Bαt directly follows from continuity of Bt We have shown that

Bαt constructed as the sum of standard Brownian motion and drift αt is indeed

Brownian motion with drift

Brownian motion with negative drift provides a better approximation of thewaiting-time of customers in a queue than does standard Brownian motion How-ever the unboundedness of Brownian motion suggests that waiting times couldperhaps be negative In order to circumvent this issue we instead use reflectedBrownian motion with drift Reflected Brownian motion acts much the same asBrownian motion except on a given boundary off of which the Brownian motionismdashunsurprisinglymdashreflected In the one-dimensional case the boundaries may beany interval on the reals that provide a lower and upper bound to the Brownianmotion For our purposes we only use a single boundary zero that gives a lowerbound to Brownian motion That is for all t ge 0 we want Bα

t ge 0

Definition 313 Let Bαt be a standard Brownian motion with drift parameter α

and let Mαt be the running maximum of Bα

t

Mαt = sup

0lesletBα

s

Then reflected Brownian motion with drift parameter minusα and boundary 0 definedon R+ can be constructed as follows

Rα0t equiv max0Mα

t minusBαt

Note that max0 Sαt ge 0 and Mα

t ge Bαt B

α0t ge 0 for all t Most importantly

the local behavior of Bα0t is exactly like Brownian motion with drift parameter minusα

since max0 Sαt is simply a constant when Bα

t ∕= 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 17

We have now provided sufficient modifications of Brownian motion and we nextmove onmdashwhat we will ultimately use to approximate heavy traffic waiting timesThe next example provides us with the distribution of the supremum of a Brownianmotion with negative drift

Example 314Let Bt be a Brownian motion with drift α lt 0 and variance parameter σ2 Define

Mt = sup0leslet

Bs

to be the running maximum then

Minfin sim exp

9830612|α|σ2

983062

that is

PW ge w = eminusminus2|α|σ2 w w ge 0

E [Minfin] =σ2

2|α|

Prooffor constants a b gt 0 Let T (minusa b) be the first time that B minus t hits minusa or b

T (minusa b) = inft Bt = minusa or Bt = bIn [7] it is proved that

P983051BT (minusab)=b

983052=

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)

Since α lt 0 exp(2αaσ2) rarr 0 we have

limararrinfin

PBT (minusab) = b = limararrinfin

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)= exp(2αbσ2)

The left-hand side becomes the probability that the process will reach b somewherealong its path (ie that the maximum of the process exceeds b somewhere alongits path) Therefore we have

PW ge b = exp(2αbσ2) = expminus2|α|bσ2

This example shows us that the supremum of a Brownian motion observed ldquofor-everrdquo (ie on the interval [0infin)) is exponentially distributed with rate 2|α|σ2

4 Heavy Traffic Approximation

We now have all we need to provide a heuristic argument for the approximationof single-servers queue in heavy traffic using Brownian Motion For a rigorousalbeit opaque treatment of this approximation see [8] the seminal work on thissubject by Kingman

Theorem 41 The waiting time of the nth customer in an MM1 queue canbe approximated by the supremum of reflected Brownian motion with negative driftand boundary at 0

18 HARVEY BARNHARD

ProofRecall that an MM1 queue is a process where the number of customer arrivalsfollows a poisson process with rate λ and service times are exponentially distributedwith rate micro From Lindleyrsquos equation in Section 2 we can write the waiting timefor the nth customer as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

where U (n) = S(n)minus I(n) is the difference between the nth service time and the nthinter-arrival time Define the partial sum P

(n)k to be zero when k = 0 and

P(n)k equiv

k983131

i=1

U (nminusi) for k ge 1

We can then rewrite the waiting time of the nth customer as

W (n)q = max

0lekle(nminus1)P

(n)k

The fact that P(n)k is the kth value of a random walk follows immediately from

U (1) U (nminus1) being iid random variables Let α and σ2 be the expectationand variance of U (i) We calculate α and σ2 by first noting that S(i) and I(i) areexponentially distributed with rates micro and λ Recall that ρ = λmicro is the trafficintensity as defined in the first section of this paper

α = E983147U (i)

983148= E

983147S(i)

983148minus E

983147I(i)

983148=

1

microminus 1

λ=

ρminus 1

λ

σ2 = Var983153U (i)

983154= Var

983153S(i)

983154+Var

983153I(i)

983154=

1

micro2+

1

λ2

We next compute the expectation and variance of P(n)k using the fact that U (i) are

iid random variables

E983147P

(n)k

983148=

k983131

i=1

E983147U (nminusi)

983148= kα

Var983153P

(n)k

983154=

k983131

i=1

Var983153U (nminusi)

983154= kσ2

For a finite n W(n)q is therefore the maximum value of a random walk with n steps

We now show that Pnk can be approximated by Brownian motion with drift

parameter α and variance parameter σ2 when n is sufficiently large We restate theproperties of such a Brownian motion Bt below

(1) B0 = 0(2) Bt has stationary and independent increments(3) Bt sim N(αtσ2t)(4) t 983041rarr Bt is continuous with probability one

The first property is clear as P(n)0 = 0 for all n The second property follows from

the fact that U (i) are iid random variables The third property is a result of thecentral limit theorem for iid sequences the result of which is provided below

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

4 HARVEY BARNHARD

in state j after ∆t units of time have elapsed (here we have assumed differentiabilityof pij(t) at ∆t = 0)

qij = lim∆trarr0

=PXt+∆t = j | Xt = i

∆t

The matrix P(t) is the transition matrix and its (i j)th element is pij(t)

P(t) =

983093

983097983095p11(t) p12(t) middot middot middotp21(t) p22(t) middot middot middot

983094

983098983096

The matrix Q is called the infinitesimal generator and its (i j)th element is qij Note that if zero time has elapsed then there is zero probability of transitioningout of the original state so pii(0) = 1 and P(0) = I We can therefore write

Q = Pprime(0)

and

P(t) = I+Q∆t

Example 25 will illustrate a deeper connection between qij and pij(t) but we mustfirst define what it means for two states to be communicable

Definition 24 Let i and j be two states of a Markov chain We say that i and jcommunicate if there is a positive probability that one state will lead to the otherstate given a certain amount of time In notation i and j communicate if thereexists time durations r gt 0 and s gt 0 such that

PXt+r = i | Xt = j PXt+s = j | Xt = i

A communication class is the collection of all such states that communicate withone another The set of communication classes partition the state space into disjointsets If there exists only one communication class then the Markov chain is said tobe irreducible In this paper the state-space of queues comprises the non-negativeintegers and is irreducible

Example 25 Let Xt be an irreducible continuous-time Markov chain Showthat for each i j and every t gt 0

P Xt = j | X0 = i gt 0

ProofFix i and j Since Xt is irreducible there exists some time t such that

P Xt = j|X0 = i gt 0

We want to show that this holds for all t Consider the sequence of events thatoccur between i and j and denote them as k1 kn where k1 occurs immediatelyafter i and kn occurs immediately before j It is clear that

qklkl+1gt 0 for all l

Or equivalently for some times s lt t

P Xt = kl+1 | Xs = kl gt 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 5

Now pick any time t gt 0 Note that there are potentially many paths from i to jand k1 kn represents one such path Therefore there is a non-zero probabilitythat i will go to j by this transition path In notation

P Xt = j | X0 = i genminus1983132

l=1

P983153X (l+1)t

n= kl+1|X lt

n= kl

983154gt 0

We now extend the result of the example to show how each continuous-timeMarkov chain induces a discrete-time Markov chain Let Xttge0 be an irreduciblecontinuous-time Markov chain Let S be a countable state-space and i j be statesin S Let t0 t1 t2 be distinct times with t0 lt t1 lt t2 Define

pij(t0 t2) equiv P Xt2 = j | Xt0 = ito be the probability of Xt being in state j at time t2 given that the process wasin state i at time t0 Then for each transition probability we can write

pij(t0 t2) =983131

jisinS

pij(t0 t1)pjk(t1 t2)

Enumerate the sequence of states in order of occurrence

n0 = 0

n1 = inft Xt ∕= X0n2 = inft ge n1 Xt ∕= Xn1

nm = inft ge nmminus1 Xt ∕= Xnmminus1then Xn equiv Xn n isin n0 n1 forms a discrete-time Markov chain out of acontinuous-time Markov chain This process is known as embedding and Xn iscalled an embedded Markov chain A similar procedure will be used later in Section33 when we construct a random walk a discrete-time process out of a Brownianmotion a continuous-time process Note that the embedding process causes us tolose information about the holding times of each state the amount of time spent ina state before transitioning to another state We therefore do not lose any furtherinformation by normalizing the time between states to some constant c gt 0

nm minus nmminus1 = c for all m ge 1

So after normalization the transition-rate matrix for Xn simply becomes P(c)as defined above

22 Birth-Death ProcessesA birth-death process is a non-negative integer valued continuous-time Markovchain That is a stochastic process Nt fulfilling the following two conditions

(1) Nt isin 0 1 2 3 (2) as s rarr 0 (Nt+s minusNt) isin 0minus1 1

The second condition states that for an infinitesimal change in time the popula-tion of the system will either increase by one decrease by one or remain the sameA birth-death process is a system where the rate of exit out of and entrance intothe system are known and the collection of such processes forms a large family of

6 HARVEY BARNHARD

stochastic processes For instance a Poisson process is one of the simplest birth-death processes where the ldquobirthrdquo rate is constant and the ldquodeathrdquo rate is zero Inparticular a queuing process is a birth-death process where customer arrivals areldquobirthsrdquo and service completions are ldquodeathsrdquo

23 Queuing Models

Example 26 (MMc) In this model inter-arrival times and service times areexponentially distributed with rates λ and micro respectively In this queue λ and microare constant and independent of the current state That is λN = λ and microN = micro forall N Service is received on a ldquofirst come first servedrdquo basis and c customers maybe serviced at a time The simplest case is when c = 1 where only one customeris serviced at a time In an MMc queue where customers are serviced as soon asthey arrive regardless of the population of the system we set c = infin The MMcmodel assumes that no customers leave the system between arrival and completionof service

This paper focuses on the MM1 queue but we introduce a few more modelsas examples of the breadth of situations that can be analyzed through the lens ofqueuing theory

Example 27 (GGc) This model is the same as the MMc model exceptinter-arrival times and service times follow some general (represented by G) notnecessarily exponential distribution

Example 28 (GGcK) This model is the same as the GGc model but withan upper bound K on the number of customers that can occupy the system at agiven time This type of queue is called a truncated queue

Example 29 (ABCDE) The above queuing models are identified usingKendall Notation a combination of letters and slashes Each position in the nota-tion corresponds to a certain queuing characteristic (eg inter-arrival time distri-bution) and the letter specifies the characteristic (eg exponentially distributedinter-arrival times) The following table covers a wide selection of queuing modelsthat are easily be described using Kendall notation

Characteristic Symbol Explanation(A) Inter-arrival time distribution M Exponential(B) Service time distribution D Deterministic

Ek Erlang type kG General

(C) of parallel servers 1 2 infin(D) Max system capacity 1 2 infin(E) Order of service FCFS First come first served

LCFS Last come first servedRSS Random selectionPR PriorityGD General

The table above is by no means a definitive list of queuing characteristics Forinstance inter-arrival and service times may be state-dependent as introduced inthe beginning of this section with rates λN and microN Queues can also be cyclic in

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 7

the sense that once a customer is serviced the customer immediately returns to aposition in line to await service once again

24 Properties of Queuing ModelsThe first two properties one may ask of a queue are the expected number of cus-tomers in line and the duration of time a customer must wait before being ser-viced Let Nt be the number of customers in a GGc queue at time t If we letpn = PNt = n the expected population of the system is

E[Nt] =

infin983131

n=0

npn

and the expected population of those in the queue (and not being serviced) is

E[Ntq] =

infin983131

n=0

(nminus c)pn

Definition 210 The nth waiting time is the amount of time the nth customerspends waiting in line prior to entering service We denote waiting time as the

random variable W(n)q and the total time a customer is in the system including

service time as W (n) The nth customer is the nth overall customer to enter thesystem not the customer in the nth position in line

We now consider the waiting time of the nth customer to enter a queuing systemwhere the initial population is zero that is N0 = 0 Let S(n) be the nth servicetime and let I(n) be the nth inter-arrival time ie the time between the (nminus 1)stcustomer and the nth customer arriving in the system Define

U (n) equiv S(n) minus I(n)

to be the time between the nth inter-arrival time and the nth service time

Theorem 211 (Lindleyrsquos Equation) In a single-server queue where customersare serviced on a first-come first-served basis the waiting time of the (n + 1)thcustomer is recursively given by

W (n+1)q = max

9831530W (n)

q + S(n) minus I(n)983154

ProofSince the initial population of the system is zero the first customer to arrive will

be serviced immediately so W(1)q = 0 The waiting time for the second customer

will be the time it takes for the first customer to finish being serviced If the firstcustomer has already been serviced by the time the second customer arrives thenthe waiting time for the second customer will be zero

W (2)q = max

9831530 S(1) minus I(1)

983154= max

9831530W (1)

q + S(1) minus I(1)983154

For n ge 3 the waiting time of the nth customer is simply the waiting time of the(nminus 1)th customer and the amount of time it takes for the (nminus 1)th customer tobe serviced or zero if the (nminus 1)th customer has already been serviced by the timethe nth customer enters the system

Lemma 212 Under the same conditions as the preceding theorem W(n)q may be

rewritten as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

8 HARVEY BARNHARD

Proof

We prove the lemma by iterating over W(n)q

W (n)q = max

983153W (nminus1)

q + U (nminus1) 0983154

= max983153max

983153W (nminus2)

q + U (nminus2) 0983154+ U (nminus1) 0

983154

= max983153max

983153W (nminus2)

q + U (nminus2) + U (nminus1) U (nminus1)983154 0983154

= max983153W (nminus2)

q + U (nminus2) + U (nminus1) U (nminus1) 0983154

Continue this process in a recursive manner until we have

W (n)q = max

983153W (1)

q + U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1)983154

The result of the lemma is now immediate since we have assumed thatW(1)q = 0

The goal of this paper is to approximate W(n)q the waiting time of the nth

customer From the preceding lemma it is clear that the waiting time depends onthe waiting times of all preceding customers in line More precisely the waitingtime of the nth customer is the maximum of the partial sums of the decreasingsequence

983051U (nminusi)

983052n

i=1 an iid sequence of random variables Section 3 will show

that this decreasing sequence is a random walk and when n is sufficiently large thissequence can be approximated by Brownian motion For such an approximation tobe accurate however each U (nminusi) must be sufficiently small small This situationarises in queues that exhibit heavy traffic

Definition 213 The traffic intensity ρ of a given queue is defined as the ratioof arrival times to service times So for an MMc queue the traffic intensity is

ρ =λ

cmicro

where λ and micro are the arrival and service rate respectively of customers in thequeue For a single server MM1 queue the traffic intensity is

ρ =λ

micro

Remark 214 When ρ is close to zero waiting times approach zero as queue lengthshortens and new customers are serviced immediately upon arrival When ρ isgreater than one waiting times successively increase as the queue population ldquoex-plodesrdquo to infinity The interesting case occurs when ρ is close tomdashbut does notexceedmdashone A system is said to be in heavy traffic when ρ isin (1minus ε 1) and ε gt 0is small

3 Brownian Motion

This section introduces the stochastic process of Brownian motion viewed as thelimit of random walks In this paper we confine our study of Brownian motion tothe one-dimensional case

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 9

31 Random Walks

Definition 31 A one-dimensional random walk is a stochastic process constructedas the sum of iid random variables That is if Xi is a sequence of iid randomvariables with Xi isin R (regarded as steps) then the sum of the first n such randomvariables

Rn =

n983131

i=1

Xi

is the n-th value of a random walk A random walk is said to be symmetric ifXi isin R is symmetrically distributed about zero A random walk is called simple ifPXi = 1+ PXi = minus1 = 1

Example 32 Figure 3 depicts a simple random walk simulated using the statisti-cal computing software R with PXi = minus1 = PXi = 1 = 1

2 and 15 total stepsTaking R0 = X0 = 0 the values of Xi and Ri are

Xi15i=1 = 1minus1 1minus1minus1minus1minus1 1minus1minus1 1 1minus1 1minus1Ri15i=1 = 0 1 0 1 0minus1minus2minus3minus2minus3minus4minus3minus2minus3minus2minus3

0 5 10 15

minus4minus3

minus2minus1

01

n

S n

Figure 3

An important property of random walks is that they exhibit independent incre-ments meaning that for any selection of positive integers t1 lt t2 lt middot middot middot lt tn therandom variables

Rt2 minusRt1 Rtn minusRtnminus1

are independent This property immediately follow from the fact that each incre-ment is the sum of iid random variables

10 HARVEY BARNHARD

32 Brownian Motion DefinitionWe now introduce the continuous-time analog to a random walkmdashBrownian motionA one-dimensional standard Brownian motion with variance parameter σ2 is a real-valued process Bttge0 defined in the probability space (ΩF P) that has thefollowing properties

(I) If t0 lt t1 lt middot middot middot lt tn then Bt0 Bt1 minusBt0 Btn minusBtnminus1are independent

(II) If s t ge 0 then

P (Bs+t minusBs isin A) =

983133

A

1radic2πσ2t

exp(minusx22σ2t)

(III) With probability 1 t 983041rarr Bt is continuous

The first two conditions may be concisely reworded as

(I) Independent increments(II) Increment lengths are normally distributed with mean 0 and variance σ2t

Bs+t minusBs sim N(0σ2t)

Remark 33Brownian motion is a collection of stochastic processes When we say a Brownianmotion or the Brownian motion we are talking about a particular realization ofBrownian motion in the space of continuous functions C[0 T ] where T is a stoppingtime

The above definition establishes the necessary conditions for a stochastic process tobe Brownian but the existence of such a process requires further work Throughan application of the Kolmogorov extension theorem existence is proved in [4]However such a proof is bereft of intuition for queueing theory and approximatingwaiting times Instead we take the existence of Brownian motion as given andshow that Brownian motion can be constructed as a limit of random walksmdashtheresult of Donskerrsquos Theorem Brownian motion constructed in such a manner iscalled the standard Wiener process

Before constructing Brownian motion we must first establish what type of con-vergence we are discussing

Definition 34 A sequence of random variables Xiinfini=1 converges almost surelyto a random variable X in the underlying probability space (ΩF P) (denoted

Xnasminusminusrarr X) if

P983153ω isin Ω lim

nrarrinfinXn(ω) = X(ω)

983154= 1

We aim to show almost sure convergence of a scaled random walk to a Brownianmotion in (C[0 T ] d) the metric space of continuous functions on the interval [0 T ]with d as the supremum norm

d(f g) = suptisin[0T ]

|f(t)minus g(t)|

Note that random walks and Brownian motions are stochastic processes while dmeasures the distance between two continuous functions In order to measure thedistance between random walks and Brownian motion we instead consider the pathstaken by these stochastic processes

t 983041rarr Rt and t 983041rarr Bt

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 11

33 Brownian Motion Construction In this paper the construction of Brow-nian motion comprises the following steps

(1) Construct a simple random walk out of a Brownian motion path usingSkorokhod embedding

(2) Linearly interpolate and scale the embedded random walk creating a con-tinuous function from [0 n] 983041rarr R

(3) Take the uniform limit of this scaled random walk to derive the originalBrownian motion path

The paper presents an alternative formulation to the canonical construction ofBrownian motion which involves defining Brownian motion on the dyadic rationalsand using continuity to extend the definition to the reals To motivate this particu-lar construction we will proceed out of order we will begin with step (2) backtrackto step (1) and conclude by stringing together steps (1) through (3) in subsection333 (Strong Approximation)

331 Scaled Random WalksLet Rm be a random walk where E[Xi] = 0 VarXi = σ2 and

Rm =

m983131

i=1

Xi

Definition 35 A scaled random walk with scaling parameter n is a real valuedrandom process

983051R(n)

983052 defined on t ge 0 For t such that nt is an integer

R(n)t =

1radicnσ2

Rnt

For t such that nt is not an integer S(n)(t) is defined by linear interpolation

R(n)t =

1radicnσ2

983063Rlfloorntrfloor + (ntminus lfloorntrfloor)

983043Rlceilntrceil minusRlfloorntrfloor

983044 983064

This method of scaling operates on two aspects of random walks as n increasesthe time between state transitions decreases and increment lengths decrease Lett0 = 0 and tk = inftkt isin Z nt gt nt(kminus1) then as n rarr infin

tk minus t(kminus1) rarr 0 and R(n)tk

minusR(n)t(kminus1)

rarr 0

As a result R(n)(t) is continuous with probability 1 on t ge 0 for all n The scalingfactor 1radic

nσ2ensures constant variance for a given t

Var

9830691radicnσ2

Rnt

983070= Var

9830831radicnσ2

nt983131

i=1

Xi

983084=

9830611radicnσ2

9830622

ntVarXi = tVar Xi

σ2= t

By observing the limiting behavior of R(n)t (ie as n rarr infin) we establish a con-

struction of Brownian motion

12 HARVEY BARNHARD

332 Skorokhod EmbeddingTo construct Brownian motion out of random walks we must first construct randomwalks out of Brownian motion Structuring the proof in this matter is not withoutreason there are uncountably many realizations of Brownian motion so we want toensure that the random walks to which we are taking the limit do in fact convergeto the Brownian motion of interest Such a procedure of deriving a random walkfrom a Brownian motion is called Skorokhod embedding The procedure is given inthe following pseudocode

Set t0 = 0 and n = 0 Let T be a stopping time of Brownian motion Btso that Bt is defined on [0 T ] Let R0 = 0 be the initial value

of the Skorokhod embedding

While tn + t le TInitialize t = 0

While |Bt minusBtn | le 1 Continue Bt until |Bt minusBtn | = 1 for the first time

Set t(n+1) = inft |Bt minusBtn | = 1If Bt(n+1)

minusBtn = 1 then set X(n+1) = 1

Otherwise Bt(n+1)minusBtn = minus1 so set X(n+1) = minus1

Let the (n+ 1)th value of the Skorokhod embedding be

R(n+1) = Rn +Xn+1

Increment to the next step of the inner while loop n = n+ 1

The resulting sequence Rnn=12 is the Skorokhod embedding of Bttisin[0T ]

Theorem 36 Let Bt be a standard Brownian motion Set t0 = 0 and let tn bethe stopping time where |Bt| = 1 for the nth time

tn equiv inft ge 0 |Bt minusBtn | = 1

If we define Rn equiv Btn then Rn is a simple random walk

ProofTo prove that B0 Bt1 Bt2 is a simple random walk we merely need to showthat

(1) PBtn+1minusBtn = 1 = PBtn+1

minusBtn = minus1 = 12 for all n isin N

(2) tn+1 minus tn are iid random variables

The first item follows from the symmetric property of Brownian increments prop-erty (II) in the definition of Brownian motion Bs+t minusBs sim N(0σ2t)

The second item follows from the strong Markov property of Brownian motionwhich states that if τ is a stopping time with respect to Bt then Bt minus Bτ is

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 13

also a Brownian motion We do not include a rigorous treatment of stopping timesor a proof of the strong Markov property

We now state a theorem that places a probabilistic bound on the differencebetween a Brownian motion and its Skorokhod embedding

Theorem 37 Define Θ to be the maximum distance between a Brownian motionand its Skorokhod embedding

Θ(BRT ) = max0letleT

|Bt minusRt|

Note that Θ is equivalent to the distance between Bt and Rt in the space ofcontinuous functions on the time interval [0 T ] equipped with the supremum normThere exist c a isin [0infin) such that for all r le n14 and all integers n ge 3

P983153Θ(BRT ) ge rn14

983155log n

983154le ceminusar

ProofIt will suffice to prove the theorem for r ge 9c2 where c is the constant such that

P983153osc(B δ T ) gt r

983155δ log(1δ)

983154le cT δ(rc)

2

andosc(B δ T ) = sup |Bt minusBs| s t isin D s t isin [0 T ] |sminus t| le δ

is the oscillation of Bt restricted to t isin D the dyadic rationals The proof thatsuch a c exists can be found in [3] page 68 where it is also shown that for n isin N

Θ(BSn) le 1 + osc(B 1 n) + max|Bj minusBτj | j = 1 nNow suppose 9c2 le r le n14 If |Bn minusBτn | is large then either |nminus τn| is large

or the oscillation of B is large Consider the three events

(1)983051Θ(BRn) ge rn14

radiclog n

983052

(2)983051osc(B r

radicn 2n) ge (r3)n14

radiclog n

983052

(3) max1lejlen |τj minus j| ge rradicn

From above we know that event (1) is contained in the union of events (2) and (3)As a consequence we need only prove the result of the theorem for events (2) and(3) We first tackle event (2)

P983153osc(B r

radicn 2n) gt (r3)n(14)

983155log n

983154le 3P

983153osc(B r

radicn n) gt (r3)n(14)

983155log n

983154

= 3P983153osc(B rnminus12) gt (r3)nminus(14)

983155log n

983154

le 3P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log

983043n12r

983044983070

Ifradicr3 ge c and r le n14 we can conclude that there exist c and a such that

P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log(n12)

983070le ceminusar logn

For event (3) we refer to a proof on page 266 in the appendix of [3] to show thatthere exist c a such that

P983069

max1lejlen

|τj minus j| gt rradicn

983070le ceminusar2

14 HARVEY BARNHARD

333 Strong ApproximationWith preliminaries in order we now show that Brownian motion can be thought ofas a limit of simple random walks a construction known as the strong approximationof Brownian motion Let Bt be a standard Brownian motion with variance

parameter 1 as defined in the probability space (ΩF P) and let B(n)t be the

scaled Brownian motion

B(n)t =

1radicnBnt

From B(n) we use the Skorokhod embedding to derive the simple random walkS(n) And from S(n) we derive the scaled simple random walk R(n)

R(n)t =

1radicnS(n)nt

From Theorem 37 we know that there exists c a isin [0infin) such that for all positiveintegers T

P983069

max0letleTn

983055983055983055S(n)t minusB

(n)t

983055983055983055 ge cr(Tn)14983155log (Tn)

983070le ceminusar

multiplying by 1radicn this becomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge crT 14nminus14983155log (Tn)

983070le ceminusar

Letting r = c log n where c is sufficiently large for the given T the inequalitybecomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

983070le c

n2

To proceed we require the use of the Borel-Cantelli lemma the result of which isprovided below

Lemma 38 (Borel-Cantelli)If An is a sequence of events in Ω and

983123infini=1 PAn lt infin then

PAn io = 0

where An io = lim supnrarrinfin An is the event that infinitely many An occur

If we set

An = max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

we can then apply the Borel-Cantelli lemma to the above inequality to concludethat

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 le cnminus14 log32 n

with probability one for all n sufficiently large Since

limnrarrinfin

cnminus14 log32 n = 0

we conclude that

R(n)t

asminusminusrarr Bt in C[0 T ]The following theorem condenses the results of this section up to this point

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 15

Theorem 39 (Donskerrsquos Theorem) Let Xi be a sequence of iid random vari-

ables with mean 0 and variance σ2 If R(n)t is a scaled random walk as defined

above then R(n)t rArr Bt where Bt is a standard Brownian motion This result is

also known as the functional central limit theorem

34 Brownian Motion VariantsThe preceding construction of Brownian motion will prove particularly useful in ourapproximation of heavy traffic waiting times However this process requires furthermodification Recall that when traffic intensity ρ is slightly less than 1 customersare being serviced at a slightly faster rate then they are entering the queue Thusgiven enough time traffic will eventually ldquoclear outrdquo due to faster service timesHowever this particular construction of Brownian motion has increments of meanzero

E[Bs+t minusBs] = 0

meaning that the approximate number of customers in line wouldmdashon averagemdashremain unchanged regardless of how long the line is observed Such a propertyis called the martingale property Instead we want a process that exhibits thesupermartingale property

E[Bs+t minusBs] lt 0

Definition 310 Let Fn be an increasing sequence of σ-fields (Fn is called afiltration) Let Xn be a sequence of random variables with finite mean andXn isin Fn for all n Martingales supermartingales and submartingales are thendefined as follows for all n

Martingale E [Xn+1 | Fn] = Xn

Supermartingale E [Xn+1 | Fn] le Xn

Submartingale E [Xn+1 | Fn] ge Xn

In words a sequence that we expect to remain the same over time is a martingalea sequence we expect to increase is a submartingale and a sequence we expect todecrease is a supermartingale To approximate heavy traffic we want a Brownianmotion-like process that exhibits the supermartingale property

Definition 311 A Brownian motion with drift parameter α and variance param-eter σ2 Bα

t is defined by the following properties

(1) Independent increments(2) Increment lengths are normally distributed with mean αt and variance σ2t(3) the path t 983041rarr Bα

t is continuous

Note that these properties are the same as standard Brownian motion exceptincrement lengths now have non-zero means Since we have already shown theexistence of standard Brownian motion the construction of Brownian motion withdrift is now easy

Lemma 312 Let Bt be a standard Brownian motion with variance parameterσ2 then Bα

t = Bt + αt is Brownian motion with drift parameter α and varianceparameter σ2 Moreover Bα

t fulfills the supermartingale property when α lt 0

16 HARVEY BARNHARD

ProofIndependence of increments follows directly from independence of increments forBt and the fact that αt is a constant for a given t To prove that Bα

t has stationaryincrements we have to show that the distribution of increments Bα

s+tminusBαs depends

only the time interval t and not the absolute time Recall from the definition of Bt

that Bs+t minusBt are normal random variables with mean 0 and variance t for s gt 0

E983045Bα

s+t minusBαs

983046= E

983045Bα

s+t

983046minus E [Bα

s ]

= E983045Bα

s+t + α(s+ t)983046minus E [Bs + αs]

= α(s+ t)minus αs

= αt

Var983051Bα

s+t minusBαs

983052= Var Bs+t minusBs + αt= Var Bs+t minusBs= σ2t

And since for fixed t αt is just a constant we use the property of normal randomvariables to conclude that

Bαs+t minusBα

s sim N(αtσ2t) for all s gt 0

Continuity of Bαt directly follows from continuity of Bt We have shown that

Bαt constructed as the sum of standard Brownian motion and drift αt is indeed

Brownian motion with drift

Brownian motion with negative drift provides a better approximation of thewaiting-time of customers in a queue than does standard Brownian motion How-ever the unboundedness of Brownian motion suggests that waiting times couldperhaps be negative In order to circumvent this issue we instead use reflectedBrownian motion with drift Reflected Brownian motion acts much the same asBrownian motion except on a given boundary off of which the Brownian motionismdashunsurprisinglymdashreflected In the one-dimensional case the boundaries may beany interval on the reals that provide a lower and upper bound to the Brownianmotion For our purposes we only use a single boundary zero that gives a lowerbound to Brownian motion That is for all t ge 0 we want Bα

t ge 0

Definition 313 Let Bαt be a standard Brownian motion with drift parameter α

and let Mαt be the running maximum of Bα

t

Mαt = sup

0lesletBα

s

Then reflected Brownian motion with drift parameter minusα and boundary 0 definedon R+ can be constructed as follows

Rα0t equiv max0Mα

t minusBαt

Note that max0 Sαt ge 0 and Mα

t ge Bαt B

α0t ge 0 for all t Most importantly

the local behavior of Bα0t is exactly like Brownian motion with drift parameter minusα

since max0 Sαt is simply a constant when Bα

t ∕= 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 17

We have now provided sufficient modifications of Brownian motion and we nextmove onmdashwhat we will ultimately use to approximate heavy traffic waiting timesThe next example provides us with the distribution of the supremum of a Brownianmotion with negative drift

Example 314Let Bt be a Brownian motion with drift α lt 0 and variance parameter σ2 Define

Mt = sup0leslet

Bs

to be the running maximum then

Minfin sim exp

9830612|α|σ2

983062

that is

PW ge w = eminusminus2|α|σ2 w w ge 0

E [Minfin] =σ2

2|α|

Prooffor constants a b gt 0 Let T (minusa b) be the first time that B minus t hits minusa or b

T (minusa b) = inft Bt = minusa or Bt = bIn [7] it is proved that

P983051BT (minusab)=b

983052=

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)

Since α lt 0 exp(2αaσ2) rarr 0 we have

limararrinfin

PBT (minusab) = b = limararrinfin

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)= exp(2αbσ2)

The left-hand side becomes the probability that the process will reach b somewherealong its path (ie that the maximum of the process exceeds b somewhere alongits path) Therefore we have

PW ge b = exp(2αbσ2) = expminus2|α|bσ2

This example shows us that the supremum of a Brownian motion observed ldquofor-everrdquo (ie on the interval [0infin)) is exponentially distributed with rate 2|α|σ2

4 Heavy Traffic Approximation

We now have all we need to provide a heuristic argument for the approximationof single-servers queue in heavy traffic using Brownian Motion For a rigorousalbeit opaque treatment of this approximation see [8] the seminal work on thissubject by Kingman

Theorem 41 The waiting time of the nth customer in an MM1 queue canbe approximated by the supremum of reflected Brownian motion with negative driftand boundary at 0

18 HARVEY BARNHARD

ProofRecall that an MM1 queue is a process where the number of customer arrivalsfollows a poisson process with rate λ and service times are exponentially distributedwith rate micro From Lindleyrsquos equation in Section 2 we can write the waiting timefor the nth customer as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

where U (n) = S(n)minus I(n) is the difference between the nth service time and the nthinter-arrival time Define the partial sum P

(n)k to be zero when k = 0 and

P(n)k equiv

k983131

i=1

U (nminusi) for k ge 1

We can then rewrite the waiting time of the nth customer as

W (n)q = max

0lekle(nminus1)P

(n)k

The fact that P(n)k is the kth value of a random walk follows immediately from

U (1) U (nminus1) being iid random variables Let α and σ2 be the expectationand variance of U (i) We calculate α and σ2 by first noting that S(i) and I(i) areexponentially distributed with rates micro and λ Recall that ρ = λmicro is the trafficintensity as defined in the first section of this paper

α = E983147U (i)

983148= E

983147S(i)

983148minus E

983147I(i)

983148=

1

microminus 1

λ=

ρminus 1

λ

σ2 = Var983153U (i)

983154= Var

983153S(i)

983154+Var

983153I(i)

983154=

1

micro2+

1

λ2

We next compute the expectation and variance of P(n)k using the fact that U (i) are

iid random variables

E983147P

(n)k

983148=

k983131

i=1

E983147U (nminusi)

983148= kα

Var983153P

(n)k

983154=

k983131

i=1

Var983153U (nminusi)

983154= kσ2

For a finite n W(n)q is therefore the maximum value of a random walk with n steps

We now show that Pnk can be approximated by Brownian motion with drift

parameter α and variance parameter σ2 when n is sufficiently large We restate theproperties of such a Brownian motion Bt below

(1) B0 = 0(2) Bt has stationary and independent increments(3) Bt sim N(αtσ2t)(4) t 983041rarr Bt is continuous with probability one

The first property is clear as P(n)0 = 0 for all n The second property follows from

the fact that U (i) are iid random variables The third property is a result of thecentral limit theorem for iid sequences the result of which is provided below

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 5

Now pick any time t gt 0 Note that there are potentially many paths from i to jand k1 kn represents one such path Therefore there is a non-zero probabilitythat i will go to j by this transition path In notation

P Xt = j | X0 = i genminus1983132

l=1

P983153X (l+1)t

n= kl+1|X lt

n= kl

983154gt 0

We now extend the result of the example to show how each continuous-timeMarkov chain induces a discrete-time Markov chain Let Xttge0 be an irreduciblecontinuous-time Markov chain Let S be a countable state-space and i j be statesin S Let t0 t1 t2 be distinct times with t0 lt t1 lt t2 Define

pij(t0 t2) equiv P Xt2 = j | Xt0 = ito be the probability of Xt being in state j at time t2 given that the process wasin state i at time t0 Then for each transition probability we can write

pij(t0 t2) =983131

jisinS

pij(t0 t1)pjk(t1 t2)

Enumerate the sequence of states in order of occurrence

n0 = 0

n1 = inft Xt ∕= X0n2 = inft ge n1 Xt ∕= Xn1

nm = inft ge nmminus1 Xt ∕= Xnmminus1then Xn equiv Xn n isin n0 n1 forms a discrete-time Markov chain out of acontinuous-time Markov chain This process is known as embedding and Xn iscalled an embedded Markov chain A similar procedure will be used later in Section33 when we construct a random walk a discrete-time process out of a Brownianmotion a continuous-time process Note that the embedding process causes us tolose information about the holding times of each state the amount of time spent ina state before transitioning to another state We therefore do not lose any furtherinformation by normalizing the time between states to some constant c gt 0

nm minus nmminus1 = c for all m ge 1

So after normalization the transition-rate matrix for Xn simply becomes P(c)as defined above

22 Birth-Death ProcessesA birth-death process is a non-negative integer valued continuous-time Markovchain That is a stochastic process Nt fulfilling the following two conditions

(1) Nt isin 0 1 2 3 (2) as s rarr 0 (Nt+s minusNt) isin 0minus1 1

The second condition states that for an infinitesimal change in time the popula-tion of the system will either increase by one decrease by one or remain the sameA birth-death process is a system where the rate of exit out of and entrance intothe system are known and the collection of such processes forms a large family of

6 HARVEY BARNHARD

stochastic processes For instance a Poisson process is one of the simplest birth-death processes where the ldquobirthrdquo rate is constant and the ldquodeathrdquo rate is zero Inparticular a queuing process is a birth-death process where customer arrivals areldquobirthsrdquo and service completions are ldquodeathsrdquo

23 Queuing Models

Example 26 (MMc) In this model inter-arrival times and service times areexponentially distributed with rates λ and micro respectively In this queue λ and microare constant and independent of the current state That is λN = λ and microN = micro forall N Service is received on a ldquofirst come first servedrdquo basis and c customers maybe serviced at a time The simplest case is when c = 1 where only one customeris serviced at a time In an MMc queue where customers are serviced as soon asthey arrive regardless of the population of the system we set c = infin The MMcmodel assumes that no customers leave the system between arrival and completionof service

This paper focuses on the MM1 queue but we introduce a few more modelsas examples of the breadth of situations that can be analyzed through the lens ofqueuing theory

Example 27 (GGc) This model is the same as the MMc model exceptinter-arrival times and service times follow some general (represented by G) notnecessarily exponential distribution

Example 28 (GGcK) This model is the same as the GGc model but withan upper bound K on the number of customers that can occupy the system at agiven time This type of queue is called a truncated queue

Example 29 (ABCDE) The above queuing models are identified usingKendall Notation a combination of letters and slashes Each position in the nota-tion corresponds to a certain queuing characteristic (eg inter-arrival time distri-bution) and the letter specifies the characteristic (eg exponentially distributedinter-arrival times) The following table covers a wide selection of queuing modelsthat are easily be described using Kendall notation

Characteristic Symbol Explanation(A) Inter-arrival time distribution M Exponential(B) Service time distribution D Deterministic

Ek Erlang type kG General

(C) of parallel servers 1 2 infin(D) Max system capacity 1 2 infin(E) Order of service FCFS First come first served

LCFS Last come first servedRSS Random selectionPR PriorityGD General

The table above is by no means a definitive list of queuing characteristics Forinstance inter-arrival and service times may be state-dependent as introduced inthe beginning of this section with rates λN and microN Queues can also be cyclic in

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 7

the sense that once a customer is serviced the customer immediately returns to aposition in line to await service once again

24 Properties of Queuing ModelsThe first two properties one may ask of a queue are the expected number of cus-tomers in line and the duration of time a customer must wait before being ser-viced Let Nt be the number of customers in a GGc queue at time t If we letpn = PNt = n the expected population of the system is

E[Nt] =

infin983131

n=0

npn

and the expected population of those in the queue (and not being serviced) is

E[Ntq] =

infin983131

n=0

(nminus c)pn

Definition 210 The nth waiting time is the amount of time the nth customerspends waiting in line prior to entering service We denote waiting time as the

random variable W(n)q and the total time a customer is in the system including

service time as W (n) The nth customer is the nth overall customer to enter thesystem not the customer in the nth position in line

We now consider the waiting time of the nth customer to enter a queuing systemwhere the initial population is zero that is N0 = 0 Let S(n) be the nth servicetime and let I(n) be the nth inter-arrival time ie the time between the (nminus 1)stcustomer and the nth customer arriving in the system Define

U (n) equiv S(n) minus I(n)

to be the time between the nth inter-arrival time and the nth service time

Theorem 211 (Lindleyrsquos Equation) In a single-server queue where customersare serviced on a first-come first-served basis the waiting time of the (n + 1)thcustomer is recursively given by

W (n+1)q = max

9831530W (n)

q + S(n) minus I(n)983154

ProofSince the initial population of the system is zero the first customer to arrive will

be serviced immediately so W(1)q = 0 The waiting time for the second customer

will be the time it takes for the first customer to finish being serviced If the firstcustomer has already been serviced by the time the second customer arrives thenthe waiting time for the second customer will be zero

W (2)q = max

9831530 S(1) minus I(1)

983154= max

9831530W (1)

q + S(1) minus I(1)983154

For n ge 3 the waiting time of the nth customer is simply the waiting time of the(nminus 1)th customer and the amount of time it takes for the (nminus 1)th customer tobe serviced or zero if the (nminus 1)th customer has already been serviced by the timethe nth customer enters the system

Lemma 212 Under the same conditions as the preceding theorem W(n)q may be

rewritten as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

8 HARVEY BARNHARD

Proof

We prove the lemma by iterating over W(n)q

W (n)q = max

983153W (nminus1)

q + U (nminus1) 0983154

= max983153max

983153W (nminus2)

q + U (nminus2) 0983154+ U (nminus1) 0

983154

= max983153max

983153W (nminus2)

q + U (nminus2) + U (nminus1) U (nminus1)983154 0983154

= max983153W (nminus2)

q + U (nminus2) + U (nminus1) U (nminus1) 0983154

Continue this process in a recursive manner until we have

W (n)q = max

983153W (1)

q + U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1)983154

The result of the lemma is now immediate since we have assumed thatW(1)q = 0

The goal of this paper is to approximate W(n)q the waiting time of the nth

customer From the preceding lemma it is clear that the waiting time depends onthe waiting times of all preceding customers in line More precisely the waitingtime of the nth customer is the maximum of the partial sums of the decreasingsequence

983051U (nminusi)

983052n

i=1 an iid sequence of random variables Section 3 will show

that this decreasing sequence is a random walk and when n is sufficiently large thissequence can be approximated by Brownian motion For such an approximation tobe accurate however each U (nminusi) must be sufficiently small small This situationarises in queues that exhibit heavy traffic

Definition 213 The traffic intensity ρ of a given queue is defined as the ratioof arrival times to service times So for an MMc queue the traffic intensity is

ρ =λ

cmicro

where λ and micro are the arrival and service rate respectively of customers in thequeue For a single server MM1 queue the traffic intensity is

ρ =λ

micro

Remark 214 When ρ is close to zero waiting times approach zero as queue lengthshortens and new customers are serviced immediately upon arrival When ρ isgreater than one waiting times successively increase as the queue population ldquoex-plodesrdquo to infinity The interesting case occurs when ρ is close tomdashbut does notexceedmdashone A system is said to be in heavy traffic when ρ isin (1minus ε 1) and ε gt 0is small

3 Brownian Motion

This section introduces the stochastic process of Brownian motion viewed as thelimit of random walks In this paper we confine our study of Brownian motion tothe one-dimensional case

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 9

31 Random Walks

Definition 31 A one-dimensional random walk is a stochastic process constructedas the sum of iid random variables That is if Xi is a sequence of iid randomvariables with Xi isin R (regarded as steps) then the sum of the first n such randomvariables

Rn =

n983131

i=1

Xi

is the n-th value of a random walk A random walk is said to be symmetric ifXi isin R is symmetrically distributed about zero A random walk is called simple ifPXi = 1+ PXi = minus1 = 1

Example 32 Figure 3 depicts a simple random walk simulated using the statisti-cal computing software R with PXi = minus1 = PXi = 1 = 1

2 and 15 total stepsTaking R0 = X0 = 0 the values of Xi and Ri are

Xi15i=1 = 1minus1 1minus1minus1minus1minus1 1minus1minus1 1 1minus1 1minus1Ri15i=1 = 0 1 0 1 0minus1minus2minus3minus2minus3minus4minus3minus2minus3minus2minus3

0 5 10 15

minus4minus3

minus2minus1

01

n

S n

Figure 3

An important property of random walks is that they exhibit independent incre-ments meaning that for any selection of positive integers t1 lt t2 lt middot middot middot lt tn therandom variables

Rt2 minusRt1 Rtn minusRtnminus1

are independent This property immediately follow from the fact that each incre-ment is the sum of iid random variables

10 HARVEY BARNHARD

32 Brownian Motion DefinitionWe now introduce the continuous-time analog to a random walkmdashBrownian motionA one-dimensional standard Brownian motion with variance parameter σ2 is a real-valued process Bttge0 defined in the probability space (ΩF P) that has thefollowing properties

(I) If t0 lt t1 lt middot middot middot lt tn then Bt0 Bt1 minusBt0 Btn minusBtnminus1are independent

(II) If s t ge 0 then

P (Bs+t minusBs isin A) =

983133

A

1radic2πσ2t

exp(minusx22σ2t)

(III) With probability 1 t 983041rarr Bt is continuous

The first two conditions may be concisely reworded as

(I) Independent increments(II) Increment lengths are normally distributed with mean 0 and variance σ2t

Bs+t minusBs sim N(0σ2t)

Remark 33Brownian motion is a collection of stochastic processes When we say a Brownianmotion or the Brownian motion we are talking about a particular realization ofBrownian motion in the space of continuous functions C[0 T ] where T is a stoppingtime

The above definition establishes the necessary conditions for a stochastic process tobe Brownian but the existence of such a process requires further work Throughan application of the Kolmogorov extension theorem existence is proved in [4]However such a proof is bereft of intuition for queueing theory and approximatingwaiting times Instead we take the existence of Brownian motion as given andshow that Brownian motion can be constructed as a limit of random walksmdashtheresult of Donskerrsquos Theorem Brownian motion constructed in such a manner iscalled the standard Wiener process

Before constructing Brownian motion we must first establish what type of con-vergence we are discussing

Definition 34 A sequence of random variables Xiinfini=1 converges almost surelyto a random variable X in the underlying probability space (ΩF P) (denoted

Xnasminusminusrarr X) if

P983153ω isin Ω lim

nrarrinfinXn(ω) = X(ω)

983154= 1

We aim to show almost sure convergence of a scaled random walk to a Brownianmotion in (C[0 T ] d) the metric space of continuous functions on the interval [0 T ]with d as the supremum norm

d(f g) = suptisin[0T ]

|f(t)minus g(t)|

Note that random walks and Brownian motions are stochastic processes while dmeasures the distance between two continuous functions In order to measure thedistance between random walks and Brownian motion we instead consider the pathstaken by these stochastic processes

t 983041rarr Rt and t 983041rarr Bt

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 11

33 Brownian Motion Construction In this paper the construction of Brow-nian motion comprises the following steps

(1) Construct a simple random walk out of a Brownian motion path usingSkorokhod embedding

(2) Linearly interpolate and scale the embedded random walk creating a con-tinuous function from [0 n] 983041rarr R

(3) Take the uniform limit of this scaled random walk to derive the originalBrownian motion path

The paper presents an alternative formulation to the canonical construction ofBrownian motion which involves defining Brownian motion on the dyadic rationalsand using continuity to extend the definition to the reals To motivate this particu-lar construction we will proceed out of order we will begin with step (2) backtrackto step (1) and conclude by stringing together steps (1) through (3) in subsection333 (Strong Approximation)

331 Scaled Random WalksLet Rm be a random walk where E[Xi] = 0 VarXi = σ2 and

Rm =

m983131

i=1

Xi

Definition 35 A scaled random walk with scaling parameter n is a real valuedrandom process

983051R(n)

983052 defined on t ge 0 For t such that nt is an integer

R(n)t =

1radicnσ2

Rnt

For t such that nt is not an integer S(n)(t) is defined by linear interpolation

R(n)t =

1radicnσ2

983063Rlfloorntrfloor + (ntminus lfloorntrfloor)

983043Rlceilntrceil minusRlfloorntrfloor

983044 983064

This method of scaling operates on two aspects of random walks as n increasesthe time between state transitions decreases and increment lengths decrease Lett0 = 0 and tk = inftkt isin Z nt gt nt(kminus1) then as n rarr infin

tk minus t(kminus1) rarr 0 and R(n)tk

minusR(n)t(kminus1)

rarr 0

As a result R(n)(t) is continuous with probability 1 on t ge 0 for all n The scalingfactor 1radic

nσ2ensures constant variance for a given t

Var

9830691radicnσ2

Rnt

983070= Var

9830831radicnσ2

nt983131

i=1

Xi

983084=

9830611radicnσ2

9830622

ntVarXi = tVar Xi

σ2= t

By observing the limiting behavior of R(n)t (ie as n rarr infin) we establish a con-

struction of Brownian motion

12 HARVEY BARNHARD

332 Skorokhod EmbeddingTo construct Brownian motion out of random walks we must first construct randomwalks out of Brownian motion Structuring the proof in this matter is not withoutreason there are uncountably many realizations of Brownian motion so we want toensure that the random walks to which we are taking the limit do in fact convergeto the Brownian motion of interest Such a procedure of deriving a random walkfrom a Brownian motion is called Skorokhod embedding The procedure is given inthe following pseudocode

Set t0 = 0 and n = 0 Let T be a stopping time of Brownian motion Btso that Bt is defined on [0 T ] Let R0 = 0 be the initial value

of the Skorokhod embedding

While tn + t le TInitialize t = 0

While |Bt minusBtn | le 1 Continue Bt until |Bt minusBtn | = 1 for the first time

Set t(n+1) = inft |Bt minusBtn | = 1If Bt(n+1)

minusBtn = 1 then set X(n+1) = 1

Otherwise Bt(n+1)minusBtn = minus1 so set X(n+1) = minus1

Let the (n+ 1)th value of the Skorokhod embedding be

R(n+1) = Rn +Xn+1

Increment to the next step of the inner while loop n = n+ 1

The resulting sequence Rnn=12 is the Skorokhod embedding of Bttisin[0T ]

Theorem 36 Let Bt be a standard Brownian motion Set t0 = 0 and let tn bethe stopping time where |Bt| = 1 for the nth time

tn equiv inft ge 0 |Bt minusBtn | = 1

If we define Rn equiv Btn then Rn is a simple random walk

ProofTo prove that B0 Bt1 Bt2 is a simple random walk we merely need to showthat

(1) PBtn+1minusBtn = 1 = PBtn+1

minusBtn = minus1 = 12 for all n isin N

(2) tn+1 minus tn are iid random variables

The first item follows from the symmetric property of Brownian increments prop-erty (II) in the definition of Brownian motion Bs+t minusBs sim N(0σ2t)

The second item follows from the strong Markov property of Brownian motionwhich states that if τ is a stopping time with respect to Bt then Bt minus Bτ is

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 13

also a Brownian motion We do not include a rigorous treatment of stopping timesor a proof of the strong Markov property

We now state a theorem that places a probabilistic bound on the differencebetween a Brownian motion and its Skorokhod embedding

Theorem 37 Define Θ to be the maximum distance between a Brownian motionand its Skorokhod embedding

Θ(BRT ) = max0letleT

|Bt minusRt|

Note that Θ is equivalent to the distance between Bt and Rt in the space ofcontinuous functions on the time interval [0 T ] equipped with the supremum normThere exist c a isin [0infin) such that for all r le n14 and all integers n ge 3

P983153Θ(BRT ) ge rn14

983155log n

983154le ceminusar

ProofIt will suffice to prove the theorem for r ge 9c2 where c is the constant such that

P983153osc(B δ T ) gt r

983155δ log(1δ)

983154le cT δ(rc)

2

andosc(B δ T ) = sup |Bt minusBs| s t isin D s t isin [0 T ] |sminus t| le δ

is the oscillation of Bt restricted to t isin D the dyadic rationals The proof thatsuch a c exists can be found in [3] page 68 where it is also shown that for n isin N

Θ(BSn) le 1 + osc(B 1 n) + max|Bj minusBτj | j = 1 nNow suppose 9c2 le r le n14 If |Bn minusBτn | is large then either |nminus τn| is large

or the oscillation of B is large Consider the three events

(1)983051Θ(BRn) ge rn14

radiclog n

983052

(2)983051osc(B r

radicn 2n) ge (r3)n14

radiclog n

983052

(3) max1lejlen |τj minus j| ge rradicn

From above we know that event (1) is contained in the union of events (2) and (3)As a consequence we need only prove the result of the theorem for events (2) and(3) We first tackle event (2)

P983153osc(B r

radicn 2n) gt (r3)n(14)

983155log n

983154le 3P

983153osc(B r

radicn n) gt (r3)n(14)

983155log n

983154

= 3P983153osc(B rnminus12) gt (r3)nminus(14)

983155log n

983154

le 3P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log

983043n12r

983044983070

Ifradicr3 ge c and r le n14 we can conclude that there exist c and a such that

P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log(n12)

983070le ceminusar logn

For event (3) we refer to a proof on page 266 in the appendix of [3] to show thatthere exist c a such that

P983069

max1lejlen

|τj minus j| gt rradicn

983070le ceminusar2

14 HARVEY BARNHARD

333 Strong ApproximationWith preliminaries in order we now show that Brownian motion can be thought ofas a limit of simple random walks a construction known as the strong approximationof Brownian motion Let Bt be a standard Brownian motion with variance

parameter 1 as defined in the probability space (ΩF P) and let B(n)t be the

scaled Brownian motion

B(n)t =

1radicnBnt

From B(n) we use the Skorokhod embedding to derive the simple random walkS(n) And from S(n) we derive the scaled simple random walk R(n)

R(n)t =

1radicnS(n)nt

From Theorem 37 we know that there exists c a isin [0infin) such that for all positiveintegers T

P983069

max0letleTn

983055983055983055S(n)t minusB

(n)t

983055983055983055 ge cr(Tn)14983155log (Tn)

983070le ceminusar

multiplying by 1radicn this becomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge crT 14nminus14983155log (Tn)

983070le ceminusar

Letting r = c log n where c is sufficiently large for the given T the inequalitybecomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

983070le c

n2

To proceed we require the use of the Borel-Cantelli lemma the result of which isprovided below

Lemma 38 (Borel-Cantelli)If An is a sequence of events in Ω and

983123infini=1 PAn lt infin then

PAn io = 0

where An io = lim supnrarrinfin An is the event that infinitely many An occur

If we set

An = max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

we can then apply the Borel-Cantelli lemma to the above inequality to concludethat

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 le cnminus14 log32 n

with probability one for all n sufficiently large Since

limnrarrinfin

cnminus14 log32 n = 0

we conclude that

R(n)t

asminusminusrarr Bt in C[0 T ]The following theorem condenses the results of this section up to this point

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 15

Theorem 39 (Donskerrsquos Theorem) Let Xi be a sequence of iid random vari-

ables with mean 0 and variance σ2 If R(n)t is a scaled random walk as defined

above then R(n)t rArr Bt where Bt is a standard Brownian motion This result is

also known as the functional central limit theorem

34 Brownian Motion VariantsThe preceding construction of Brownian motion will prove particularly useful in ourapproximation of heavy traffic waiting times However this process requires furthermodification Recall that when traffic intensity ρ is slightly less than 1 customersare being serviced at a slightly faster rate then they are entering the queue Thusgiven enough time traffic will eventually ldquoclear outrdquo due to faster service timesHowever this particular construction of Brownian motion has increments of meanzero

E[Bs+t minusBs] = 0

meaning that the approximate number of customers in line wouldmdashon averagemdashremain unchanged regardless of how long the line is observed Such a propertyis called the martingale property Instead we want a process that exhibits thesupermartingale property

E[Bs+t minusBs] lt 0

Definition 310 Let Fn be an increasing sequence of σ-fields (Fn is called afiltration) Let Xn be a sequence of random variables with finite mean andXn isin Fn for all n Martingales supermartingales and submartingales are thendefined as follows for all n

Martingale E [Xn+1 | Fn] = Xn

Supermartingale E [Xn+1 | Fn] le Xn

Submartingale E [Xn+1 | Fn] ge Xn

In words a sequence that we expect to remain the same over time is a martingalea sequence we expect to increase is a submartingale and a sequence we expect todecrease is a supermartingale To approximate heavy traffic we want a Brownianmotion-like process that exhibits the supermartingale property

Definition 311 A Brownian motion with drift parameter α and variance param-eter σ2 Bα

t is defined by the following properties

(1) Independent increments(2) Increment lengths are normally distributed with mean αt and variance σ2t(3) the path t 983041rarr Bα

t is continuous

Note that these properties are the same as standard Brownian motion exceptincrement lengths now have non-zero means Since we have already shown theexistence of standard Brownian motion the construction of Brownian motion withdrift is now easy

Lemma 312 Let Bt be a standard Brownian motion with variance parameterσ2 then Bα

t = Bt + αt is Brownian motion with drift parameter α and varianceparameter σ2 Moreover Bα

t fulfills the supermartingale property when α lt 0

16 HARVEY BARNHARD

ProofIndependence of increments follows directly from independence of increments forBt and the fact that αt is a constant for a given t To prove that Bα

t has stationaryincrements we have to show that the distribution of increments Bα

s+tminusBαs depends

only the time interval t and not the absolute time Recall from the definition of Bt

that Bs+t minusBt are normal random variables with mean 0 and variance t for s gt 0

E983045Bα

s+t minusBαs

983046= E

983045Bα

s+t

983046minus E [Bα

s ]

= E983045Bα

s+t + α(s+ t)983046minus E [Bs + αs]

= α(s+ t)minus αs

= αt

Var983051Bα

s+t minusBαs

983052= Var Bs+t minusBs + αt= Var Bs+t minusBs= σ2t

And since for fixed t αt is just a constant we use the property of normal randomvariables to conclude that

Bαs+t minusBα

s sim N(αtσ2t) for all s gt 0

Continuity of Bαt directly follows from continuity of Bt We have shown that

Bαt constructed as the sum of standard Brownian motion and drift αt is indeed

Brownian motion with drift

Brownian motion with negative drift provides a better approximation of thewaiting-time of customers in a queue than does standard Brownian motion How-ever the unboundedness of Brownian motion suggests that waiting times couldperhaps be negative In order to circumvent this issue we instead use reflectedBrownian motion with drift Reflected Brownian motion acts much the same asBrownian motion except on a given boundary off of which the Brownian motionismdashunsurprisinglymdashreflected In the one-dimensional case the boundaries may beany interval on the reals that provide a lower and upper bound to the Brownianmotion For our purposes we only use a single boundary zero that gives a lowerbound to Brownian motion That is for all t ge 0 we want Bα

t ge 0

Definition 313 Let Bαt be a standard Brownian motion with drift parameter α

and let Mαt be the running maximum of Bα

t

Mαt = sup

0lesletBα

s

Then reflected Brownian motion with drift parameter minusα and boundary 0 definedon R+ can be constructed as follows

Rα0t equiv max0Mα

t minusBαt

Note that max0 Sαt ge 0 and Mα

t ge Bαt B

α0t ge 0 for all t Most importantly

the local behavior of Bα0t is exactly like Brownian motion with drift parameter minusα

since max0 Sαt is simply a constant when Bα

t ∕= 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 17

We have now provided sufficient modifications of Brownian motion and we nextmove onmdashwhat we will ultimately use to approximate heavy traffic waiting timesThe next example provides us with the distribution of the supremum of a Brownianmotion with negative drift

Example 314Let Bt be a Brownian motion with drift α lt 0 and variance parameter σ2 Define

Mt = sup0leslet

Bs

to be the running maximum then

Minfin sim exp

9830612|α|σ2

983062

that is

PW ge w = eminusminus2|α|σ2 w w ge 0

E [Minfin] =σ2

2|α|

Prooffor constants a b gt 0 Let T (minusa b) be the first time that B minus t hits minusa or b

T (minusa b) = inft Bt = minusa or Bt = bIn [7] it is proved that

P983051BT (minusab)=b

983052=

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)

Since α lt 0 exp(2αaσ2) rarr 0 we have

limararrinfin

PBT (minusab) = b = limararrinfin

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)= exp(2αbσ2)

The left-hand side becomes the probability that the process will reach b somewherealong its path (ie that the maximum of the process exceeds b somewhere alongits path) Therefore we have

PW ge b = exp(2αbσ2) = expminus2|α|bσ2

This example shows us that the supremum of a Brownian motion observed ldquofor-everrdquo (ie on the interval [0infin)) is exponentially distributed with rate 2|α|σ2

4 Heavy Traffic Approximation

We now have all we need to provide a heuristic argument for the approximationof single-servers queue in heavy traffic using Brownian Motion For a rigorousalbeit opaque treatment of this approximation see [8] the seminal work on thissubject by Kingman

Theorem 41 The waiting time of the nth customer in an MM1 queue canbe approximated by the supremum of reflected Brownian motion with negative driftand boundary at 0

18 HARVEY BARNHARD

ProofRecall that an MM1 queue is a process where the number of customer arrivalsfollows a poisson process with rate λ and service times are exponentially distributedwith rate micro From Lindleyrsquos equation in Section 2 we can write the waiting timefor the nth customer as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

where U (n) = S(n)minus I(n) is the difference between the nth service time and the nthinter-arrival time Define the partial sum P

(n)k to be zero when k = 0 and

P(n)k equiv

k983131

i=1

U (nminusi) for k ge 1

We can then rewrite the waiting time of the nth customer as

W (n)q = max

0lekle(nminus1)P

(n)k

The fact that P(n)k is the kth value of a random walk follows immediately from

U (1) U (nminus1) being iid random variables Let α and σ2 be the expectationand variance of U (i) We calculate α and σ2 by first noting that S(i) and I(i) areexponentially distributed with rates micro and λ Recall that ρ = λmicro is the trafficintensity as defined in the first section of this paper

α = E983147U (i)

983148= E

983147S(i)

983148minus E

983147I(i)

983148=

1

microminus 1

λ=

ρminus 1

λ

σ2 = Var983153U (i)

983154= Var

983153S(i)

983154+Var

983153I(i)

983154=

1

micro2+

1

λ2

We next compute the expectation and variance of P(n)k using the fact that U (i) are

iid random variables

E983147P

(n)k

983148=

k983131

i=1

E983147U (nminusi)

983148= kα

Var983153P

(n)k

983154=

k983131

i=1

Var983153U (nminusi)

983154= kσ2

For a finite n W(n)q is therefore the maximum value of a random walk with n steps

We now show that Pnk can be approximated by Brownian motion with drift

parameter α and variance parameter σ2 when n is sufficiently large We restate theproperties of such a Brownian motion Bt below

(1) B0 = 0(2) Bt has stationary and independent increments(3) Bt sim N(αtσ2t)(4) t 983041rarr Bt is continuous with probability one

The first property is clear as P(n)0 = 0 for all n The second property follows from

the fact that U (i) are iid random variables The third property is a result of thecentral limit theorem for iid sequences the result of which is provided below

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

6 HARVEY BARNHARD

stochastic processes For instance a Poisson process is one of the simplest birth-death processes where the ldquobirthrdquo rate is constant and the ldquodeathrdquo rate is zero Inparticular a queuing process is a birth-death process where customer arrivals areldquobirthsrdquo and service completions are ldquodeathsrdquo

23 Queuing Models

Example 26 (MMc) In this model inter-arrival times and service times areexponentially distributed with rates λ and micro respectively In this queue λ and microare constant and independent of the current state That is λN = λ and microN = micro forall N Service is received on a ldquofirst come first servedrdquo basis and c customers maybe serviced at a time The simplest case is when c = 1 where only one customeris serviced at a time In an MMc queue where customers are serviced as soon asthey arrive regardless of the population of the system we set c = infin The MMcmodel assumes that no customers leave the system between arrival and completionof service

This paper focuses on the MM1 queue but we introduce a few more modelsas examples of the breadth of situations that can be analyzed through the lens ofqueuing theory

Example 27 (GGc) This model is the same as the MMc model exceptinter-arrival times and service times follow some general (represented by G) notnecessarily exponential distribution

Example 28 (GGcK) This model is the same as the GGc model but withan upper bound K on the number of customers that can occupy the system at agiven time This type of queue is called a truncated queue

Example 29 (ABCDE) The above queuing models are identified usingKendall Notation a combination of letters and slashes Each position in the nota-tion corresponds to a certain queuing characteristic (eg inter-arrival time distri-bution) and the letter specifies the characteristic (eg exponentially distributedinter-arrival times) The following table covers a wide selection of queuing modelsthat are easily be described using Kendall notation

Characteristic Symbol Explanation(A) Inter-arrival time distribution M Exponential(B) Service time distribution D Deterministic

Ek Erlang type kG General

(C) of parallel servers 1 2 infin(D) Max system capacity 1 2 infin(E) Order of service FCFS First come first served

LCFS Last come first servedRSS Random selectionPR PriorityGD General

The table above is by no means a definitive list of queuing characteristics Forinstance inter-arrival and service times may be state-dependent as introduced inthe beginning of this section with rates λN and microN Queues can also be cyclic in

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 7

the sense that once a customer is serviced the customer immediately returns to aposition in line to await service once again

24 Properties of Queuing ModelsThe first two properties one may ask of a queue are the expected number of cus-tomers in line and the duration of time a customer must wait before being ser-viced Let Nt be the number of customers in a GGc queue at time t If we letpn = PNt = n the expected population of the system is

E[Nt] =

infin983131

n=0

npn

and the expected population of those in the queue (and not being serviced) is

E[Ntq] =

infin983131

n=0

(nminus c)pn

Definition 210 The nth waiting time is the amount of time the nth customerspends waiting in line prior to entering service We denote waiting time as the

random variable W(n)q and the total time a customer is in the system including

service time as W (n) The nth customer is the nth overall customer to enter thesystem not the customer in the nth position in line

We now consider the waiting time of the nth customer to enter a queuing systemwhere the initial population is zero that is N0 = 0 Let S(n) be the nth servicetime and let I(n) be the nth inter-arrival time ie the time between the (nminus 1)stcustomer and the nth customer arriving in the system Define

U (n) equiv S(n) minus I(n)

to be the time between the nth inter-arrival time and the nth service time

Theorem 211 (Lindleyrsquos Equation) In a single-server queue where customersare serviced on a first-come first-served basis the waiting time of the (n + 1)thcustomer is recursively given by

W (n+1)q = max

9831530W (n)

q + S(n) minus I(n)983154

ProofSince the initial population of the system is zero the first customer to arrive will

be serviced immediately so W(1)q = 0 The waiting time for the second customer

will be the time it takes for the first customer to finish being serviced If the firstcustomer has already been serviced by the time the second customer arrives thenthe waiting time for the second customer will be zero

W (2)q = max

9831530 S(1) minus I(1)

983154= max

9831530W (1)

q + S(1) minus I(1)983154

For n ge 3 the waiting time of the nth customer is simply the waiting time of the(nminus 1)th customer and the amount of time it takes for the (nminus 1)th customer tobe serviced or zero if the (nminus 1)th customer has already been serviced by the timethe nth customer enters the system

Lemma 212 Under the same conditions as the preceding theorem W(n)q may be

rewritten as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

8 HARVEY BARNHARD

Proof

We prove the lemma by iterating over W(n)q

W (n)q = max

983153W (nminus1)

q + U (nminus1) 0983154

= max983153max

983153W (nminus2)

q + U (nminus2) 0983154+ U (nminus1) 0

983154

= max983153max

983153W (nminus2)

q + U (nminus2) + U (nminus1) U (nminus1)983154 0983154

= max983153W (nminus2)

q + U (nminus2) + U (nminus1) U (nminus1) 0983154

Continue this process in a recursive manner until we have

W (n)q = max

983153W (1)

q + U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1)983154

The result of the lemma is now immediate since we have assumed thatW(1)q = 0

The goal of this paper is to approximate W(n)q the waiting time of the nth

customer From the preceding lemma it is clear that the waiting time depends onthe waiting times of all preceding customers in line More precisely the waitingtime of the nth customer is the maximum of the partial sums of the decreasingsequence

983051U (nminusi)

983052n

i=1 an iid sequence of random variables Section 3 will show

that this decreasing sequence is a random walk and when n is sufficiently large thissequence can be approximated by Brownian motion For such an approximation tobe accurate however each U (nminusi) must be sufficiently small small This situationarises in queues that exhibit heavy traffic

Definition 213 The traffic intensity ρ of a given queue is defined as the ratioof arrival times to service times So for an MMc queue the traffic intensity is

ρ =λ

cmicro

where λ and micro are the arrival and service rate respectively of customers in thequeue For a single server MM1 queue the traffic intensity is

ρ =λ

micro

Remark 214 When ρ is close to zero waiting times approach zero as queue lengthshortens and new customers are serviced immediately upon arrival When ρ isgreater than one waiting times successively increase as the queue population ldquoex-plodesrdquo to infinity The interesting case occurs when ρ is close tomdashbut does notexceedmdashone A system is said to be in heavy traffic when ρ isin (1minus ε 1) and ε gt 0is small

3 Brownian Motion

This section introduces the stochastic process of Brownian motion viewed as thelimit of random walks In this paper we confine our study of Brownian motion tothe one-dimensional case

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 9

31 Random Walks

Definition 31 A one-dimensional random walk is a stochastic process constructedas the sum of iid random variables That is if Xi is a sequence of iid randomvariables with Xi isin R (regarded as steps) then the sum of the first n such randomvariables

Rn =

n983131

i=1

Xi

is the n-th value of a random walk A random walk is said to be symmetric ifXi isin R is symmetrically distributed about zero A random walk is called simple ifPXi = 1+ PXi = minus1 = 1

Example 32 Figure 3 depicts a simple random walk simulated using the statisti-cal computing software R with PXi = minus1 = PXi = 1 = 1

2 and 15 total stepsTaking R0 = X0 = 0 the values of Xi and Ri are

Xi15i=1 = 1minus1 1minus1minus1minus1minus1 1minus1minus1 1 1minus1 1minus1Ri15i=1 = 0 1 0 1 0minus1minus2minus3minus2minus3minus4minus3minus2minus3minus2minus3

0 5 10 15

minus4minus3

minus2minus1

01

n

S n

Figure 3

An important property of random walks is that they exhibit independent incre-ments meaning that for any selection of positive integers t1 lt t2 lt middot middot middot lt tn therandom variables

Rt2 minusRt1 Rtn minusRtnminus1

are independent This property immediately follow from the fact that each incre-ment is the sum of iid random variables

10 HARVEY BARNHARD

32 Brownian Motion DefinitionWe now introduce the continuous-time analog to a random walkmdashBrownian motionA one-dimensional standard Brownian motion with variance parameter σ2 is a real-valued process Bttge0 defined in the probability space (ΩF P) that has thefollowing properties

(I) If t0 lt t1 lt middot middot middot lt tn then Bt0 Bt1 minusBt0 Btn minusBtnminus1are independent

(II) If s t ge 0 then

P (Bs+t minusBs isin A) =

983133

A

1radic2πσ2t

exp(minusx22σ2t)

(III) With probability 1 t 983041rarr Bt is continuous

The first two conditions may be concisely reworded as

(I) Independent increments(II) Increment lengths are normally distributed with mean 0 and variance σ2t

Bs+t minusBs sim N(0σ2t)

Remark 33Brownian motion is a collection of stochastic processes When we say a Brownianmotion or the Brownian motion we are talking about a particular realization ofBrownian motion in the space of continuous functions C[0 T ] where T is a stoppingtime

The above definition establishes the necessary conditions for a stochastic process tobe Brownian but the existence of such a process requires further work Throughan application of the Kolmogorov extension theorem existence is proved in [4]However such a proof is bereft of intuition for queueing theory and approximatingwaiting times Instead we take the existence of Brownian motion as given andshow that Brownian motion can be constructed as a limit of random walksmdashtheresult of Donskerrsquos Theorem Brownian motion constructed in such a manner iscalled the standard Wiener process

Before constructing Brownian motion we must first establish what type of con-vergence we are discussing

Definition 34 A sequence of random variables Xiinfini=1 converges almost surelyto a random variable X in the underlying probability space (ΩF P) (denoted

Xnasminusminusrarr X) if

P983153ω isin Ω lim

nrarrinfinXn(ω) = X(ω)

983154= 1

We aim to show almost sure convergence of a scaled random walk to a Brownianmotion in (C[0 T ] d) the metric space of continuous functions on the interval [0 T ]with d as the supremum norm

d(f g) = suptisin[0T ]

|f(t)minus g(t)|

Note that random walks and Brownian motions are stochastic processes while dmeasures the distance between two continuous functions In order to measure thedistance between random walks and Brownian motion we instead consider the pathstaken by these stochastic processes

t 983041rarr Rt and t 983041rarr Bt

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 11

33 Brownian Motion Construction In this paper the construction of Brow-nian motion comprises the following steps

(1) Construct a simple random walk out of a Brownian motion path usingSkorokhod embedding

(2) Linearly interpolate and scale the embedded random walk creating a con-tinuous function from [0 n] 983041rarr R

(3) Take the uniform limit of this scaled random walk to derive the originalBrownian motion path

The paper presents an alternative formulation to the canonical construction ofBrownian motion which involves defining Brownian motion on the dyadic rationalsand using continuity to extend the definition to the reals To motivate this particu-lar construction we will proceed out of order we will begin with step (2) backtrackto step (1) and conclude by stringing together steps (1) through (3) in subsection333 (Strong Approximation)

331 Scaled Random WalksLet Rm be a random walk where E[Xi] = 0 VarXi = σ2 and

Rm =

m983131

i=1

Xi

Definition 35 A scaled random walk with scaling parameter n is a real valuedrandom process

983051R(n)

983052 defined on t ge 0 For t such that nt is an integer

R(n)t =

1radicnσ2

Rnt

For t such that nt is not an integer S(n)(t) is defined by linear interpolation

R(n)t =

1radicnσ2

983063Rlfloorntrfloor + (ntminus lfloorntrfloor)

983043Rlceilntrceil minusRlfloorntrfloor

983044 983064

This method of scaling operates on two aspects of random walks as n increasesthe time between state transitions decreases and increment lengths decrease Lett0 = 0 and tk = inftkt isin Z nt gt nt(kminus1) then as n rarr infin

tk minus t(kminus1) rarr 0 and R(n)tk

minusR(n)t(kminus1)

rarr 0

As a result R(n)(t) is continuous with probability 1 on t ge 0 for all n The scalingfactor 1radic

nσ2ensures constant variance for a given t

Var

9830691radicnσ2

Rnt

983070= Var

9830831radicnσ2

nt983131

i=1

Xi

983084=

9830611radicnσ2

9830622

ntVarXi = tVar Xi

σ2= t

By observing the limiting behavior of R(n)t (ie as n rarr infin) we establish a con-

struction of Brownian motion

12 HARVEY BARNHARD

332 Skorokhod EmbeddingTo construct Brownian motion out of random walks we must first construct randomwalks out of Brownian motion Structuring the proof in this matter is not withoutreason there are uncountably many realizations of Brownian motion so we want toensure that the random walks to which we are taking the limit do in fact convergeto the Brownian motion of interest Such a procedure of deriving a random walkfrom a Brownian motion is called Skorokhod embedding The procedure is given inthe following pseudocode

Set t0 = 0 and n = 0 Let T be a stopping time of Brownian motion Btso that Bt is defined on [0 T ] Let R0 = 0 be the initial value

of the Skorokhod embedding

While tn + t le TInitialize t = 0

While |Bt minusBtn | le 1 Continue Bt until |Bt minusBtn | = 1 for the first time

Set t(n+1) = inft |Bt minusBtn | = 1If Bt(n+1)

minusBtn = 1 then set X(n+1) = 1

Otherwise Bt(n+1)minusBtn = minus1 so set X(n+1) = minus1

Let the (n+ 1)th value of the Skorokhod embedding be

R(n+1) = Rn +Xn+1

Increment to the next step of the inner while loop n = n+ 1

The resulting sequence Rnn=12 is the Skorokhod embedding of Bttisin[0T ]

Theorem 36 Let Bt be a standard Brownian motion Set t0 = 0 and let tn bethe stopping time where |Bt| = 1 for the nth time

tn equiv inft ge 0 |Bt minusBtn | = 1

If we define Rn equiv Btn then Rn is a simple random walk

ProofTo prove that B0 Bt1 Bt2 is a simple random walk we merely need to showthat

(1) PBtn+1minusBtn = 1 = PBtn+1

minusBtn = minus1 = 12 for all n isin N

(2) tn+1 minus tn are iid random variables

The first item follows from the symmetric property of Brownian increments prop-erty (II) in the definition of Brownian motion Bs+t minusBs sim N(0σ2t)

The second item follows from the strong Markov property of Brownian motionwhich states that if τ is a stopping time with respect to Bt then Bt minus Bτ is

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 13

also a Brownian motion We do not include a rigorous treatment of stopping timesor a proof of the strong Markov property

We now state a theorem that places a probabilistic bound on the differencebetween a Brownian motion and its Skorokhod embedding

Theorem 37 Define Θ to be the maximum distance between a Brownian motionand its Skorokhod embedding

Θ(BRT ) = max0letleT

|Bt minusRt|

Note that Θ is equivalent to the distance between Bt and Rt in the space ofcontinuous functions on the time interval [0 T ] equipped with the supremum normThere exist c a isin [0infin) such that for all r le n14 and all integers n ge 3

P983153Θ(BRT ) ge rn14

983155log n

983154le ceminusar

ProofIt will suffice to prove the theorem for r ge 9c2 where c is the constant such that

P983153osc(B δ T ) gt r

983155δ log(1δ)

983154le cT δ(rc)

2

andosc(B δ T ) = sup |Bt minusBs| s t isin D s t isin [0 T ] |sminus t| le δ

is the oscillation of Bt restricted to t isin D the dyadic rationals The proof thatsuch a c exists can be found in [3] page 68 where it is also shown that for n isin N

Θ(BSn) le 1 + osc(B 1 n) + max|Bj minusBτj | j = 1 nNow suppose 9c2 le r le n14 If |Bn minusBτn | is large then either |nminus τn| is large

or the oscillation of B is large Consider the three events

(1)983051Θ(BRn) ge rn14

radiclog n

983052

(2)983051osc(B r

radicn 2n) ge (r3)n14

radiclog n

983052

(3) max1lejlen |τj minus j| ge rradicn

From above we know that event (1) is contained in the union of events (2) and (3)As a consequence we need only prove the result of the theorem for events (2) and(3) We first tackle event (2)

P983153osc(B r

radicn 2n) gt (r3)n(14)

983155log n

983154le 3P

983153osc(B r

radicn n) gt (r3)n(14)

983155log n

983154

= 3P983153osc(B rnminus12) gt (r3)nminus(14)

983155log n

983154

le 3P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log

983043n12r

983044983070

Ifradicr3 ge c and r le n14 we can conclude that there exist c and a such that

P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log(n12)

983070le ceminusar logn

For event (3) we refer to a proof on page 266 in the appendix of [3] to show thatthere exist c a such that

P983069

max1lejlen

|τj minus j| gt rradicn

983070le ceminusar2

14 HARVEY BARNHARD

333 Strong ApproximationWith preliminaries in order we now show that Brownian motion can be thought ofas a limit of simple random walks a construction known as the strong approximationof Brownian motion Let Bt be a standard Brownian motion with variance

parameter 1 as defined in the probability space (ΩF P) and let B(n)t be the

scaled Brownian motion

B(n)t =

1radicnBnt

From B(n) we use the Skorokhod embedding to derive the simple random walkS(n) And from S(n) we derive the scaled simple random walk R(n)

R(n)t =

1radicnS(n)nt

From Theorem 37 we know that there exists c a isin [0infin) such that for all positiveintegers T

P983069

max0letleTn

983055983055983055S(n)t minusB

(n)t

983055983055983055 ge cr(Tn)14983155log (Tn)

983070le ceminusar

multiplying by 1radicn this becomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge crT 14nminus14983155log (Tn)

983070le ceminusar

Letting r = c log n where c is sufficiently large for the given T the inequalitybecomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

983070le c

n2

To proceed we require the use of the Borel-Cantelli lemma the result of which isprovided below

Lemma 38 (Borel-Cantelli)If An is a sequence of events in Ω and

983123infini=1 PAn lt infin then

PAn io = 0

where An io = lim supnrarrinfin An is the event that infinitely many An occur

If we set

An = max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

we can then apply the Borel-Cantelli lemma to the above inequality to concludethat

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 le cnminus14 log32 n

with probability one for all n sufficiently large Since

limnrarrinfin

cnminus14 log32 n = 0

we conclude that

R(n)t

asminusminusrarr Bt in C[0 T ]The following theorem condenses the results of this section up to this point

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 15

Theorem 39 (Donskerrsquos Theorem) Let Xi be a sequence of iid random vari-

ables with mean 0 and variance σ2 If R(n)t is a scaled random walk as defined

above then R(n)t rArr Bt where Bt is a standard Brownian motion This result is

also known as the functional central limit theorem

34 Brownian Motion VariantsThe preceding construction of Brownian motion will prove particularly useful in ourapproximation of heavy traffic waiting times However this process requires furthermodification Recall that when traffic intensity ρ is slightly less than 1 customersare being serviced at a slightly faster rate then they are entering the queue Thusgiven enough time traffic will eventually ldquoclear outrdquo due to faster service timesHowever this particular construction of Brownian motion has increments of meanzero

E[Bs+t minusBs] = 0

meaning that the approximate number of customers in line wouldmdashon averagemdashremain unchanged regardless of how long the line is observed Such a propertyis called the martingale property Instead we want a process that exhibits thesupermartingale property

E[Bs+t minusBs] lt 0

Definition 310 Let Fn be an increasing sequence of σ-fields (Fn is called afiltration) Let Xn be a sequence of random variables with finite mean andXn isin Fn for all n Martingales supermartingales and submartingales are thendefined as follows for all n

Martingale E [Xn+1 | Fn] = Xn

Supermartingale E [Xn+1 | Fn] le Xn

Submartingale E [Xn+1 | Fn] ge Xn

In words a sequence that we expect to remain the same over time is a martingalea sequence we expect to increase is a submartingale and a sequence we expect todecrease is a supermartingale To approximate heavy traffic we want a Brownianmotion-like process that exhibits the supermartingale property

Definition 311 A Brownian motion with drift parameter α and variance param-eter σ2 Bα

t is defined by the following properties

(1) Independent increments(2) Increment lengths are normally distributed with mean αt and variance σ2t(3) the path t 983041rarr Bα

t is continuous

Note that these properties are the same as standard Brownian motion exceptincrement lengths now have non-zero means Since we have already shown theexistence of standard Brownian motion the construction of Brownian motion withdrift is now easy

Lemma 312 Let Bt be a standard Brownian motion with variance parameterσ2 then Bα

t = Bt + αt is Brownian motion with drift parameter α and varianceparameter σ2 Moreover Bα

t fulfills the supermartingale property when α lt 0

16 HARVEY BARNHARD

ProofIndependence of increments follows directly from independence of increments forBt and the fact that αt is a constant for a given t To prove that Bα

t has stationaryincrements we have to show that the distribution of increments Bα

s+tminusBαs depends

only the time interval t and not the absolute time Recall from the definition of Bt

that Bs+t minusBt are normal random variables with mean 0 and variance t for s gt 0

E983045Bα

s+t minusBαs

983046= E

983045Bα

s+t

983046minus E [Bα

s ]

= E983045Bα

s+t + α(s+ t)983046minus E [Bs + αs]

= α(s+ t)minus αs

= αt

Var983051Bα

s+t minusBαs

983052= Var Bs+t minusBs + αt= Var Bs+t minusBs= σ2t

And since for fixed t αt is just a constant we use the property of normal randomvariables to conclude that

Bαs+t minusBα

s sim N(αtσ2t) for all s gt 0

Continuity of Bαt directly follows from continuity of Bt We have shown that

Bαt constructed as the sum of standard Brownian motion and drift αt is indeed

Brownian motion with drift

Brownian motion with negative drift provides a better approximation of thewaiting-time of customers in a queue than does standard Brownian motion How-ever the unboundedness of Brownian motion suggests that waiting times couldperhaps be negative In order to circumvent this issue we instead use reflectedBrownian motion with drift Reflected Brownian motion acts much the same asBrownian motion except on a given boundary off of which the Brownian motionismdashunsurprisinglymdashreflected In the one-dimensional case the boundaries may beany interval on the reals that provide a lower and upper bound to the Brownianmotion For our purposes we only use a single boundary zero that gives a lowerbound to Brownian motion That is for all t ge 0 we want Bα

t ge 0

Definition 313 Let Bαt be a standard Brownian motion with drift parameter α

and let Mαt be the running maximum of Bα

t

Mαt = sup

0lesletBα

s

Then reflected Brownian motion with drift parameter minusα and boundary 0 definedon R+ can be constructed as follows

Rα0t equiv max0Mα

t minusBαt

Note that max0 Sαt ge 0 and Mα

t ge Bαt B

α0t ge 0 for all t Most importantly

the local behavior of Bα0t is exactly like Brownian motion with drift parameter minusα

since max0 Sαt is simply a constant when Bα

t ∕= 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 17

We have now provided sufficient modifications of Brownian motion and we nextmove onmdashwhat we will ultimately use to approximate heavy traffic waiting timesThe next example provides us with the distribution of the supremum of a Brownianmotion with negative drift

Example 314Let Bt be a Brownian motion with drift α lt 0 and variance parameter σ2 Define

Mt = sup0leslet

Bs

to be the running maximum then

Minfin sim exp

9830612|α|σ2

983062

that is

PW ge w = eminusminus2|α|σ2 w w ge 0

E [Minfin] =σ2

2|α|

Prooffor constants a b gt 0 Let T (minusa b) be the first time that B minus t hits minusa or b

T (minusa b) = inft Bt = minusa or Bt = bIn [7] it is proved that

P983051BT (minusab)=b

983052=

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)

Since α lt 0 exp(2αaσ2) rarr 0 we have

limararrinfin

PBT (minusab) = b = limararrinfin

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)= exp(2αbσ2)

The left-hand side becomes the probability that the process will reach b somewherealong its path (ie that the maximum of the process exceeds b somewhere alongits path) Therefore we have

PW ge b = exp(2αbσ2) = expminus2|α|bσ2

This example shows us that the supremum of a Brownian motion observed ldquofor-everrdquo (ie on the interval [0infin)) is exponentially distributed with rate 2|α|σ2

4 Heavy Traffic Approximation

We now have all we need to provide a heuristic argument for the approximationof single-servers queue in heavy traffic using Brownian Motion For a rigorousalbeit opaque treatment of this approximation see [8] the seminal work on thissubject by Kingman

Theorem 41 The waiting time of the nth customer in an MM1 queue canbe approximated by the supremum of reflected Brownian motion with negative driftand boundary at 0

18 HARVEY BARNHARD

ProofRecall that an MM1 queue is a process where the number of customer arrivalsfollows a poisson process with rate λ and service times are exponentially distributedwith rate micro From Lindleyrsquos equation in Section 2 we can write the waiting timefor the nth customer as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

where U (n) = S(n)minus I(n) is the difference between the nth service time and the nthinter-arrival time Define the partial sum P

(n)k to be zero when k = 0 and

P(n)k equiv

k983131

i=1

U (nminusi) for k ge 1

We can then rewrite the waiting time of the nth customer as

W (n)q = max

0lekle(nminus1)P

(n)k

The fact that P(n)k is the kth value of a random walk follows immediately from

U (1) U (nminus1) being iid random variables Let α and σ2 be the expectationand variance of U (i) We calculate α and σ2 by first noting that S(i) and I(i) areexponentially distributed with rates micro and λ Recall that ρ = λmicro is the trafficintensity as defined in the first section of this paper

α = E983147U (i)

983148= E

983147S(i)

983148minus E

983147I(i)

983148=

1

microminus 1

λ=

ρminus 1

λ

σ2 = Var983153U (i)

983154= Var

983153S(i)

983154+Var

983153I(i)

983154=

1

micro2+

1

λ2

We next compute the expectation and variance of P(n)k using the fact that U (i) are

iid random variables

E983147P

(n)k

983148=

k983131

i=1

E983147U (nminusi)

983148= kα

Var983153P

(n)k

983154=

k983131

i=1

Var983153U (nminusi)

983154= kσ2

For a finite n W(n)q is therefore the maximum value of a random walk with n steps

We now show that Pnk can be approximated by Brownian motion with drift

parameter α and variance parameter σ2 when n is sufficiently large We restate theproperties of such a Brownian motion Bt below

(1) B0 = 0(2) Bt has stationary and independent increments(3) Bt sim N(αtσ2t)(4) t 983041rarr Bt is continuous with probability one

The first property is clear as P(n)0 = 0 for all n The second property follows from

the fact that U (i) are iid random variables The third property is a result of thecentral limit theorem for iid sequences the result of which is provided below

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 7

the sense that once a customer is serviced the customer immediately returns to aposition in line to await service once again

24 Properties of Queuing ModelsThe first two properties one may ask of a queue are the expected number of cus-tomers in line and the duration of time a customer must wait before being ser-viced Let Nt be the number of customers in a GGc queue at time t If we letpn = PNt = n the expected population of the system is

E[Nt] =

infin983131

n=0

npn

and the expected population of those in the queue (and not being serviced) is

E[Ntq] =

infin983131

n=0

(nminus c)pn

Definition 210 The nth waiting time is the amount of time the nth customerspends waiting in line prior to entering service We denote waiting time as the

random variable W(n)q and the total time a customer is in the system including

service time as W (n) The nth customer is the nth overall customer to enter thesystem not the customer in the nth position in line

We now consider the waiting time of the nth customer to enter a queuing systemwhere the initial population is zero that is N0 = 0 Let S(n) be the nth servicetime and let I(n) be the nth inter-arrival time ie the time between the (nminus 1)stcustomer and the nth customer arriving in the system Define

U (n) equiv S(n) minus I(n)

to be the time between the nth inter-arrival time and the nth service time

Theorem 211 (Lindleyrsquos Equation) In a single-server queue where customersare serviced on a first-come first-served basis the waiting time of the (n + 1)thcustomer is recursively given by

W (n+1)q = max

9831530W (n)

q + S(n) minus I(n)983154

ProofSince the initial population of the system is zero the first customer to arrive will

be serviced immediately so W(1)q = 0 The waiting time for the second customer

will be the time it takes for the first customer to finish being serviced If the firstcustomer has already been serviced by the time the second customer arrives thenthe waiting time for the second customer will be zero

W (2)q = max

9831530 S(1) minus I(1)

983154= max

9831530W (1)

q + S(1) minus I(1)983154

For n ge 3 the waiting time of the nth customer is simply the waiting time of the(nminus 1)th customer and the amount of time it takes for the (nminus 1)th customer tobe serviced or zero if the (nminus 1)th customer has already been serviced by the timethe nth customer enters the system

Lemma 212 Under the same conditions as the preceding theorem W(n)q may be

rewritten as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

8 HARVEY BARNHARD

Proof

We prove the lemma by iterating over W(n)q

W (n)q = max

983153W (nminus1)

q + U (nminus1) 0983154

= max983153max

983153W (nminus2)

q + U (nminus2) 0983154+ U (nminus1) 0

983154

= max983153max

983153W (nminus2)

q + U (nminus2) + U (nminus1) U (nminus1)983154 0983154

= max983153W (nminus2)

q + U (nminus2) + U (nminus1) U (nminus1) 0983154

Continue this process in a recursive manner until we have

W (n)q = max

983153W (1)

q + U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1)983154

The result of the lemma is now immediate since we have assumed thatW(1)q = 0

The goal of this paper is to approximate W(n)q the waiting time of the nth

customer From the preceding lemma it is clear that the waiting time depends onthe waiting times of all preceding customers in line More precisely the waitingtime of the nth customer is the maximum of the partial sums of the decreasingsequence

983051U (nminusi)

983052n

i=1 an iid sequence of random variables Section 3 will show

that this decreasing sequence is a random walk and when n is sufficiently large thissequence can be approximated by Brownian motion For such an approximation tobe accurate however each U (nminusi) must be sufficiently small small This situationarises in queues that exhibit heavy traffic

Definition 213 The traffic intensity ρ of a given queue is defined as the ratioof arrival times to service times So for an MMc queue the traffic intensity is

ρ =λ

cmicro

where λ and micro are the arrival and service rate respectively of customers in thequeue For a single server MM1 queue the traffic intensity is

ρ =λ

micro

Remark 214 When ρ is close to zero waiting times approach zero as queue lengthshortens and new customers are serviced immediately upon arrival When ρ isgreater than one waiting times successively increase as the queue population ldquoex-plodesrdquo to infinity The interesting case occurs when ρ is close tomdashbut does notexceedmdashone A system is said to be in heavy traffic when ρ isin (1minus ε 1) and ε gt 0is small

3 Brownian Motion

This section introduces the stochastic process of Brownian motion viewed as thelimit of random walks In this paper we confine our study of Brownian motion tothe one-dimensional case

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 9

31 Random Walks

Definition 31 A one-dimensional random walk is a stochastic process constructedas the sum of iid random variables That is if Xi is a sequence of iid randomvariables with Xi isin R (regarded as steps) then the sum of the first n such randomvariables

Rn =

n983131

i=1

Xi

is the n-th value of a random walk A random walk is said to be symmetric ifXi isin R is symmetrically distributed about zero A random walk is called simple ifPXi = 1+ PXi = minus1 = 1

Example 32 Figure 3 depicts a simple random walk simulated using the statisti-cal computing software R with PXi = minus1 = PXi = 1 = 1

2 and 15 total stepsTaking R0 = X0 = 0 the values of Xi and Ri are

Xi15i=1 = 1minus1 1minus1minus1minus1minus1 1minus1minus1 1 1minus1 1minus1Ri15i=1 = 0 1 0 1 0minus1minus2minus3minus2minus3minus4minus3minus2minus3minus2minus3

0 5 10 15

minus4minus3

minus2minus1

01

n

S n

Figure 3

An important property of random walks is that they exhibit independent incre-ments meaning that for any selection of positive integers t1 lt t2 lt middot middot middot lt tn therandom variables

Rt2 minusRt1 Rtn minusRtnminus1

are independent This property immediately follow from the fact that each incre-ment is the sum of iid random variables

10 HARVEY BARNHARD

32 Brownian Motion DefinitionWe now introduce the continuous-time analog to a random walkmdashBrownian motionA one-dimensional standard Brownian motion with variance parameter σ2 is a real-valued process Bttge0 defined in the probability space (ΩF P) that has thefollowing properties

(I) If t0 lt t1 lt middot middot middot lt tn then Bt0 Bt1 minusBt0 Btn minusBtnminus1are independent

(II) If s t ge 0 then

P (Bs+t minusBs isin A) =

983133

A

1radic2πσ2t

exp(minusx22σ2t)

(III) With probability 1 t 983041rarr Bt is continuous

The first two conditions may be concisely reworded as

(I) Independent increments(II) Increment lengths are normally distributed with mean 0 and variance σ2t

Bs+t minusBs sim N(0σ2t)

Remark 33Brownian motion is a collection of stochastic processes When we say a Brownianmotion or the Brownian motion we are talking about a particular realization ofBrownian motion in the space of continuous functions C[0 T ] where T is a stoppingtime

The above definition establishes the necessary conditions for a stochastic process tobe Brownian but the existence of such a process requires further work Throughan application of the Kolmogorov extension theorem existence is proved in [4]However such a proof is bereft of intuition for queueing theory and approximatingwaiting times Instead we take the existence of Brownian motion as given andshow that Brownian motion can be constructed as a limit of random walksmdashtheresult of Donskerrsquos Theorem Brownian motion constructed in such a manner iscalled the standard Wiener process

Before constructing Brownian motion we must first establish what type of con-vergence we are discussing

Definition 34 A sequence of random variables Xiinfini=1 converges almost surelyto a random variable X in the underlying probability space (ΩF P) (denoted

Xnasminusminusrarr X) if

P983153ω isin Ω lim

nrarrinfinXn(ω) = X(ω)

983154= 1

We aim to show almost sure convergence of a scaled random walk to a Brownianmotion in (C[0 T ] d) the metric space of continuous functions on the interval [0 T ]with d as the supremum norm

d(f g) = suptisin[0T ]

|f(t)minus g(t)|

Note that random walks and Brownian motions are stochastic processes while dmeasures the distance between two continuous functions In order to measure thedistance between random walks and Brownian motion we instead consider the pathstaken by these stochastic processes

t 983041rarr Rt and t 983041rarr Bt

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 11

33 Brownian Motion Construction In this paper the construction of Brow-nian motion comprises the following steps

(1) Construct a simple random walk out of a Brownian motion path usingSkorokhod embedding

(2) Linearly interpolate and scale the embedded random walk creating a con-tinuous function from [0 n] 983041rarr R

(3) Take the uniform limit of this scaled random walk to derive the originalBrownian motion path

The paper presents an alternative formulation to the canonical construction ofBrownian motion which involves defining Brownian motion on the dyadic rationalsand using continuity to extend the definition to the reals To motivate this particu-lar construction we will proceed out of order we will begin with step (2) backtrackto step (1) and conclude by stringing together steps (1) through (3) in subsection333 (Strong Approximation)

331 Scaled Random WalksLet Rm be a random walk where E[Xi] = 0 VarXi = σ2 and

Rm =

m983131

i=1

Xi

Definition 35 A scaled random walk with scaling parameter n is a real valuedrandom process

983051R(n)

983052 defined on t ge 0 For t such that nt is an integer

R(n)t =

1radicnσ2

Rnt

For t such that nt is not an integer S(n)(t) is defined by linear interpolation

R(n)t =

1radicnσ2

983063Rlfloorntrfloor + (ntminus lfloorntrfloor)

983043Rlceilntrceil minusRlfloorntrfloor

983044 983064

This method of scaling operates on two aspects of random walks as n increasesthe time between state transitions decreases and increment lengths decrease Lett0 = 0 and tk = inftkt isin Z nt gt nt(kminus1) then as n rarr infin

tk minus t(kminus1) rarr 0 and R(n)tk

minusR(n)t(kminus1)

rarr 0

As a result R(n)(t) is continuous with probability 1 on t ge 0 for all n The scalingfactor 1radic

nσ2ensures constant variance for a given t

Var

9830691radicnσ2

Rnt

983070= Var

9830831radicnσ2

nt983131

i=1

Xi

983084=

9830611radicnσ2

9830622

ntVarXi = tVar Xi

σ2= t

By observing the limiting behavior of R(n)t (ie as n rarr infin) we establish a con-

struction of Brownian motion

12 HARVEY BARNHARD

332 Skorokhod EmbeddingTo construct Brownian motion out of random walks we must first construct randomwalks out of Brownian motion Structuring the proof in this matter is not withoutreason there are uncountably many realizations of Brownian motion so we want toensure that the random walks to which we are taking the limit do in fact convergeto the Brownian motion of interest Such a procedure of deriving a random walkfrom a Brownian motion is called Skorokhod embedding The procedure is given inthe following pseudocode

Set t0 = 0 and n = 0 Let T be a stopping time of Brownian motion Btso that Bt is defined on [0 T ] Let R0 = 0 be the initial value

of the Skorokhod embedding

While tn + t le TInitialize t = 0

While |Bt minusBtn | le 1 Continue Bt until |Bt minusBtn | = 1 for the first time

Set t(n+1) = inft |Bt minusBtn | = 1If Bt(n+1)

minusBtn = 1 then set X(n+1) = 1

Otherwise Bt(n+1)minusBtn = minus1 so set X(n+1) = minus1

Let the (n+ 1)th value of the Skorokhod embedding be

R(n+1) = Rn +Xn+1

Increment to the next step of the inner while loop n = n+ 1

The resulting sequence Rnn=12 is the Skorokhod embedding of Bttisin[0T ]

Theorem 36 Let Bt be a standard Brownian motion Set t0 = 0 and let tn bethe stopping time where |Bt| = 1 for the nth time

tn equiv inft ge 0 |Bt minusBtn | = 1

If we define Rn equiv Btn then Rn is a simple random walk

ProofTo prove that B0 Bt1 Bt2 is a simple random walk we merely need to showthat

(1) PBtn+1minusBtn = 1 = PBtn+1

minusBtn = minus1 = 12 for all n isin N

(2) tn+1 minus tn are iid random variables

The first item follows from the symmetric property of Brownian increments prop-erty (II) in the definition of Brownian motion Bs+t minusBs sim N(0σ2t)

The second item follows from the strong Markov property of Brownian motionwhich states that if τ is a stopping time with respect to Bt then Bt minus Bτ is

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 13

also a Brownian motion We do not include a rigorous treatment of stopping timesor a proof of the strong Markov property

We now state a theorem that places a probabilistic bound on the differencebetween a Brownian motion and its Skorokhod embedding

Theorem 37 Define Θ to be the maximum distance between a Brownian motionand its Skorokhod embedding

Θ(BRT ) = max0letleT

|Bt minusRt|

Note that Θ is equivalent to the distance between Bt and Rt in the space ofcontinuous functions on the time interval [0 T ] equipped with the supremum normThere exist c a isin [0infin) such that for all r le n14 and all integers n ge 3

P983153Θ(BRT ) ge rn14

983155log n

983154le ceminusar

ProofIt will suffice to prove the theorem for r ge 9c2 where c is the constant such that

P983153osc(B δ T ) gt r

983155δ log(1δ)

983154le cT δ(rc)

2

andosc(B δ T ) = sup |Bt minusBs| s t isin D s t isin [0 T ] |sminus t| le δ

is the oscillation of Bt restricted to t isin D the dyadic rationals The proof thatsuch a c exists can be found in [3] page 68 where it is also shown that for n isin N

Θ(BSn) le 1 + osc(B 1 n) + max|Bj minusBτj | j = 1 nNow suppose 9c2 le r le n14 If |Bn minusBτn | is large then either |nminus τn| is large

or the oscillation of B is large Consider the three events

(1)983051Θ(BRn) ge rn14

radiclog n

983052

(2)983051osc(B r

radicn 2n) ge (r3)n14

radiclog n

983052

(3) max1lejlen |τj minus j| ge rradicn

From above we know that event (1) is contained in the union of events (2) and (3)As a consequence we need only prove the result of the theorem for events (2) and(3) We first tackle event (2)

P983153osc(B r

radicn 2n) gt (r3)n(14)

983155log n

983154le 3P

983153osc(B r

radicn n) gt (r3)n(14)

983155log n

983154

= 3P983153osc(B rnminus12) gt (r3)nminus(14)

983155log n

983154

le 3P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log

983043n12r

983044983070

Ifradicr3 ge c and r le n14 we can conclude that there exist c and a such that

P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log(n12)

983070le ceminusar logn

For event (3) we refer to a proof on page 266 in the appendix of [3] to show thatthere exist c a such that

P983069

max1lejlen

|τj minus j| gt rradicn

983070le ceminusar2

14 HARVEY BARNHARD

333 Strong ApproximationWith preliminaries in order we now show that Brownian motion can be thought ofas a limit of simple random walks a construction known as the strong approximationof Brownian motion Let Bt be a standard Brownian motion with variance

parameter 1 as defined in the probability space (ΩF P) and let B(n)t be the

scaled Brownian motion

B(n)t =

1radicnBnt

From B(n) we use the Skorokhod embedding to derive the simple random walkS(n) And from S(n) we derive the scaled simple random walk R(n)

R(n)t =

1radicnS(n)nt

From Theorem 37 we know that there exists c a isin [0infin) such that for all positiveintegers T

P983069

max0letleTn

983055983055983055S(n)t minusB

(n)t

983055983055983055 ge cr(Tn)14983155log (Tn)

983070le ceminusar

multiplying by 1radicn this becomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge crT 14nminus14983155log (Tn)

983070le ceminusar

Letting r = c log n where c is sufficiently large for the given T the inequalitybecomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

983070le c

n2

To proceed we require the use of the Borel-Cantelli lemma the result of which isprovided below

Lemma 38 (Borel-Cantelli)If An is a sequence of events in Ω and

983123infini=1 PAn lt infin then

PAn io = 0

where An io = lim supnrarrinfin An is the event that infinitely many An occur

If we set

An = max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

we can then apply the Borel-Cantelli lemma to the above inequality to concludethat

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 le cnminus14 log32 n

with probability one for all n sufficiently large Since

limnrarrinfin

cnminus14 log32 n = 0

we conclude that

R(n)t

asminusminusrarr Bt in C[0 T ]The following theorem condenses the results of this section up to this point

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 15

Theorem 39 (Donskerrsquos Theorem) Let Xi be a sequence of iid random vari-

ables with mean 0 and variance σ2 If R(n)t is a scaled random walk as defined

above then R(n)t rArr Bt where Bt is a standard Brownian motion This result is

also known as the functional central limit theorem

34 Brownian Motion VariantsThe preceding construction of Brownian motion will prove particularly useful in ourapproximation of heavy traffic waiting times However this process requires furthermodification Recall that when traffic intensity ρ is slightly less than 1 customersare being serviced at a slightly faster rate then they are entering the queue Thusgiven enough time traffic will eventually ldquoclear outrdquo due to faster service timesHowever this particular construction of Brownian motion has increments of meanzero

E[Bs+t minusBs] = 0

meaning that the approximate number of customers in line wouldmdashon averagemdashremain unchanged regardless of how long the line is observed Such a propertyis called the martingale property Instead we want a process that exhibits thesupermartingale property

E[Bs+t minusBs] lt 0

Definition 310 Let Fn be an increasing sequence of σ-fields (Fn is called afiltration) Let Xn be a sequence of random variables with finite mean andXn isin Fn for all n Martingales supermartingales and submartingales are thendefined as follows for all n

Martingale E [Xn+1 | Fn] = Xn

Supermartingale E [Xn+1 | Fn] le Xn

Submartingale E [Xn+1 | Fn] ge Xn

In words a sequence that we expect to remain the same over time is a martingalea sequence we expect to increase is a submartingale and a sequence we expect todecrease is a supermartingale To approximate heavy traffic we want a Brownianmotion-like process that exhibits the supermartingale property

Definition 311 A Brownian motion with drift parameter α and variance param-eter σ2 Bα

t is defined by the following properties

(1) Independent increments(2) Increment lengths are normally distributed with mean αt and variance σ2t(3) the path t 983041rarr Bα

t is continuous

Note that these properties are the same as standard Brownian motion exceptincrement lengths now have non-zero means Since we have already shown theexistence of standard Brownian motion the construction of Brownian motion withdrift is now easy

Lemma 312 Let Bt be a standard Brownian motion with variance parameterσ2 then Bα

t = Bt + αt is Brownian motion with drift parameter α and varianceparameter σ2 Moreover Bα

t fulfills the supermartingale property when α lt 0

16 HARVEY BARNHARD

ProofIndependence of increments follows directly from independence of increments forBt and the fact that αt is a constant for a given t To prove that Bα

t has stationaryincrements we have to show that the distribution of increments Bα

s+tminusBαs depends

only the time interval t and not the absolute time Recall from the definition of Bt

that Bs+t minusBt are normal random variables with mean 0 and variance t for s gt 0

E983045Bα

s+t minusBαs

983046= E

983045Bα

s+t

983046minus E [Bα

s ]

= E983045Bα

s+t + α(s+ t)983046minus E [Bs + αs]

= α(s+ t)minus αs

= αt

Var983051Bα

s+t minusBαs

983052= Var Bs+t minusBs + αt= Var Bs+t minusBs= σ2t

And since for fixed t αt is just a constant we use the property of normal randomvariables to conclude that

Bαs+t minusBα

s sim N(αtσ2t) for all s gt 0

Continuity of Bαt directly follows from continuity of Bt We have shown that

Bαt constructed as the sum of standard Brownian motion and drift αt is indeed

Brownian motion with drift

Brownian motion with negative drift provides a better approximation of thewaiting-time of customers in a queue than does standard Brownian motion How-ever the unboundedness of Brownian motion suggests that waiting times couldperhaps be negative In order to circumvent this issue we instead use reflectedBrownian motion with drift Reflected Brownian motion acts much the same asBrownian motion except on a given boundary off of which the Brownian motionismdashunsurprisinglymdashreflected In the one-dimensional case the boundaries may beany interval on the reals that provide a lower and upper bound to the Brownianmotion For our purposes we only use a single boundary zero that gives a lowerbound to Brownian motion That is for all t ge 0 we want Bα

t ge 0

Definition 313 Let Bαt be a standard Brownian motion with drift parameter α

and let Mαt be the running maximum of Bα

t

Mαt = sup

0lesletBα

s

Then reflected Brownian motion with drift parameter minusα and boundary 0 definedon R+ can be constructed as follows

Rα0t equiv max0Mα

t minusBαt

Note that max0 Sαt ge 0 and Mα

t ge Bαt B

α0t ge 0 for all t Most importantly

the local behavior of Bα0t is exactly like Brownian motion with drift parameter minusα

since max0 Sαt is simply a constant when Bα

t ∕= 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 17

We have now provided sufficient modifications of Brownian motion and we nextmove onmdashwhat we will ultimately use to approximate heavy traffic waiting timesThe next example provides us with the distribution of the supremum of a Brownianmotion with negative drift

Example 314Let Bt be a Brownian motion with drift α lt 0 and variance parameter σ2 Define

Mt = sup0leslet

Bs

to be the running maximum then

Minfin sim exp

9830612|α|σ2

983062

that is

PW ge w = eminusminus2|α|σ2 w w ge 0

E [Minfin] =σ2

2|α|

Prooffor constants a b gt 0 Let T (minusa b) be the first time that B minus t hits minusa or b

T (minusa b) = inft Bt = minusa or Bt = bIn [7] it is proved that

P983051BT (minusab)=b

983052=

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)

Since α lt 0 exp(2αaσ2) rarr 0 we have

limararrinfin

PBT (minusab) = b = limararrinfin

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)= exp(2αbσ2)

The left-hand side becomes the probability that the process will reach b somewherealong its path (ie that the maximum of the process exceeds b somewhere alongits path) Therefore we have

PW ge b = exp(2αbσ2) = expminus2|α|bσ2

This example shows us that the supremum of a Brownian motion observed ldquofor-everrdquo (ie on the interval [0infin)) is exponentially distributed with rate 2|α|σ2

4 Heavy Traffic Approximation

We now have all we need to provide a heuristic argument for the approximationof single-servers queue in heavy traffic using Brownian Motion For a rigorousalbeit opaque treatment of this approximation see [8] the seminal work on thissubject by Kingman

Theorem 41 The waiting time of the nth customer in an MM1 queue canbe approximated by the supremum of reflected Brownian motion with negative driftand boundary at 0

18 HARVEY BARNHARD

ProofRecall that an MM1 queue is a process where the number of customer arrivalsfollows a poisson process with rate λ and service times are exponentially distributedwith rate micro From Lindleyrsquos equation in Section 2 we can write the waiting timefor the nth customer as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

where U (n) = S(n)minus I(n) is the difference between the nth service time and the nthinter-arrival time Define the partial sum P

(n)k to be zero when k = 0 and

P(n)k equiv

k983131

i=1

U (nminusi) for k ge 1

We can then rewrite the waiting time of the nth customer as

W (n)q = max

0lekle(nminus1)P

(n)k

The fact that P(n)k is the kth value of a random walk follows immediately from

U (1) U (nminus1) being iid random variables Let α and σ2 be the expectationand variance of U (i) We calculate α and σ2 by first noting that S(i) and I(i) areexponentially distributed with rates micro and λ Recall that ρ = λmicro is the trafficintensity as defined in the first section of this paper

α = E983147U (i)

983148= E

983147S(i)

983148minus E

983147I(i)

983148=

1

microminus 1

λ=

ρminus 1

λ

σ2 = Var983153U (i)

983154= Var

983153S(i)

983154+Var

983153I(i)

983154=

1

micro2+

1

λ2

We next compute the expectation and variance of P(n)k using the fact that U (i) are

iid random variables

E983147P

(n)k

983148=

k983131

i=1

E983147U (nminusi)

983148= kα

Var983153P

(n)k

983154=

k983131

i=1

Var983153U (nminusi)

983154= kσ2

For a finite n W(n)q is therefore the maximum value of a random walk with n steps

We now show that Pnk can be approximated by Brownian motion with drift

parameter α and variance parameter σ2 when n is sufficiently large We restate theproperties of such a Brownian motion Bt below

(1) B0 = 0(2) Bt has stationary and independent increments(3) Bt sim N(αtσ2t)(4) t 983041rarr Bt is continuous with probability one

The first property is clear as P(n)0 = 0 for all n The second property follows from

the fact that U (i) are iid random variables The third property is a result of thecentral limit theorem for iid sequences the result of which is provided below

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

8 HARVEY BARNHARD

Proof

We prove the lemma by iterating over W(n)q

W (n)q = max

983153W (nminus1)

q + U (nminus1) 0983154

= max983153max

983153W (nminus2)

q + U (nminus2) 0983154+ U (nminus1) 0

983154

= max983153max

983153W (nminus2)

q + U (nminus2) + U (nminus1) U (nminus1)983154 0983154

= max983153W (nminus2)

q + U (nminus2) + U (nminus1) U (nminus1) 0983154

Continue this process in a recursive manner until we have

W (n)q = max

983153W (1)

q + U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1)983154

The result of the lemma is now immediate since we have assumed thatW(1)q = 0

The goal of this paper is to approximate W(n)q the waiting time of the nth

customer From the preceding lemma it is clear that the waiting time depends onthe waiting times of all preceding customers in line More precisely the waitingtime of the nth customer is the maximum of the partial sums of the decreasingsequence

983051U (nminusi)

983052n

i=1 an iid sequence of random variables Section 3 will show

that this decreasing sequence is a random walk and when n is sufficiently large thissequence can be approximated by Brownian motion For such an approximation tobe accurate however each U (nminusi) must be sufficiently small small This situationarises in queues that exhibit heavy traffic

Definition 213 The traffic intensity ρ of a given queue is defined as the ratioof arrival times to service times So for an MMc queue the traffic intensity is

ρ =λ

cmicro

where λ and micro are the arrival and service rate respectively of customers in thequeue For a single server MM1 queue the traffic intensity is

ρ =λ

micro

Remark 214 When ρ is close to zero waiting times approach zero as queue lengthshortens and new customers are serviced immediately upon arrival When ρ isgreater than one waiting times successively increase as the queue population ldquoex-plodesrdquo to infinity The interesting case occurs when ρ is close tomdashbut does notexceedmdashone A system is said to be in heavy traffic when ρ isin (1minus ε 1) and ε gt 0is small

3 Brownian Motion

This section introduces the stochastic process of Brownian motion viewed as thelimit of random walks In this paper we confine our study of Brownian motion tothe one-dimensional case

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 9

31 Random Walks

Definition 31 A one-dimensional random walk is a stochastic process constructedas the sum of iid random variables That is if Xi is a sequence of iid randomvariables with Xi isin R (regarded as steps) then the sum of the first n such randomvariables

Rn =

n983131

i=1

Xi

is the n-th value of a random walk A random walk is said to be symmetric ifXi isin R is symmetrically distributed about zero A random walk is called simple ifPXi = 1+ PXi = minus1 = 1

Example 32 Figure 3 depicts a simple random walk simulated using the statisti-cal computing software R with PXi = minus1 = PXi = 1 = 1

2 and 15 total stepsTaking R0 = X0 = 0 the values of Xi and Ri are

Xi15i=1 = 1minus1 1minus1minus1minus1minus1 1minus1minus1 1 1minus1 1minus1Ri15i=1 = 0 1 0 1 0minus1minus2minus3minus2minus3minus4minus3minus2minus3minus2minus3

0 5 10 15

minus4minus3

minus2minus1

01

n

S n

Figure 3

An important property of random walks is that they exhibit independent incre-ments meaning that for any selection of positive integers t1 lt t2 lt middot middot middot lt tn therandom variables

Rt2 minusRt1 Rtn minusRtnminus1

are independent This property immediately follow from the fact that each incre-ment is the sum of iid random variables

10 HARVEY BARNHARD

32 Brownian Motion DefinitionWe now introduce the continuous-time analog to a random walkmdashBrownian motionA one-dimensional standard Brownian motion with variance parameter σ2 is a real-valued process Bttge0 defined in the probability space (ΩF P) that has thefollowing properties

(I) If t0 lt t1 lt middot middot middot lt tn then Bt0 Bt1 minusBt0 Btn minusBtnminus1are independent

(II) If s t ge 0 then

P (Bs+t minusBs isin A) =

983133

A

1radic2πσ2t

exp(minusx22σ2t)

(III) With probability 1 t 983041rarr Bt is continuous

The first two conditions may be concisely reworded as

(I) Independent increments(II) Increment lengths are normally distributed with mean 0 and variance σ2t

Bs+t minusBs sim N(0σ2t)

Remark 33Brownian motion is a collection of stochastic processes When we say a Brownianmotion or the Brownian motion we are talking about a particular realization ofBrownian motion in the space of continuous functions C[0 T ] where T is a stoppingtime

The above definition establishes the necessary conditions for a stochastic process tobe Brownian but the existence of such a process requires further work Throughan application of the Kolmogorov extension theorem existence is proved in [4]However such a proof is bereft of intuition for queueing theory and approximatingwaiting times Instead we take the existence of Brownian motion as given andshow that Brownian motion can be constructed as a limit of random walksmdashtheresult of Donskerrsquos Theorem Brownian motion constructed in such a manner iscalled the standard Wiener process

Before constructing Brownian motion we must first establish what type of con-vergence we are discussing

Definition 34 A sequence of random variables Xiinfini=1 converges almost surelyto a random variable X in the underlying probability space (ΩF P) (denoted

Xnasminusminusrarr X) if

P983153ω isin Ω lim

nrarrinfinXn(ω) = X(ω)

983154= 1

We aim to show almost sure convergence of a scaled random walk to a Brownianmotion in (C[0 T ] d) the metric space of continuous functions on the interval [0 T ]with d as the supremum norm

d(f g) = suptisin[0T ]

|f(t)minus g(t)|

Note that random walks and Brownian motions are stochastic processes while dmeasures the distance between two continuous functions In order to measure thedistance between random walks and Brownian motion we instead consider the pathstaken by these stochastic processes

t 983041rarr Rt and t 983041rarr Bt

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 11

33 Brownian Motion Construction In this paper the construction of Brow-nian motion comprises the following steps

(1) Construct a simple random walk out of a Brownian motion path usingSkorokhod embedding

(2) Linearly interpolate and scale the embedded random walk creating a con-tinuous function from [0 n] 983041rarr R

(3) Take the uniform limit of this scaled random walk to derive the originalBrownian motion path

The paper presents an alternative formulation to the canonical construction ofBrownian motion which involves defining Brownian motion on the dyadic rationalsand using continuity to extend the definition to the reals To motivate this particu-lar construction we will proceed out of order we will begin with step (2) backtrackto step (1) and conclude by stringing together steps (1) through (3) in subsection333 (Strong Approximation)

331 Scaled Random WalksLet Rm be a random walk where E[Xi] = 0 VarXi = σ2 and

Rm =

m983131

i=1

Xi

Definition 35 A scaled random walk with scaling parameter n is a real valuedrandom process

983051R(n)

983052 defined on t ge 0 For t such that nt is an integer

R(n)t =

1radicnσ2

Rnt

For t such that nt is not an integer S(n)(t) is defined by linear interpolation

R(n)t =

1radicnσ2

983063Rlfloorntrfloor + (ntminus lfloorntrfloor)

983043Rlceilntrceil minusRlfloorntrfloor

983044 983064

This method of scaling operates on two aspects of random walks as n increasesthe time between state transitions decreases and increment lengths decrease Lett0 = 0 and tk = inftkt isin Z nt gt nt(kminus1) then as n rarr infin

tk minus t(kminus1) rarr 0 and R(n)tk

minusR(n)t(kminus1)

rarr 0

As a result R(n)(t) is continuous with probability 1 on t ge 0 for all n The scalingfactor 1radic

nσ2ensures constant variance for a given t

Var

9830691radicnσ2

Rnt

983070= Var

9830831radicnσ2

nt983131

i=1

Xi

983084=

9830611radicnσ2

9830622

ntVarXi = tVar Xi

σ2= t

By observing the limiting behavior of R(n)t (ie as n rarr infin) we establish a con-

struction of Brownian motion

12 HARVEY BARNHARD

332 Skorokhod EmbeddingTo construct Brownian motion out of random walks we must first construct randomwalks out of Brownian motion Structuring the proof in this matter is not withoutreason there are uncountably many realizations of Brownian motion so we want toensure that the random walks to which we are taking the limit do in fact convergeto the Brownian motion of interest Such a procedure of deriving a random walkfrom a Brownian motion is called Skorokhod embedding The procedure is given inthe following pseudocode

Set t0 = 0 and n = 0 Let T be a stopping time of Brownian motion Btso that Bt is defined on [0 T ] Let R0 = 0 be the initial value

of the Skorokhod embedding

While tn + t le TInitialize t = 0

While |Bt minusBtn | le 1 Continue Bt until |Bt minusBtn | = 1 for the first time

Set t(n+1) = inft |Bt minusBtn | = 1If Bt(n+1)

minusBtn = 1 then set X(n+1) = 1

Otherwise Bt(n+1)minusBtn = minus1 so set X(n+1) = minus1

Let the (n+ 1)th value of the Skorokhod embedding be

R(n+1) = Rn +Xn+1

Increment to the next step of the inner while loop n = n+ 1

The resulting sequence Rnn=12 is the Skorokhod embedding of Bttisin[0T ]

Theorem 36 Let Bt be a standard Brownian motion Set t0 = 0 and let tn bethe stopping time where |Bt| = 1 for the nth time

tn equiv inft ge 0 |Bt minusBtn | = 1

If we define Rn equiv Btn then Rn is a simple random walk

ProofTo prove that B0 Bt1 Bt2 is a simple random walk we merely need to showthat

(1) PBtn+1minusBtn = 1 = PBtn+1

minusBtn = minus1 = 12 for all n isin N

(2) tn+1 minus tn are iid random variables

The first item follows from the symmetric property of Brownian increments prop-erty (II) in the definition of Brownian motion Bs+t minusBs sim N(0σ2t)

The second item follows from the strong Markov property of Brownian motionwhich states that if τ is a stopping time with respect to Bt then Bt minus Bτ is

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 13

also a Brownian motion We do not include a rigorous treatment of stopping timesor a proof of the strong Markov property

We now state a theorem that places a probabilistic bound on the differencebetween a Brownian motion and its Skorokhod embedding

Theorem 37 Define Θ to be the maximum distance between a Brownian motionand its Skorokhod embedding

Θ(BRT ) = max0letleT

|Bt minusRt|

Note that Θ is equivalent to the distance between Bt and Rt in the space ofcontinuous functions on the time interval [0 T ] equipped with the supremum normThere exist c a isin [0infin) such that for all r le n14 and all integers n ge 3

P983153Θ(BRT ) ge rn14

983155log n

983154le ceminusar

ProofIt will suffice to prove the theorem for r ge 9c2 where c is the constant such that

P983153osc(B δ T ) gt r

983155δ log(1δ)

983154le cT δ(rc)

2

andosc(B δ T ) = sup |Bt minusBs| s t isin D s t isin [0 T ] |sminus t| le δ

is the oscillation of Bt restricted to t isin D the dyadic rationals The proof thatsuch a c exists can be found in [3] page 68 where it is also shown that for n isin N

Θ(BSn) le 1 + osc(B 1 n) + max|Bj minusBτj | j = 1 nNow suppose 9c2 le r le n14 If |Bn minusBτn | is large then either |nminus τn| is large

or the oscillation of B is large Consider the three events

(1)983051Θ(BRn) ge rn14

radiclog n

983052

(2)983051osc(B r

radicn 2n) ge (r3)n14

radiclog n

983052

(3) max1lejlen |τj minus j| ge rradicn

From above we know that event (1) is contained in the union of events (2) and (3)As a consequence we need only prove the result of the theorem for events (2) and(3) We first tackle event (2)

P983153osc(B r

radicn 2n) gt (r3)n(14)

983155log n

983154le 3P

983153osc(B r

radicn n) gt (r3)n(14)

983155log n

983154

= 3P983153osc(B rnminus12) gt (r3)nminus(14)

983155log n

983154

le 3P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log

983043n12r

983044983070

Ifradicr3 ge c and r le n14 we can conclude that there exist c and a such that

P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log(n12)

983070le ceminusar logn

For event (3) we refer to a proof on page 266 in the appendix of [3] to show thatthere exist c a such that

P983069

max1lejlen

|τj minus j| gt rradicn

983070le ceminusar2

14 HARVEY BARNHARD

333 Strong ApproximationWith preliminaries in order we now show that Brownian motion can be thought ofas a limit of simple random walks a construction known as the strong approximationof Brownian motion Let Bt be a standard Brownian motion with variance

parameter 1 as defined in the probability space (ΩF P) and let B(n)t be the

scaled Brownian motion

B(n)t =

1radicnBnt

From B(n) we use the Skorokhod embedding to derive the simple random walkS(n) And from S(n) we derive the scaled simple random walk R(n)

R(n)t =

1radicnS(n)nt

From Theorem 37 we know that there exists c a isin [0infin) such that for all positiveintegers T

P983069

max0letleTn

983055983055983055S(n)t minusB

(n)t

983055983055983055 ge cr(Tn)14983155log (Tn)

983070le ceminusar

multiplying by 1radicn this becomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge crT 14nminus14983155log (Tn)

983070le ceminusar

Letting r = c log n where c is sufficiently large for the given T the inequalitybecomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

983070le c

n2

To proceed we require the use of the Borel-Cantelli lemma the result of which isprovided below

Lemma 38 (Borel-Cantelli)If An is a sequence of events in Ω and

983123infini=1 PAn lt infin then

PAn io = 0

where An io = lim supnrarrinfin An is the event that infinitely many An occur

If we set

An = max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

we can then apply the Borel-Cantelli lemma to the above inequality to concludethat

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 le cnminus14 log32 n

with probability one for all n sufficiently large Since

limnrarrinfin

cnminus14 log32 n = 0

we conclude that

R(n)t

asminusminusrarr Bt in C[0 T ]The following theorem condenses the results of this section up to this point

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 15

Theorem 39 (Donskerrsquos Theorem) Let Xi be a sequence of iid random vari-

ables with mean 0 and variance σ2 If R(n)t is a scaled random walk as defined

above then R(n)t rArr Bt where Bt is a standard Brownian motion This result is

also known as the functional central limit theorem

34 Brownian Motion VariantsThe preceding construction of Brownian motion will prove particularly useful in ourapproximation of heavy traffic waiting times However this process requires furthermodification Recall that when traffic intensity ρ is slightly less than 1 customersare being serviced at a slightly faster rate then they are entering the queue Thusgiven enough time traffic will eventually ldquoclear outrdquo due to faster service timesHowever this particular construction of Brownian motion has increments of meanzero

E[Bs+t minusBs] = 0

meaning that the approximate number of customers in line wouldmdashon averagemdashremain unchanged regardless of how long the line is observed Such a propertyis called the martingale property Instead we want a process that exhibits thesupermartingale property

E[Bs+t minusBs] lt 0

Definition 310 Let Fn be an increasing sequence of σ-fields (Fn is called afiltration) Let Xn be a sequence of random variables with finite mean andXn isin Fn for all n Martingales supermartingales and submartingales are thendefined as follows for all n

Martingale E [Xn+1 | Fn] = Xn

Supermartingale E [Xn+1 | Fn] le Xn

Submartingale E [Xn+1 | Fn] ge Xn

In words a sequence that we expect to remain the same over time is a martingalea sequence we expect to increase is a submartingale and a sequence we expect todecrease is a supermartingale To approximate heavy traffic we want a Brownianmotion-like process that exhibits the supermartingale property

Definition 311 A Brownian motion with drift parameter α and variance param-eter σ2 Bα

t is defined by the following properties

(1) Independent increments(2) Increment lengths are normally distributed with mean αt and variance σ2t(3) the path t 983041rarr Bα

t is continuous

Note that these properties are the same as standard Brownian motion exceptincrement lengths now have non-zero means Since we have already shown theexistence of standard Brownian motion the construction of Brownian motion withdrift is now easy

Lemma 312 Let Bt be a standard Brownian motion with variance parameterσ2 then Bα

t = Bt + αt is Brownian motion with drift parameter α and varianceparameter σ2 Moreover Bα

t fulfills the supermartingale property when α lt 0

16 HARVEY BARNHARD

ProofIndependence of increments follows directly from independence of increments forBt and the fact that αt is a constant for a given t To prove that Bα

t has stationaryincrements we have to show that the distribution of increments Bα

s+tminusBαs depends

only the time interval t and not the absolute time Recall from the definition of Bt

that Bs+t minusBt are normal random variables with mean 0 and variance t for s gt 0

E983045Bα

s+t minusBαs

983046= E

983045Bα

s+t

983046minus E [Bα

s ]

= E983045Bα

s+t + α(s+ t)983046minus E [Bs + αs]

= α(s+ t)minus αs

= αt

Var983051Bα

s+t minusBαs

983052= Var Bs+t minusBs + αt= Var Bs+t minusBs= σ2t

And since for fixed t αt is just a constant we use the property of normal randomvariables to conclude that

Bαs+t minusBα

s sim N(αtσ2t) for all s gt 0

Continuity of Bαt directly follows from continuity of Bt We have shown that

Bαt constructed as the sum of standard Brownian motion and drift αt is indeed

Brownian motion with drift

Brownian motion with negative drift provides a better approximation of thewaiting-time of customers in a queue than does standard Brownian motion How-ever the unboundedness of Brownian motion suggests that waiting times couldperhaps be negative In order to circumvent this issue we instead use reflectedBrownian motion with drift Reflected Brownian motion acts much the same asBrownian motion except on a given boundary off of which the Brownian motionismdashunsurprisinglymdashreflected In the one-dimensional case the boundaries may beany interval on the reals that provide a lower and upper bound to the Brownianmotion For our purposes we only use a single boundary zero that gives a lowerbound to Brownian motion That is for all t ge 0 we want Bα

t ge 0

Definition 313 Let Bαt be a standard Brownian motion with drift parameter α

and let Mαt be the running maximum of Bα

t

Mαt = sup

0lesletBα

s

Then reflected Brownian motion with drift parameter minusα and boundary 0 definedon R+ can be constructed as follows

Rα0t equiv max0Mα

t minusBαt

Note that max0 Sαt ge 0 and Mα

t ge Bαt B

α0t ge 0 for all t Most importantly

the local behavior of Bα0t is exactly like Brownian motion with drift parameter minusα

since max0 Sαt is simply a constant when Bα

t ∕= 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 17

We have now provided sufficient modifications of Brownian motion and we nextmove onmdashwhat we will ultimately use to approximate heavy traffic waiting timesThe next example provides us with the distribution of the supremum of a Brownianmotion with negative drift

Example 314Let Bt be a Brownian motion with drift α lt 0 and variance parameter σ2 Define

Mt = sup0leslet

Bs

to be the running maximum then

Minfin sim exp

9830612|α|σ2

983062

that is

PW ge w = eminusminus2|α|σ2 w w ge 0

E [Minfin] =σ2

2|α|

Prooffor constants a b gt 0 Let T (minusa b) be the first time that B minus t hits minusa or b

T (minusa b) = inft Bt = minusa or Bt = bIn [7] it is proved that

P983051BT (minusab)=b

983052=

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)

Since α lt 0 exp(2αaσ2) rarr 0 we have

limararrinfin

PBT (minusab) = b = limararrinfin

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)= exp(2αbσ2)

The left-hand side becomes the probability that the process will reach b somewherealong its path (ie that the maximum of the process exceeds b somewhere alongits path) Therefore we have

PW ge b = exp(2αbσ2) = expminus2|α|bσ2

This example shows us that the supremum of a Brownian motion observed ldquofor-everrdquo (ie on the interval [0infin)) is exponentially distributed with rate 2|α|σ2

4 Heavy Traffic Approximation

We now have all we need to provide a heuristic argument for the approximationof single-servers queue in heavy traffic using Brownian Motion For a rigorousalbeit opaque treatment of this approximation see [8] the seminal work on thissubject by Kingman

Theorem 41 The waiting time of the nth customer in an MM1 queue canbe approximated by the supremum of reflected Brownian motion with negative driftand boundary at 0

18 HARVEY BARNHARD

ProofRecall that an MM1 queue is a process where the number of customer arrivalsfollows a poisson process with rate λ and service times are exponentially distributedwith rate micro From Lindleyrsquos equation in Section 2 we can write the waiting timefor the nth customer as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

where U (n) = S(n)minus I(n) is the difference between the nth service time and the nthinter-arrival time Define the partial sum P

(n)k to be zero when k = 0 and

P(n)k equiv

k983131

i=1

U (nminusi) for k ge 1

We can then rewrite the waiting time of the nth customer as

W (n)q = max

0lekle(nminus1)P

(n)k

The fact that P(n)k is the kth value of a random walk follows immediately from

U (1) U (nminus1) being iid random variables Let α and σ2 be the expectationand variance of U (i) We calculate α and σ2 by first noting that S(i) and I(i) areexponentially distributed with rates micro and λ Recall that ρ = λmicro is the trafficintensity as defined in the first section of this paper

α = E983147U (i)

983148= E

983147S(i)

983148minus E

983147I(i)

983148=

1

microminus 1

λ=

ρminus 1

λ

σ2 = Var983153U (i)

983154= Var

983153S(i)

983154+Var

983153I(i)

983154=

1

micro2+

1

λ2

We next compute the expectation and variance of P(n)k using the fact that U (i) are

iid random variables

E983147P

(n)k

983148=

k983131

i=1

E983147U (nminusi)

983148= kα

Var983153P

(n)k

983154=

k983131

i=1

Var983153U (nminusi)

983154= kσ2

For a finite n W(n)q is therefore the maximum value of a random walk with n steps

We now show that Pnk can be approximated by Brownian motion with drift

parameter α and variance parameter σ2 when n is sufficiently large We restate theproperties of such a Brownian motion Bt below

(1) B0 = 0(2) Bt has stationary and independent increments(3) Bt sim N(αtσ2t)(4) t 983041rarr Bt is continuous with probability one

The first property is clear as P(n)0 = 0 for all n The second property follows from

the fact that U (i) are iid random variables The third property is a result of thecentral limit theorem for iid sequences the result of which is provided below

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 9

31 Random Walks

Definition 31 A one-dimensional random walk is a stochastic process constructedas the sum of iid random variables That is if Xi is a sequence of iid randomvariables with Xi isin R (regarded as steps) then the sum of the first n such randomvariables

Rn =

n983131

i=1

Xi

is the n-th value of a random walk A random walk is said to be symmetric ifXi isin R is symmetrically distributed about zero A random walk is called simple ifPXi = 1+ PXi = minus1 = 1

Example 32 Figure 3 depicts a simple random walk simulated using the statisti-cal computing software R with PXi = minus1 = PXi = 1 = 1

2 and 15 total stepsTaking R0 = X0 = 0 the values of Xi and Ri are

Xi15i=1 = 1minus1 1minus1minus1minus1minus1 1minus1minus1 1 1minus1 1minus1Ri15i=1 = 0 1 0 1 0minus1minus2minus3minus2minus3minus4minus3minus2minus3minus2minus3

0 5 10 15

minus4minus3

minus2minus1

01

n

S n

Figure 3

An important property of random walks is that they exhibit independent incre-ments meaning that for any selection of positive integers t1 lt t2 lt middot middot middot lt tn therandom variables

Rt2 minusRt1 Rtn minusRtnminus1

are independent This property immediately follow from the fact that each incre-ment is the sum of iid random variables

10 HARVEY BARNHARD

32 Brownian Motion DefinitionWe now introduce the continuous-time analog to a random walkmdashBrownian motionA one-dimensional standard Brownian motion with variance parameter σ2 is a real-valued process Bttge0 defined in the probability space (ΩF P) that has thefollowing properties

(I) If t0 lt t1 lt middot middot middot lt tn then Bt0 Bt1 minusBt0 Btn minusBtnminus1are independent

(II) If s t ge 0 then

P (Bs+t minusBs isin A) =

983133

A

1radic2πσ2t

exp(minusx22σ2t)

(III) With probability 1 t 983041rarr Bt is continuous

The first two conditions may be concisely reworded as

(I) Independent increments(II) Increment lengths are normally distributed with mean 0 and variance σ2t

Bs+t minusBs sim N(0σ2t)

Remark 33Brownian motion is a collection of stochastic processes When we say a Brownianmotion or the Brownian motion we are talking about a particular realization ofBrownian motion in the space of continuous functions C[0 T ] where T is a stoppingtime

The above definition establishes the necessary conditions for a stochastic process tobe Brownian but the existence of such a process requires further work Throughan application of the Kolmogorov extension theorem existence is proved in [4]However such a proof is bereft of intuition for queueing theory and approximatingwaiting times Instead we take the existence of Brownian motion as given andshow that Brownian motion can be constructed as a limit of random walksmdashtheresult of Donskerrsquos Theorem Brownian motion constructed in such a manner iscalled the standard Wiener process

Before constructing Brownian motion we must first establish what type of con-vergence we are discussing

Definition 34 A sequence of random variables Xiinfini=1 converges almost surelyto a random variable X in the underlying probability space (ΩF P) (denoted

Xnasminusminusrarr X) if

P983153ω isin Ω lim

nrarrinfinXn(ω) = X(ω)

983154= 1

We aim to show almost sure convergence of a scaled random walk to a Brownianmotion in (C[0 T ] d) the metric space of continuous functions on the interval [0 T ]with d as the supremum norm

d(f g) = suptisin[0T ]

|f(t)minus g(t)|

Note that random walks and Brownian motions are stochastic processes while dmeasures the distance between two continuous functions In order to measure thedistance between random walks and Brownian motion we instead consider the pathstaken by these stochastic processes

t 983041rarr Rt and t 983041rarr Bt

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 11

33 Brownian Motion Construction In this paper the construction of Brow-nian motion comprises the following steps

(1) Construct a simple random walk out of a Brownian motion path usingSkorokhod embedding

(2) Linearly interpolate and scale the embedded random walk creating a con-tinuous function from [0 n] 983041rarr R

(3) Take the uniform limit of this scaled random walk to derive the originalBrownian motion path

The paper presents an alternative formulation to the canonical construction ofBrownian motion which involves defining Brownian motion on the dyadic rationalsand using continuity to extend the definition to the reals To motivate this particu-lar construction we will proceed out of order we will begin with step (2) backtrackto step (1) and conclude by stringing together steps (1) through (3) in subsection333 (Strong Approximation)

331 Scaled Random WalksLet Rm be a random walk where E[Xi] = 0 VarXi = σ2 and

Rm =

m983131

i=1

Xi

Definition 35 A scaled random walk with scaling parameter n is a real valuedrandom process

983051R(n)

983052 defined on t ge 0 For t such that nt is an integer

R(n)t =

1radicnσ2

Rnt

For t such that nt is not an integer S(n)(t) is defined by linear interpolation

R(n)t =

1radicnσ2

983063Rlfloorntrfloor + (ntminus lfloorntrfloor)

983043Rlceilntrceil minusRlfloorntrfloor

983044 983064

This method of scaling operates on two aspects of random walks as n increasesthe time between state transitions decreases and increment lengths decrease Lett0 = 0 and tk = inftkt isin Z nt gt nt(kminus1) then as n rarr infin

tk minus t(kminus1) rarr 0 and R(n)tk

minusR(n)t(kminus1)

rarr 0

As a result R(n)(t) is continuous with probability 1 on t ge 0 for all n The scalingfactor 1radic

nσ2ensures constant variance for a given t

Var

9830691radicnσ2

Rnt

983070= Var

9830831radicnσ2

nt983131

i=1

Xi

983084=

9830611radicnσ2

9830622

ntVarXi = tVar Xi

σ2= t

By observing the limiting behavior of R(n)t (ie as n rarr infin) we establish a con-

struction of Brownian motion

12 HARVEY BARNHARD

332 Skorokhod EmbeddingTo construct Brownian motion out of random walks we must first construct randomwalks out of Brownian motion Structuring the proof in this matter is not withoutreason there are uncountably many realizations of Brownian motion so we want toensure that the random walks to which we are taking the limit do in fact convergeto the Brownian motion of interest Such a procedure of deriving a random walkfrom a Brownian motion is called Skorokhod embedding The procedure is given inthe following pseudocode

Set t0 = 0 and n = 0 Let T be a stopping time of Brownian motion Btso that Bt is defined on [0 T ] Let R0 = 0 be the initial value

of the Skorokhod embedding

While tn + t le TInitialize t = 0

While |Bt minusBtn | le 1 Continue Bt until |Bt minusBtn | = 1 for the first time

Set t(n+1) = inft |Bt minusBtn | = 1If Bt(n+1)

minusBtn = 1 then set X(n+1) = 1

Otherwise Bt(n+1)minusBtn = minus1 so set X(n+1) = minus1

Let the (n+ 1)th value of the Skorokhod embedding be

R(n+1) = Rn +Xn+1

Increment to the next step of the inner while loop n = n+ 1

The resulting sequence Rnn=12 is the Skorokhod embedding of Bttisin[0T ]

Theorem 36 Let Bt be a standard Brownian motion Set t0 = 0 and let tn bethe stopping time where |Bt| = 1 for the nth time

tn equiv inft ge 0 |Bt minusBtn | = 1

If we define Rn equiv Btn then Rn is a simple random walk

ProofTo prove that B0 Bt1 Bt2 is a simple random walk we merely need to showthat

(1) PBtn+1minusBtn = 1 = PBtn+1

minusBtn = minus1 = 12 for all n isin N

(2) tn+1 minus tn are iid random variables

The first item follows from the symmetric property of Brownian increments prop-erty (II) in the definition of Brownian motion Bs+t minusBs sim N(0σ2t)

The second item follows from the strong Markov property of Brownian motionwhich states that if τ is a stopping time with respect to Bt then Bt minus Bτ is

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 13

also a Brownian motion We do not include a rigorous treatment of stopping timesor a proof of the strong Markov property

We now state a theorem that places a probabilistic bound on the differencebetween a Brownian motion and its Skorokhod embedding

Theorem 37 Define Θ to be the maximum distance between a Brownian motionand its Skorokhod embedding

Θ(BRT ) = max0letleT

|Bt minusRt|

Note that Θ is equivalent to the distance between Bt and Rt in the space ofcontinuous functions on the time interval [0 T ] equipped with the supremum normThere exist c a isin [0infin) such that for all r le n14 and all integers n ge 3

P983153Θ(BRT ) ge rn14

983155log n

983154le ceminusar

ProofIt will suffice to prove the theorem for r ge 9c2 where c is the constant such that

P983153osc(B δ T ) gt r

983155δ log(1δ)

983154le cT δ(rc)

2

andosc(B δ T ) = sup |Bt minusBs| s t isin D s t isin [0 T ] |sminus t| le δ

is the oscillation of Bt restricted to t isin D the dyadic rationals The proof thatsuch a c exists can be found in [3] page 68 where it is also shown that for n isin N

Θ(BSn) le 1 + osc(B 1 n) + max|Bj minusBτj | j = 1 nNow suppose 9c2 le r le n14 If |Bn minusBτn | is large then either |nminus τn| is large

or the oscillation of B is large Consider the three events

(1)983051Θ(BRn) ge rn14

radiclog n

983052

(2)983051osc(B r

radicn 2n) ge (r3)n14

radiclog n

983052

(3) max1lejlen |τj minus j| ge rradicn

From above we know that event (1) is contained in the union of events (2) and (3)As a consequence we need only prove the result of the theorem for events (2) and(3) We first tackle event (2)

P983153osc(B r

radicn 2n) gt (r3)n(14)

983155log n

983154le 3P

983153osc(B r

radicn n) gt (r3)n(14)

983155log n

983154

= 3P983153osc(B rnminus12) gt (r3)nminus(14)

983155log n

983154

le 3P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log

983043n12r

983044983070

Ifradicr3 ge c and r le n14 we can conclude that there exist c and a such that

P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log(n12)

983070le ceminusar logn

For event (3) we refer to a proof on page 266 in the appendix of [3] to show thatthere exist c a such that

P983069

max1lejlen

|τj minus j| gt rradicn

983070le ceminusar2

14 HARVEY BARNHARD

333 Strong ApproximationWith preliminaries in order we now show that Brownian motion can be thought ofas a limit of simple random walks a construction known as the strong approximationof Brownian motion Let Bt be a standard Brownian motion with variance

parameter 1 as defined in the probability space (ΩF P) and let B(n)t be the

scaled Brownian motion

B(n)t =

1radicnBnt

From B(n) we use the Skorokhod embedding to derive the simple random walkS(n) And from S(n) we derive the scaled simple random walk R(n)

R(n)t =

1radicnS(n)nt

From Theorem 37 we know that there exists c a isin [0infin) such that for all positiveintegers T

P983069

max0letleTn

983055983055983055S(n)t minusB

(n)t

983055983055983055 ge cr(Tn)14983155log (Tn)

983070le ceminusar

multiplying by 1radicn this becomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge crT 14nminus14983155log (Tn)

983070le ceminusar

Letting r = c log n where c is sufficiently large for the given T the inequalitybecomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

983070le c

n2

To proceed we require the use of the Borel-Cantelli lemma the result of which isprovided below

Lemma 38 (Borel-Cantelli)If An is a sequence of events in Ω and

983123infini=1 PAn lt infin then

PAn io = 0

where An io = lim supnrarrinfin An is the event that infinitely many An occur

If we set

An = max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

we can then apply the Borel-Cantelli lemma to the above inequality to concludethat

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 le cnminus14 log32 n

with probability one for all n sufficiently large Since

limnrarrinfin

cnminus14 log32 n = 0

we conclude that

R(n)t

asminusminusrarr Bt in C[0 T ]The following theorem condenses the results of this section up to this point

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 15

Theorem 39 (Donskerrsquos Theorem) Let Xi be a sequence of iid random vari-

ables with mean 0 and variance σ2 If R(n)t is a scaled random walk as defined

above then R(n)t rArr Bt where Bt is a standard Brownian motion This result is

also known as the functional central limit theorem

34 Brownian Motion VariantsThe preceding construction of Brownian motion will prove particularly useful in ourapproximation of heavy traffic waiting times However this process requires furthermodification Recall that when traffic intensity ρ is slightly less than 1 customersare being serviced at a slightly faster rate then they are entering the queue Thusgiven enough time traffic will eventually ldquoclear outrdquo due to faster service timesHowever this particular construction of Brownian motion has increments of meanzero

E[Bs+t minusBs] = 0

meaning that the approximate number of customers in line wouldmdashon averagemdashremain unchanged regardless of how long the line is observed Such a propertyis called the martingale property Instead we want a process that exhibits thesupermartingale property

E[Bs+t minusBs] lt 0

Definition 310 Let Fn be an increasing sequence of σ-fields (Fn is called afiltration) Let Xn be a sequence of random variables with finite mean andXn isin Fn for all n Martingales supermartingales and submartingales are thendefined as follows for all n

Martingale E [Xn+1 | Fn] = Xn

Supermartingale E [Xn+1 | Fn] le Xn

Submartingale E [Xn+1 | Fn] ge Xn

In words a sequence that we expect to remain the same over time is a martingalea sequence we expect to increase is a submartingale and a sequence we expect todecrease is a supermartingale To approximate heavy traffic we want a Brownianmotion-like process that exhibits the supermartingale property

Definition 311 A Brownian motion with drift parameter α and variance param-eter σ2 Bα

t is defined by the following properties

(1) Independent increments(2) Increment lengths are normally distributed with mean αt and variance σ2t(3) the path t 983041rarr Bα

t is continuous

Note that these properties are the same as standard Brownian motion exceptincrement lengths now have non-zero means Since we have already shown theexistence of standard Brownian motion the construction of Brownian motion withdrift is now easy

Lemma 312 Let Bt be a standard Brownian motion with variance parameterσ2 then Bα

t = Bt + αt is Brownian motion with drift parameter α and varianceparameter σ2 Moreover Bα

t fulfills the supermartingale property when α lt 0

16 HARVEY BARNHARD

ProofIndependence of increments follows directly from independence of increments forBt and the fact that αt is a constant for a given t To prove that Bα

t has stationaryincrements we have to show that the distribution of increments Bα

s+tminusBαs depends

only the time interval t and not the absolute time Recall from the definition of Bt

that Bs+t minusBt are normal random variables with mean 0 and variance t for s gt 0

E983045Bα

s+t minusBαs

983046= E

983045Bα

s+t

983046minus E [Bα

s ]

= E983045Bα

s+t + α(s+ t)983046minus E [Bs + αs]

= α(s+ t)minus αs

= αt

Var983051Bα

s+t minusBαs

983052= Var Bs+t minusBs + αt= Var Bs+t minusBs= σ2t

And since for fixed t αt is just a constant we use the property of normal randomvariables to conclude that

Bαs+t minusBα

s sim N(αtσ2t) for all s gt 0

Continuity of Bαt directly follows from continuity of Bt We have shown that

Bαt constructed as the sum of standard Brownian motion and drift αt is indeed

Brownian motion with drift

Brownian motion with negative drift provides a better approximation of thewaiting-time of customers in a queue than does standard Brownian motion How-ever the unboundedness of Brownian motion suggests that waiting times couldperhaps be negative In order to circumvent this issue we instead use reflectedBrownian motion with drift Reflected Brownian motion acts much the same asBrownian motion except on a given boundary off of which the Brownian motionismdashunsurprisinglymdashreflected In the one-dimensional case the boundaries may beany interval on the reals that provide a lower and upper bound to the Brownianmotion For our purposes we only use a single boundary zero that gives a lowerbound to Brownian motion That is for all t ge 0 we want Bα

t ge 0

Definition 313 Let Bαt be a standard Brownian motion with drift parameter α

and let Mαt be the running maximum of Bα

t

Mαt = sup

0lesletBα

s

Then reflected Brownian motion with drift parameter minusα and boundary 0 definedon R+ can be constructed as follows

Rα0t equiv max0Mα

t minusBαt

Note that max0 Sαt ge 0 and Mα

t ge Bαt B

α0t ge 0 for all t Most importantly

the local behavior of Bα0t is exactly like Brownian motion with drift parameter minusα

since max0 Sαt is simply a constant when Bα

t ∕= 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 17

We have now provided sufficient modifications of Brownian motion and we nextmove onmdashwhat we will ultimately use to approximate heavy traffic waiting timesThe next example provides us with the distribution of the supremum of a Brownianmotion with negative drift

Example 314Let Bt be a Brownian motion with drift α lt 0 and variance parameter σ2 Define

Mt = sup0leslet

Bs

to be the running maximum then

Minfin sim exp

9830612|α|σ2

983062

that is

PW ge w = eminusminus2|α|σ2 w w ge 0

E [Minfin] =σ2

2|α|

Prooffor constants a b gt 0 Let T (minusa b) be the first time that B minus t hits minusa or b

T (minusa b) = inft Bt = minusa or Bt = bIn [7] it is proved that

P983051BT (minusab)=b

983052=

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)

Since α lt 0 exp(2αaσ2) rarr 0 we have

limararrinfin

PBT (minusab) = b = limararrinfin

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)= exp(2αbσ2)

The left-hand side becomes the probability that the process will reach b somewherealong its path (ie that the maximum of the process exceeds b somewhere alongits path) Therefore we have

PW ge b = exp(2αbσ2) = expminus2|α|bσ2

This example shows us that the supremum of a Brownian motion observed ldquofor-everrdquo (ie on the interval [0infin)) is exponentially distributed with rate 2|α|σ2

4 Heavy Traffic Approximation

We now have all we need to provide a heuristic argument for the approximationof single-servers queue in heavy traffic using Brownian Motion For a rigorousalbeit opaque treatment of this approximation see [8] the seminal work on thissubject by Kingman

Theorem 41 The waiting time of the nth customer in an MM1 queue canbe approximated by the supremum of reflected Brownian motion with negative driftand boundary at 0

18 HARVEY BARNHARD

ProofRecall that an MM1 queue is a process where the number of customer arrivalsfollows a poisson process with rate λ and service times are exponentially distributedwith rate micro From Lindleyrsquos equation in Section 2 we can write the waiting timefor the nth customer as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

where U (n) = S(n)minus I(n) is the difference between the nth service time and the nthinter-arrival time Define the partial sum P

(n)k to be zero when k = 0 and

P(n)k equiv

k983131

i=1

U (nminusi) for k ge 1

We can then rewrite the waiting time of the nth customer as

W (n)q = max

0lekle(nminus1)P

(n)k

The fact that P(n)k is the kth value of a random walk follows immediately from

U (1) U (nminus1) being iid random variables Let α and σ2 be the expectationand variance of U (i) We calculate α and σ2 by first noting that S(i) and I(i) areexponentially distributed with rates micro and λ Recall that ρ = λmicro is the trafficintensity as defined in the first section of this paper

α = E983147U (i)

983148= E

983147S(i)

983148minus E

983147I(i)

983148=

1

microminus 1

λ=

ρminus 1

λ

σ2 = Var983153U (i)

983154= Var

983153S(i)

983154+Var

983153I(i)

983154=

1

micro2+

1

λ2

We next compute the expectation and variance of P(n)k using the fact that U (i) are

iid random variables

E983147P

(n)k

983148=

k983131

i=1

E983147U (nminusi)

983148= kα

Var983153P

(n)k

983154=

k983131

i=1

Var983153U (nminusi)

983154= kσ2

For a finite n W(n)q is therefore the maximum value of a random walk with n steps

We now show that Pnk can be approximated by Brownian motion with drift

parameter α and variance parameter σ2 when n is sufficiently large We restate theproperties of such a Brownian motion Bt below

(1) B0 = 0(2) Bt has stationary and independent increments(3) Bt sim N(αtσ2t)(4) t 983041rarr Bt is continuous with probability one

The first property is clear as P(n)0 = 0 for all n The second property follows from

the fact that U (i) are iid random variables The third property is a result of thecentral limit theorem for iid sequences the result of which is provided below

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

10 HARVEY BARNHARD

32 Brownian Motion DefinitionWe now introduce the continuous-time analog to a random walkmdashBrownian motionA one-dimensional standard Brownian motion with variance parameter σ2 is a real-valued process Bttge0 defined in the probability space (ΩF P) that has thefollowing properties

(I) If t0 lt t1 lt middot middot middot lt tn then Bt0 Bt1 minusBt0 Btn minusBtnminus1are independent

(II) If s t ge 0 then

P (Bs+t minusBs isin A) =

983133

A

1radic2πσ2t

exp(minusx22σ2t)

(III) With probability 1 t 983041rarr Bt is continuous

The first two conditions may be concisely reworded as

(I) Independent increments(II) Increment lengths are normally distributed with mean 0 and variance σ2t

Bs+t minusBs sim N(0σ2t)

Remark 33Brownian motion is a collection of stochastic processes When we say a Brownianmotion or the Brownian motion we are talking about a particular realization ofBrownian motion in the space of continuous functions C[0 T ] where T is a stoppingtime

The above definition establishes the necessary conditions for a stochastic process tobe Brownian but the existence of such a process requires further work Throughan application of the Kolmogorov extension theorem existence is proved in [4]However such a proof is bereft of intuition for queueing theory and approximatingwaiting times Instead we take the existence of Brownian motion as given andshow that Brownian motion can be constructed as a limit of random walksmdashtheresult of Donskerrsquos Theorem Brownian motion constructed in such a manner iscalled the standard Wiener process

Before constructing Brownian motion we must first establish what type of con-vergence we are discussing

Definition 34 A sequence of random variables Xiinfini=1 converges almost surelyto a random variable X in the underlying probability space (ΩF P) (denoted

Xnasminusminusrarr X) if

P983153ω isin Ω lim

nrarrinfinXn(ω) = X(ω)

983154= 1

We aim to show almost sure convergence of a scaled random walk to a Brownianmotion in (C[0 T ] d) the metric space of continuous functions on the interval [0 T ]with d as the supremum norm

d(f g) = suptisin[0T ]

|f(t)minus g(t)|

Note that random walks and Brownian motions are stochastic processes while dmeasures the distance between two continuous functions In order to measure thedistance between random walks and Brownian motion we instead consider the pathstaken by these stochastic processes

t 983041rarr Rt and t 983041rarr Bt

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 11

33 Brownian Motion Construction In this paper the construction of Brow-nian motion comprises the following steps

(1) Construct a simple random walk out of a Brownian motion path usingSkorokhod embedding

(2) Linearly interpolate and scale the embedded random walk creating a con-tinuous function from [0 n] 983041rarr R

(3) Take the uniform limit of this scaled random walk to derive the originalBrownian motion path

The paper presents an alternative formulation to the canonical construction ofBrownian motion which involves defining Brownian motion on the dyadic rationalsand using continuity to extend the definition to the reals To motivate this particu-lar construction we will proceed out of order we will begin with step (2) backtrackto step (1) and conclude by stringing together steps (1) through (3) in subsection333 (Strong Approximation)

331 Scaled Random WalksLet Rm be a random walk where E[Xi] = 0 VarXi = σ2 and

Rm =

m983131

i=1

Xi

Definition 35 A scaled random walk with scaling parameter n is a real valuedrandom process

983051R(n)

983052 defined on t ge 0 For t such that nt is an integer

R(n)t =

1radicnσ2

Rnt

For t such that nt is not an integer S(n)(t) is defined by linear interpolation

R(n)t =

1radicnσ2

983063Rlfloorntrfloor + (ntminus lfloorntrfloor)

983043Rlceilntrceil minusRlfloorntrfloor

983044 983064

This method of scaling operates on two aspects of random walks as n increasesthe time between state transitions decreases and increment lengths decrease Lett0 = 0 and tk = inftkt isin Z nt gt nt(kminus1) then as n rarr infin

tk minus t(kminus1) rarr 0 and R(n)tk

minusR(n)t(kminus1)

rarr 0

As a result R(n)(t) is continuous with probability 1 on t ge 0 for all n The scalingfactor 1radic

nσ2ensures constant variance for a given t

Var

9830691radicnσ2

Rnt

983070= Var

9830831radicnσ2

nt983131

i=1

Xi

983084=

9830611radicnσ2

9830622

ntVarXi = tVar Xi

σ2= t

By observing the limiting behavior of R(n)t (ie as n rarr infin) we establish a con-

struction of Brownian motion

12 HARVEY BARNHARD

332 Skorokhod EmbeddingTo construct Brownian motion out of random walks we must first construct randomwalks out of Brownian motion Structuring the proof in this matter is not withoutreason there are uncountably many realizations of Brownian motion so we want toensure that the random walks to which we are taking the limit do in fact convergeto the Brownian motion of interest Such a procedure of deriving a random walkfrom a Brownian motion is called Skorokhod embedding The procedure is given inthe following pseudocode

Set t0 = 0 and n = 0 Let T be a stopping time of Brownian motion Btso that Bt is defined on [0 T ] Let R0 = 0 be the initial value

of the Skorokhod embedding

While tn + t le TInitialize t = 0

While |Bt minusBtn | le 1 Continue Bt until |Bt minusBtn | = 1 for the first time

Set t(n+1) = inft |Bt minusBtn | = 1If Bt(n+1)

minusBtn = 1 then set X(n+1) = 1

Otherwise Bt(n+1)minusBtn = minus1 so set X(n+1) = minus1

Let the (n+ 1)th value of the Skorokhod embedding be

R(n+1) = Rn +Xn+1

Increment to the next step of the inner while loop n = n+ 1

The resulting sequence Rnn=12 is the Skorokhod embedding of Bttisin[0T ]

Theorem 36 Let Bt be a standard Brownian motion Set t0 = 0 and let tn bethe stopping time where |Bt| = 1 for the nth time

tn equiv inft ge 0 |Bt minusBtn | = 1

If we define Rn equiv Btn then Rn is a simple random walk

ProofTo prove that B0 Bt1 Bt2 is a simple random walk we merely need to showthat

(1) PBtn+1minusBtn = 1 = PBtn+1

minusBtn = minus1 = 12 for all n isin N

(2) tn+1 minus tn are iid random variables

The first item follows from the symmetric property of Brownian increments prop-erty (II) in the definition of Brownian motion Bs+t minusBs sim N(0σ2t)

The second item follows from the strong Markov property of Brownian motionwhich states that if τ is a stopping time with respect to Bt then Bt minus Bτ is

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 13

also a Brownian motion We do not include a rigorous treatment of stopping timesor a proof of the strong Markov property

We now state a theorem that places a probabilistic bound on the differencebetween a Brownian motion and its Skorokhod embedding

Theorem 37 Define Θ to be the maximum distance between a Brownian motionand its Skorokhod embedding

Θ(BRT ) = max0letleT

|Bt minusRt|

Note that Θ is equivalent to the distance between Bt and Rt in the space ofcontinuous functions on the time interval [0 T ] equipped with the supremum normThere exist c a isin [0infin) such that for all r le n14 and all integers n ge 3

P983153Θ(BRT ) ge rn14

983155log n

983154le ceminusar

ProofIt will suffice to prove the theorem for r ge 9c2 where c is the constant such that

P983153osc(B δ T ) gt r

983155δ log(1δ)

983154le cT δ(rc)

2

andosc(B δ T ) = sup |Bt minusBs| s t isin D s t isin [0 T ] |sminus t| le δ

is the oscillation of Bt restricted to t isin D the dyadic rationals The proof thatsuch a c exists can be found in [3] page 68 where it is also shown that for n isin N

Θ(BSn) le 1 + osc(B 1 n) + max|Bj minusBτj | j = 1 nNow suppose 9c2 le r le n14 If |Bn minusBτn | is large then either |nminus τn| is large

or the oscillation of B is large Consider the three events

(1)983051Θ(BRn) ge rn14

radiclog n

983052

(2)983051osc(B r

radicn 2n) ge (r3)n14

radiclog n

983052

(3) max1lejlen |τj minus j| ge rradicn

From above we know that event (1) is contained in the union of events (2) and (3)As a consequence we need only prove the result of the theorem for events (2) and(3) We first tackle event (2)

P983153osc(B r

radicn 2n) gt (r3)n(14)

983155log n

983154le 3P

983153osc(B r

radicn n) gt (r3)n(14)

983155log n

983154

= 3P983153osc(B rnminus12) gt (r3)nminus(14)

983155log n

983154

le 3P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log

983043n12r

983044983070

Ifradicr3 ge c and r le n14 we can conclude that there exist c and a such that

P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log(n12)

983070le ceminusar logn

For event (3) we refer to a proof on page 266 in the appendix of [3] to show thatthere exist c a such that

P983069

max1lejlen

|τj minus j| gt rradicn

983070le ceminusar2

14 HARVEY BARNHARD

333 Strong ApproximationWith preliminaries in order we now show that Brownian motion can be thought ofas a limit of simple random walks a construction known as the strong approximationof Brownian motion Let Bt be a standard Brownian motion with variance

parameter 1 as defined in the probability space (ΩF P) and let B(n)t be the

scaled Brownian motion

B(n)t =

1radicnBnt

From B(n) we use the Skorokhod embedding to derive the simple random walkS(n) And from S(n) we derive the scaled simple random walk R(n)

R(n)t =

1radicnS(n)nt

From Theorem 37 we know that there exists c a isin [0infin) such that for all positiveintegers T

P983069

max0letleTn

983055983055983055S(n)t minusB

(n)t

983055983055983055 ge cr(Tn)14983155log (Tn)

983070le ceminusar

multiplying by 1radicn this becomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge crT 14nminus14983155log (Tn)

983070le ceminusar

Letting r = c log n where c is sufficiently large for the given T the inequalitybecomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

983070le c

n2

To proceed we require the use of the Borel-Cantelli lemma the result of which isprovided below

Lemma 38 (Borel-Cantelli)If An is a sequence of events in Ω and

983123infini=1 PAn lt infin then

PAn io = 0

where An io = lim supnrarrinfin An is the event that infinitely many An occur

If we set

An = max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

we can then apply the Borel-Cantelli lemma to the above inequality to concludethat

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 le cnminus14 log32 n

with probability one for all n sufficiently large Since

limnrarrinfin

cnminus14 log32 n = 0

we conclude that

R(n)t

asminusminusrarr Bt in C[0 T ]The following theorem condenses the results of this section up to this point

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 15

Theorem 39 (Donskerrsquos Theorem) Let Xi be a sequence of iid random vari-

ables with mean 0 and variance σ2 If R(n)t is a scaled random walk as defined

above then R(n)t rArr Bt where Bt is a standard Brownian motion This result is

also known as the functional central limit theorem

34 Brownian Motion VariantsThe preceding construction of Brownian motion will prove particularly useful in ourapproximation of heavy traffic waiting times However this process requires furthermodification Recall that when traffic intensity ρ is slightly less than 1 customersare being serviced at a slightly faster rate then they are entering the queue Thusgiven enough time traffic will eventually ldquoclear outrdquo due to faster service timesHowever this particular construction of Brownian motion has increments of meanzero

E[Bs+t minusBs] = 0

meaning that the approximate number of customers in line wouldmdashon averagemdashremain unchanged regardless of how long the line is observed Such a propertyis called the martingale property Instead we want a process that exhibits thesupermartingale property

E[Bs+t minusBs] lt 0

Definition 310 Let Fn be an increasing sequence of σ-fields (Fn is called afiltration) Let Xn be a sequence of random variables with finite mean andXn isin Fn for all n Martingales supermartingales and submartingales are thendefined as follows for all n

Martingale E [Xn+1 | Fn] = Xn

Supermartingale E [Xn+1 | Fn] le Xn

Submartingale E [Xn+1 | Fn] ge Xn

In words a sequence that we expect to remain the same over time is a martingalea sequence we expect to increase is a submartingale and a sequence we expect todecrease is a supermartingale To approximate heavy traffic we want a Brownianmotion-like process that exhibits the supermartingale property

Definition 311 A Brownian motion with drift parameter α and variance param-eter σ2 Bα

t is defined by the following properties

(1) Independent increments(2) Increment lengths are normally distributed with mean αt and variance σ2t(3) the path t 983041rarr Bα

t is continuous

Note that these properties are the same as standard Brownian motion exceptincrement lengths now have non-zero means Since we have already shown theexistence of standard Brownian motion the construction of Brownian motion withdrift is now easy

Lemma 312 Let Bt be a standard Brownian motion with variance parameterσ2 then Bα

t = Bt + αt is Brownian motion with drift parameter α and varianceparameter σ2 Moreover Bα

t fulfills the supermartingale property when α lt 0

16 HARVEY BARNHARD

ProofIndependence of increments follows directly from independence of increments forBt and the fact that αt is a constant for a given t To prove that Bα

t has stationaryincrements we have to show that the distribution of increments Bα

s+tminusBαs depends

only the time interval t and not the absolute time Recall from the definition of Bt

that Bs+t minusBt are normal random variables with mean 0 and variance t for s gt 0

E983045Bα

s+t minusBαs

983046= E

983045Bα

s+t

983046minus E [Bα

s ]

= E983045Bα

s+t + α(s+ t)983046minus E [Bs + αs]

= α(s+ t)minus αs

= αt

Var983051Bα

s+t minusBαs

983052= Var Bs+t minusBs + αt= Var Bs+t minusBs= σ2t

And since for fixed t αt is just a constant we use the property of normal randomvariables to conclude that

Bαs+t minusBα

s sim N(αtσ2t) for all s gt 0

Continuity of Bαt directly follows from continuity of Bt We have shown that

Bαt constructed as the sum of standard Brownian motion and drift αt is indeed

Brownian motion with drift

Brownian motion with negative drift provides a better approximation of thewaiting-time of customers in a queue than does standard Brownian motion How-ever the unboundedness of Brownian motion suggests that waiting times couldperhaps be negative In order to circumvent this issue we instead use reflectedBrownian motion with drift Reflected Brownian motion acts much the same asBrownian motion except on a given boundary off of which the Brownian motionismdashunsurprisinglymdashreflected In the one-dimensional case the boundaries may beany interval on the reals that provide a lower and upper bound to the Brownianmotion For our purposes we only use a single boundary zero that gives a lowerbound to Brownian motion That is for all t ge 0 we want Bα

t ge 0

Definition 313 Let Bαt be a standard Brownian motion with drift parameter α

and let Mαt be the running maximum of Bα

t

Mαt = sup

0lesletBα

s

Then reflected Brownian motion with drift parameter minusα and boundary 0 definedon R+ can be constructed as follows

Rα0t equiv max0Mα

t minusBαt

Note that max0 Sαt ge 0 and Mα

t ge Bαt B

α0t ge 0 for all t Most importantly

the local behavior of Bα0t is exactly like Brownian motion with drift parameter minusα

since max0 Sαt is simply a constant when Bα

t ∕= 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 17

We have now provided sufficient modifications of Brownian motion and we nextmove onmdashwhat we will ultimately use to approximate heavy traffic waiting timesThe next example provides us with the distribution of the supremum of a Brownianmotion with negative drift

Example 314Let Bt be a Brownian motion with drift α lt 0 and variance parameter σ2 Define

Mt = sup0leslet

Bs

to be the running maximum then

Minfin sim exp

9830612|α|σ2

983062

that is

PW ge w = eminusminus2|α|σ2 w w ge 0

E [Minfin] =σ2

2|α|

Prooffor constants a b gt 0 Let T (minusa b) be the first time that B minus t hits minusa or b

T (minusa b) = inft Bt = minusa or Bt = bIn [7] it is proved that

P983051BT (minusab)=b

983052=

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)

Since α lt 0 exp(2αaσ2) rarr 0 we have

limararrinfin

PBT (minusab) = b = limararrinfin

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)= exp(2αbσ2)

The left-hand side becomes the probability that the process will reach b somewherealong its path (ie that the maximum of the process exceeds b somewhere alongits path) Therefore we have

PW ge b = exp(2αbσ2) = expminus2|α|bσ2

This example shows us that the supremum of a Brownian motion observed ldquofor-everrdquo (ie on the interval [0infin)) is exponentially distributed with rate 2|α|σ2

4 Heavy Traffic Approximation

We now have all we need to provide a heuristic argument for the approximationof single-servers queue in heavy traffic using Brownian Motion For a rigorousalbeit opaque treatment of this approximation see [8] the seminal work on thissubject by Kingman

Theorem 41 The waiting time of the nth customer in an MM1 queue canbe approximated by the supremum of reflected Brownian motion with negative driftand boundary at 0

18 HARVEY BARNHARD

ProofRecall that an MM1 queue is a process where the number of customer arrivalsfollows a poisson process with rate λ and service times are exponentially distributedwith rate micro From Lindleyrsquos equation in Section 2 we can write the waiting timefor the nth customer as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

where U (n) = S(n)minus I(n) is the difference between the nth service time and the nthinter-arrival time Define the partial sum P

(n)k to be zero when k = 0 and

P(n)k equiv

k983131

i=1

U (nminusi) for k ge 1

We can then rewrite the waiting time of the nth customer as

W (n)q = max

0lekle(nminus1)P

(n)k

The fact that P(n)k is the kth value of a random walk follows immediately from

U (1) U (nminus1) being iid random variables Let α and σ2 be the expectationand variance of U (i) We calculate α and σ2 by first noting that S(i) and I(i) areexponentially distributed with rates micro and λ Recall that ρ = λmicro is the trafficintensity as defined in the first section of this paper

α = E983147U (i)

983148= E

983147S(i)

983148minus E

983147I(i)

983148=

1

microminus 1

λ=

ρminus 1

λ

σ2 = Var983153U (i)

983154= Var

983153S(i)

983154+Var

983153I(i)

983154=

1

micro2+

1

λ2

We next compute the expectation and variance of P(n)k using the fact that U (i) are

iid random variables

E983147P

(n)k

983148=

k983131

i=1

E983147U (nminusi)

983148= kα

Var983153P

(n)k

983154=

k983131

i=1

Var983153U (nminusi)

983154= kσ2

For a finite n W(n)q is therefore the maximum value of a random walk with n steps

We now show that Pnk can be approximated by Brownian motion with drift

parameter α and variance parameter σ2 when n is sufficiently large We restate theproperties of such a Brownian motion Bt below

(1) B0 = 0(2) Bt has stationary and independent increments(3) Bt sim N(αtσ2t)(4) t 983041rarr Bt is continuous with probability one

The first property is clear as P(n)0 = 0 for all n The second property follows from

the fact that U (i) are iid random variables The third property is a result of thecentral limit theorem for iid sequences the result of which is provided below

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 11

33 Brownian Motion Construction In this paper the construction of Brow-nian motion comprises the following steps

(1) Construct a simple random walk out of a Brownian motion path usingSkorokhod embedding

(2) Linearly interpolate and scale the embedded random walk creating a con-tinuous function from [0 n] 983041rarr R

(3) Take the uniform limit of this scaled random walk to derive the originalBrownian motion path

The paper presents an alternative formulation to the canonical construction ofBrownian motion which involves defining Brownian motion on the dyadic rationalsand using continuity to extend the definition to the reals To motivate this particu-lar construction we will proceed out of order we will begin with step (2) backtrackto step (1) and conclude by stringing together steps (1) through (3) in subsection333 (Strong Approximation)

331 Scaled Random WalksLet Rm be a random walk where E[Xi] = 0 VarXi = σ2 and

Rm =

m983131

i=1

Xi

Definition 35 A scaled random walk with scaling parameter n is a real valuedrandom process

983051R(n)

983052 defined on t ge 0 For t such that nt is an integer

R(n)t =

1radicnσ2

Rnt

For t such that nt is not an integer S(n)(t) is defined by linear interpolation

R(n)t =

1radicnσ2

983063Rlfloorntrfloor + (ntminus lfloorntrfloor)

983043Rlceilntrceil minusRlfloorntrfloor

983044 983064

This method of scaling operates on two aspects of random walks as n increasesthe time between state transitions decreases and increment lengths decrease Lett0 = 0 and tk = inftkt isin Z nt gt nt(kminus1) then as n rarr infin

tk minus t(kminus1) rarr 0 and R(n)tk

minusR(n)t(kminus1)

rarr 0

As a result R(n)(t) is continuous with probability 1 on t ge 0 for all n The scalingfactor 1radic

nσ2ensures constant variance for a given t

Var

9830691radicnσ2

Rnt

983070= Var

9830831radicnσ2

nt983131

i=1

Xi

983084=

9830611radicnσ2

9830622

ntVarXi = tVar Xi

σ2= t

By observing the limiting behavior of R(n)t (ie as n rarr infin) we establish a con-

struction of Brownian motion

12 HARVEY BARNHARD

332 Skorokhod EmbeddingTo construct Brownian motion out of random walks we must first construct randomwalks out of Brownian motion Structuring the proof in this matter is not withoutreason there are uncountably many realizations of Brownian motion so we want toensure that the random walks to which we are taking the limit do in fact convergeto the Brownian motion of interest Such a procedure of deriving a random walkfrom a Brownian motion is called Skorokhod embedding The procedure is given inthe following pseudocode

Set t0 = 0 and n = 0 Let T be a stopping time of Brownian motion Btso that Bt is defined on [0 T ] Let R0 = 0 be the initial value

of the Skorokhod embedding

While tn + t le TInitialize t = 0

While |Bt minusBtn | le 1 Continue Bt until |Bt minusBtn | = 1 for the first time

Set t(n+1) = inft |Bt minusBtn | = 1If Bt(n+1)

minusBtn = 1 then set X(n+1) = 1

Otherwise Bt(n+1)minusBtn = minus1 so set X(n+1) = minus1

Let the (n+ 1)th value of the Skorokhod embedding be

R(n+1) = Rn +Xn+1

Increment to the next step of the inner while loop n = n+ 1

The resulting sequence Rnn=12 is the Skorokhod embedding of Bttisin[0T ]

Theorem 36 Let Bt be a standard Brownian motion Set t0 = 0 and let tn bethe stopping time where |Bt| = 1 for the nth time

tn equiv inft ge 0 |Bt minusBtn | = 1

If we define Rn equiv Btn then Rn is a simple random walk

ProofTo prove that B0 Bt1 Bt2 is a simple random walk we merely need to showthat

(1) PBtn+1minusBtn = 1 = PBtn+1

minusBtn = minus1 = 12 for all n isin N

(2) tn+1 minus tn are iid random variables

The first item follows from the symmetric property of Brownian increments prop-erty (II) in the definition of Brownian motion Bs+t minusBs sim N(0σ2t)

The second item follows from the strong Markov property of Brownian motionwhich states that if τ is a stopping time with respect to Bt then Bt minus Bτ is

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 13

also a Brownian motion We do not include a rigorous treatment of stopping timesor a proof of the strong Markov property

We now state a theorem that places a probabilistic bound on the differencebetween a Brownian motion and its Skorokhod embedding

Theorem 37 Define Θ to be the maximum distance between a Brownian motionand its Skorokhod embedding

Θ(BRT ) = max0letleT

|Bt minusRt|

Note that Θ is equivalent to the distance between Bt and Rt in the space ofcontinuous functions on the time interval [0 T ] equipped with the supremum normThere exist c a isin [0infin) such that for all r le n14 and all integers n ge 3

P983153Θ(BRT ) ge rn14

983155log n

983154le ceminusar

ProofIt will suffice to prove the theorem for r ge 9c2 where c is the constant such that

P983153osc(B δ T ) gt r

983155δ log(1δ)

983154le cT δ(rc)

2

andosc(B δ T ) = sup |Bt minusBs| s t isin D s t isin [0 T ] |sminus t| le δ

is the oscillation of Bt restricted to t isin D the dyadic rationals The proof thatsuch a c exists can be found in [3] page 68 where it is also shown that for n isin N

Θ(BSn) le 1 + osc(B 1 n) + max|Bj minusBτj | j = 1 nNow suppose 9c2 le r le n14 If |Bn minusBτn | is large then either |nminus τn| is large

or the oscillation of B is large Consider the three events

(1)983051Θ(BRn) ge rn14

radiclog n

983052

(2)983051osc(B r

radicn 2n) ge (r3)n14

radiclog n

983052

(3) max1lejlen |τj minus j| ge rradicn

From above we know that event (1) is contained in the union of events (2) and (3)As a consequence we need only prove the result of the theorem for events (2) and(3) We first tackle event (2)

P983153osc(B r

radicn 2n) gt (r3)n(14)

983155log n

983154le 3P

983153osc(B r

radicn n) gt (r3)n(14)

983155log n

983154

= 3P983153osc(B rnminus12) gt (r3)nminus(14)

983155log n

983154

le 3P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log

983043n12r

983044983070

Ifradicr3 ge c and r le n14 we can conclude that there exist c and a such that

P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log(n12)

983070le ceminusar logn

For event (3) we refer to a proof on page 266 in the appendix of [3] to show thatthere exist c a such that

P983069

max1lejlen

|τj minus j| gt rradicn

983070le ceminusar2

14 HARVEY BARNHARD

333 Strong ApproximationWith preliminaries in order we now show that Brownian motion can be thought ofas a limit of simple random walks a construction known as the strong approximationof Brownian motion Let Bt be a standard Brownian motion with variance

parameter 1 as defined in the probability space (ΩF P) and let B(n)t be the

scaled Brownian motion

B(n)t =

1radicnBnt

From B(n) we use the Skorokhod embedding to derive the simple random walkS(n) And from S(n) we derive the scaled simple random walk R(n)

R(n)t =

1radicnS(n)nt

From Theorem 37 we know that there exists c a isin [0infin) such that for all positiveintegers T

P983069

max0letleTn

983055983055983055S(n)t minusB

(n)t

983055983055983055 ge cr(Tn)14983155log (Tn)

983070le ceminusar

multiplying by 1radicn this becomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge crT 14nminus14983155log (Tn)

983070le ceminusar

Letting r = c log n where c is sufficiently large for the given T the inequalitybecomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

983070le c

n2

To proceed we require the use of the Borel-Cantelli lemma the result of which isprovided below

Lemma 38 (Borel-Cantelli)If An is a sequence of events in Ω and

983123infini=1 PAn lt infin then

PAn io = 0

where An io = lim supnrarrinfin An is the event that infinitely many An occur

If we set

An = max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

we can then apply the Borel-Cantelli lemma to the above inequality to concludethat

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 le cnminus14 log32 n

with probability one for all n sufficiently large Since

limnrarrinfin

cnminus14 log32 n = 0

we conclude that

R(n)t

asminusminusrarr Bt in C[0 T ]The following theorem condenses the results of this section up to this point

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 15

Theorem 39 (Donskerrsquos Theorem) Let Xi be a sequence of iid random vari-

ables with mean 0 and variance σ2 If R(n)t is a scaled random walk as defined

above then R(n)t rArr Bt where Bt is a standard Brownian motion This result is

also known as the functional central limit theorem

34 Brownian Motion VariantsThe preceding construction of Brownian motion will prove particularly useful in ourapproximation of heavy traffic waiting times However this process requires furthermodification Recall that when traffic intensity ρ is slightly less than 1 customersare being serviced at a slightly faster rate then they are entering the queue Thusgiven enough time traffic will eventually ldquoclear outrdquo due to faster service timesHowever this particular construction of Brownian motion has increments of meanzero

E[Bs+t minusBs] = 0

meaning that the approximate number of customers in line wouldmdashon averagemdashremain unchanged regardless of how long the line is observed Such a propertyis called the martingale property Instead we want a process that exhibits thesupermartingale property

E[Bs+t minusBs] lt 0

Definition 310 Let Fn be an increasing sequence of σ-fields (Fn is called afiltration) Let Xn be a sequence of random variables with finite mean andXn isin Fn for all n Martingales supermartingales and submartingales are thendefined as follows for all n

Martingale E [Xn+1 | Fn] = Xn

Supermartingale E [Xn+1 | Fn] le Xn

Submartingale E [Xn+1 | Fn] ge Xn

In words a sequence that we expect to remain the same over time is a martingalea sequence we expect to increase is a submartingale and a sequence we expect todecrease is a supermartingale To approximate heavy traffic we want a Brownianmotion-like process that exhibits the supermartingale property

Definition 311 A Brownian motion with drift parameter α and variance param-eter σ2 Bα

t is defined by the following properties

(1) Independent increments(2) Increment lengths are normally distributed with mean αt and variance σ2t(3) the path t 983041rarr Bα

t is continuous

Note that these properties are the same as standard Brownian motion exceptincrement lengths now have non-zero means Since we have already shown theexistence of standard Brownian motion the construction of Brownian motion withdrift is now easy

Lemma 312 Let Bt be a standard Brownian motion with variance parameterσ2 then Bα

t = Bt + αt is Brownian motion with drift parameter α and varianceparameter σ2 Moreover Bα

t fulfills the supermartingale property when α lt 0

16 HARVEY BARNHARD

ProofIndependence of increments follows directly from independence of increments forBt and the fact that αt is a constant for a given t To prove that Bα

t has stationaryincrements we have to show that the distribution of increments Bα

s+tminusBαs depends

only the time interval t and not the absolute time Recall from the definition of Bt

that Bs+t minusBt are normal random variables with mean 0 and variance t for s gt 0

E983045Bα

s+t minusBαs

983046= E

983045Bα

s+t

983046minus E [Bα

s ]

= E983045Bα

s+t + α(s+ t)983046minus E [Bs + αs]

= α(s+ t)minus αs

= αt

Var983051Bα

s+t minusBαs

983052= Var Bs+t minusBs + αt= Var Bs+t minusBs= σ2t

And since for fixed t αt is just a constant we use the property of normal randomvariables to conclude that

Bαs+t minusBα

s sim N(αtσ2t) for all s gt 0

Continuity of Bαt directly follows from continuity of Bt We have shown that

Bαt constructed as the sum of standard Brownian motion and drift αt is indeed

Brownian motion with drift

Brownian motion with negative drift provides a better approximation of thewaiting-time of customers in a queue than does standard Brownian motion How-ever the unboundedness of Brownian motion suggests that waiting times couldperhaps be negative In order to circumvent this issue we instead use reflectedBrownian motion with drift Reflected Brownian motion acts much the same asBrownian motion except on a given boundary off of which the Brownian motionismdashunsurprisinglymdashreflected In the one-dimensional case the boundaries may beany interval on the reals that provide a lower and upper bound to the Brownianmotion For our purposes we only use a single boundary zero that gives a lowerbound to Brownian motion That is for all t ge 0 we want Bα

t ge 0

Definition 313 Let Bαt be a standard Brownian motion with drift parameter α

and let Mαt be the running maximum of Bα

t

Mαt = sup

0lesletBα

s

Then reflected Brownian motion with drift parameter minusα and boundary 0 definedon R+ can be constructed as follows

Rα0t equiv max0Mα

t minusBαt

Note that max0 Sαt ge 0 and Mα

t ge Bαt B

α0t ge 0 for all t Most importantly

the local behavior of Bα0t is exactly like Brownian motion with drift parameter minusα

since max0 Sαt is simply a constant when Bα

t ∕= 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 17

We have now provided sufficient modifications of Brownian motion and we nextmove onmdashwhat we will ultimately use to approximate heavy traffic waiting timesThe next example provides us with the distribution of the supremum of a Brownianmotion with negative drift

Example 314Let Bt be a Brownian motion with drift α lt 0 and variance parameter σ2 Define

Mt = sup0leslet

Bs

to be the running maximum then

Minfin sim exp

9830612|α|σ2

983062

that is

PW ge w = eminusminus2|α|σ2 w w ge 0

E [Minfin] =σ2

2|α|

Prooffor constants a b gt 0 Let T (minusa b) be the first time that B minus t hits minusa or b

T (minusa b) = inft Bt = minusa or Bt = bIn [7] it is proved that

P983051BT (minusab)=b

983052=

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)

Since α lt 0 exp(2αaσ2) rarr 0 we have

limararrinfin

PBT (minusab) = b = limararrinfin

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)= exp(2αbσ2)

The left-hand side becomes the probability that the process will reach b somewherealong its path (ie that the maximum of the process exceeds b somewhere alongits path) Therefore we have

PW ge b = exp(2αbσ2) = expminus2|α|bσ2

This example shows us that the supremum of a Brownian motion observed ldquofor-everrdquo (ie on the interval [0infin)) is exponentially distributed with rate 2|α|σ2

4 Heavy Traffic Approximation

We now have all we need to provide a heuristic argument for the approximationof single-servers queue in heavy traffic using Brownian Motion For a rigorousalbeit opaque treatment of this approximation see [8] the seminal work on thissubject by Kingman

Theorem 41 The waiting time of the nth customer in an MM1 queue canbe approximated by the supremum of reflected Brownian motion with negative driftand boundary at 0

18 HARVEY BARNHARD

ProofRecall that an MM1 queue is a process where the number of customer arrivalsfollows a poisson process with rate λ and service times are exponentially distributedwith rate micro From Lindleyrsquos equation in Section 2 we can write the waiting timefor the nth customer as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

where U (n) = S(n)minus I(n) is the difference between the nth service time and the nthinter-arrival time Define the partial sum P

(n)k to be zero when k = 0 and

P(n)k equiv

k983131

i=1

U (nminusi) for k ge 1

We can then rewrite the waiting time of the nth customer as

W (n)q = max

0lekle(nminus1)P

(n)k

The fact that P(n)k is the kth value of a random walk follows immediately from

U (1) U (nminus1) being iid random variables Let α and σ2 be the expectationand variance of U (i) We calculate α and σ2 by first noting that S(i) and I(i) areexponentially distributed with rates micro and λ Recall that ρ = λmicro is the trafficintensity as defined in the first section of this paper

α = E983147U (i)

983148= E

983147S(i)

983148minus E

983147I(i)

983148=

1

microminus 1

λ=

ρminus 1

λ

σ2 = Var983153U (i)

983154= Var

983153S(i)

983154+Var

983153I(i)

983154=

1

micro2+

1

λ2

We next compute the expectation and variance of P(n)k using the fact that U (i) are

iid random variables

E983147P

(n)k

983148=

k983131

i=1

E983147U (nminusi)

983148= kα

Var983153P

(n)k

983154=

k983131

i=1

Var983153U (nminusi)

983154= kσ2

For a finite n W(n)q is therefore the maximum value of a random walk with n steps

We now show that Pnk can be approximated by Brownian motion with drift

parameter α and variance parameter σ2 when n is sufficiently large We restate theproperties of such a Brownian motion Bt below

(1) B0 = 0(2) Bt has stationary and independent increments(3) Bt sim N(αtσ2t)(4) t 983041rarr Bt is continuous with probability one

The first property is clear as P(n)0 = 0 for all n The second property follows from

the fact that U (i) are iid random variables The third property is a result of thecentral limit theorem for iid sequences the result of which is provided below

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

12 HARVEY BARNHARD

332 Skorokhod EmbeddingTo construct Brownian motion out of random walks we must first construct randomwalks out of Brownian motion Structuring the proof in this matter is not withoutreason there are uncountably many realizations of Brownian motion so we want toensure that the random walks to which we are taking the limit do in fact convergeto the Brownian motion of interest Such a procedure of deriving a random walkfrom a Brownian motion is called Skorokhod embedding The procedure is given inthe following pseudocode

Set t0 = 0 and n = 0 Let T be a stopping time of Brownian motion Btso that Bt is defined on [0 T ] Let R0 = 0 be the initial value

of the Skorokhod embedding

While tn + t le TInitialize t = 0

While |Bt minusBtn | le 1 Continue Bt until |Bt minusBtn | = 1 for the first time

Set t(n+1) = inft |Bt minusBtn | = 1If Bt(n+1)

minusBtn = 1 then set X(n+1) = 1

Otherwise Bt(n+1)minusBtn = minus1 so set X(n+1) = minus1

Let the (n+ 1)th value of the Skorokhod embedding be

R(n+1) = Rn +Xn+1

Increment to the next step of the inner while loop n = n+ 1

The resulting sequence Rnn=12 is the Skorokhod embedding of Bttisin[0T ]

Theorem 36 Let Bt be a standard Brownian motion Set t0 = 0 and let tn bethe stopping time where |Bt| = 1 for the nth time

tn equiv inft ge 0 |Bt minusBtn | = 1

If we define Rn equiv Btn then Rn is a simple random walk

ProofTo prove that B0 Bt1 Bt2 is a simple random walk we merely need to showthat

(1) PBtn+1minusBtn = 1 = PBtn+1

minusBtn = minus1 = 12 for all n isin N

(2) tn+1 minus tn are iid random variables

The first item follows from the symmetric property of Brownian increments prop-erty (II) in the definition of Brownian motion Bs+t minusBs sim N(0σ2t)

The second item follows from the strong Markov property of Brownian motionwhich states that if τ is a stopping time with respect to Bt then Bt minus Bτ is

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 13

also a Brownian motion We do not include a rigorous treatment of stopping timesor a proof of the strong Markov property

We now state a theorem that places a probabilistic bound on the differencebetween a Brownian motion and its Skorokhod embedding

Theorem 37 Define Θ to be the maximum distance between a Brownian motionand its Skorokhod embedding

Θ(BRT ) = max0letleT

|Bt minusRt|

Note that Θ is equivalent to the distance between Bt and Rt in the space ofcontinuous functions on the time interval [0 T ] equipped with the supremum normThere exist c a isin [0infin) such that for all r le n14 and all integers n ge 3

P983153Θ(BRT ) ge rn14

983155log n

983154le ceminusar

ProofIt will suffice to prove the theorem for r ge 9c2 where c is the constant such that

P983153osc(B δ T ) gt r

983155δ log(1δ)

983154le cT δ(rc)

2

andosc(B δ T ) = sup |Bt minusBs| s t isin D s t isin [0 T ] |sminus t| le δ

is the oscillation of Bt restricted to t isin D the dyadic rationals The proof thatsuch a c exists can be found in [3] page 68 where it is also shown that for n isin N

Θ(BSn) le 1 + osc(B 1 n) + max|Bj minusBτj | j = 1 nNow suppose 9c2 le r le n14 If |Bn minusBτn | is large then either |nminus τn| is large

or the oscillation of B is large Consider the three events

(1)983051Θ(BRn) ge rn14

radiclog n

983052

(2)983051osc(B r

radicn 2n) ge (r3)n14

radiclog n

983052

(3) max1lejlen |τj minus j| ge rradicn

From above we know that event (1) is contained in the union of events (2) and (3)As a consequence we need only prove the result of the theorem for events (2) and(3) We first tackle event (2)

P983153osc(B r

radicn 2n) gt (r3)n(14)

983155log n

983154le 3P

983153osc(B r

radicn n) gt (r3)n(14)

983155log n

983154

= 3P983153osc(B rnminus12) gt (r3)nminus(14)

983155log n

983154

le 3P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log

983043n12r

983044983070

Ifradicr3 ge c and r le n14 we can conclude that there exist c and a such that

P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log(n12)

983070le ceminusar logn

For event (3) we refer to a proof on page 266 in the appendix of [3] to show thatthere exist c a such that

P983069

max1lejlen

|τj minus j| gt rradicn

983070le ceminusar2

14 HARVEY BARNHARD

333 Strong ApproximationWith preliminaries in order we now show that Brownian motion can be thought ofas a limit of simple random walks a construction known as the strong approximationof Brownian motion Let Bt be a standard Brownian motion with variance

parameter 1 as defined in the probability space (ΩF P) and let B(n)t be the

scaled Brownian motion

B(n)t =

1radicnBnt

From B(n) we use the Skorokhod embedding to derive the simple random walkS(n) And from S(n) we derive the scaled simple random walk R(n)

R(n)t =

1radicnS(n)nt

From Theorem 37 we know that there exists c a isin [0infin) such that for all positiveintegers T

P983069

max0letleTn

983055983055983055S(n)t minusB

(n)t

983055983055983055 ge cr(Tn)14983155log (Tn)

983070le ceminusar

multiplying by 1radicn this becomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge crT 14nminus14983155log (Tn)

983070le ceminusar

Letting r = c log n where c is sufficiently large for the given T the inequalitybecomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

983070le c

n2

To proceed we require the use of the Borel-Cantelli lemma the result of which isprovided below

Lemma 38 (Borel-Cantelli)If An is a sequence of events in Ω and

983123infini=1 PAn lt infin then

PAn io = 0

where An io = lim supnrarrinfin An is the event that infinitely many An occur

If we set

An = max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

we can then apply the Borel-Cantelli lemma to the above inequality to concludethat

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 le cnminus14 log32 n

with probability one for all n sufficiently large Since

limnrarrinfin

cnminus14 log32 n = 0

we conclude that

R(n)t

asminusminusrarr Bt in C[0 T ]The following theorem condenses the results of this section up to this point

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 15

Theorem 39 (Donskerrsquos Theorem) Let Xi be a sequence of iid random vari-

ables with mean 0 and variance σ2 If R(n)t is a scaled random walk as defined

above then R(n)t rArr Bt where Bt is a standard Brownian motion This result is

also known as the functional central limit theorem

34 Brownian Motion VariantsThe preceding construction of Brownian motion will prove particularly useful in ourapproximation of heavy traffic waiting times However this process requires furthermodification Recall that when traffic intensity ρ is slightly less than 1 customersare being serviced at a slightly faster rate then they are entering the queue Thusgiven enough time traffic will eventually ldquoclear outrdquo due to faster service timesHowever this particular construction of Brownian motion has increments of meanzero

E[Bs+t minusBs] = 0

meaning that the approximate number of customers in line wouldmdashon averagemdashremain unchanged regardless of how long the line is observed Such a propertyis called the martingale property Instead we want a process that exhibits thesupermartingale property

E[Bs+t minusBs] lt 0

Definition 310 Let Fn be an increasing sequence of σ-fields (Fn is called afiltration) Let Xn be a sequence of random variables with finite mean andXn isin Fn for all n Martingales supermartingales and submartingales are thendefined as follows for all n

Martingale E [Xn+1 | Fn] = Xn

Supermartingale E [Xn+1 | Fn] le Xn

Submartingale E [Xn+1 | Fn] ge Xn

In words a sequence that we expect to remain the same over time is a martingalea sequence we expect to increase is a submartingale and a sequence we expect todecrease is a supermartingale To approximate heavy traffic we want a Brownianmotion-like process that exhibits the supermartingale property

Definition 311 A Brownian motion with drift parameter α and variance param-eter σ2 Bα

t is defined by the following properties

(1) Independent increments(2) Increment lengths are normally distributed with mean αt and variance σ2t(3) the path t 983041rarr Bα

t is continuous

Note that these properties are the same as standard Brownian motion exceptincrement lengths now have non-zero means Since we have already shown theexistence of standard Brownian motion the construction of Brownian motion withdrift is now easy

Lemma 312 Let Bt be a standard Brownian motion with variance parameterσ2 then Bα

t = Bt + αt is Brownian motion with drift parameter α and varianceparameter σ2 Moreover Bα

t fulfills the supermartingale property when α lt 0

16 HARVEY BARNHARD

ProofIndependence of increments follows directly from independence of increments forBt and the fact that αt is a constant for a given t To prove that Bα

t has stationaryincrements we have to show that the distribution of increments Bα

s+tminusBαs depends

only the time interval t and not the absolute time Recall from the definition of Bt

that Bs+t minusBt are normal random variables with mean 0 and variance t for s gt 0

E983045Bα

s+t minusBαs

983046= E

983045Bα

s+t

983046minus E [Bα

s ]

= E983045Bα

s+t + α(s+ t)983046minus E [Bs + αs]

= α(s+ t)minus αs

= αt

Var983051Bα

s+t minusBαs

983052= Var Bs+t minusBs + αt= Var Bs+t minusBs= σ2t

And since for fixed t αt is just a constant we use the property of normal randomvariables to conclude that

Bαs+t minusBα

s sim N(αtσ2t) for all s gt 0

Continuity of Bαt directly follows from continuity of Bt We have shown that

Bαt constructed as the sum of standard Brownian motion and drift αt is indeed

Brownian motion with drift

Brownian motion with negative drift provides a better approximation of thewaiting-time of customers in a queue than does standard Brownian motion How-ever the unboundedness of Brownian motion suggests that waiting times couldperhaps be negative In order to circumvent this issue we instead use reflectedBrownian motion with drift Reflected Brownian motion acts much the same asBrownian motion except on a given boundary off of which the Brownian motionismdashunsurprisinglymdashreflected In the one-dimensional case the boundaries may beany interval on the reals that provide a lower and upper bound to the Brownianmotion For our purposes we only use a single boundary zero that gives a lowerbound to Brownian motion That is for all t ge 0 we want Bα

t ge 0

Definition 313 Let Bαt be a standard Brownian motion with drift parameter α

and let Mαt be the running maximum of Bα

t

Mαt = sup

0lesletBα

s

Then reflected Brownian motion with drift parameter minusα and boundary 0 definedon R+ can be constructed as follows

Rα0t equiv max0Mα

t minusBαt

Note that max0 Sαt ge 0 and Mα

t ge Bαt B

α0t ge 0 for all t Most importantly

the local behavior of Bα0t is exactly like Brownian motion with drift parameter minusα

since max0 Sαt is simply a constant when Bα

t ∕= 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 17

We have now provided sufficient modifications of Brownian motion and we nextmove onmdashwhat we will ultimately use to approximate heavy traffic waiting timesThe next example provides us with the distribution of the supremum of a Brownianmotion with negative drift

Example 314Let Bt be a Brownian motion with drift α lt 0 and variance parameter σ2 Define

Mt = sup0leslet

Bs

to be the running maximum then

Minfin sim exp

9830612|α|σ2

983062

that is

PW ge w = eminusminus2|α|σ2 w w ge 0

E [Minfin] =σ2

2|α|

Prooffor constants a b gt 0 Let T (minusa b) be the first time that B minus t hits minusa or b

T (minusa b) = inft Bt = minusa or Bt = bIn [7] it is proved that

P983051BT (minusab)=b

983052=

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)

Since α lt 0 exp(2αaσ2) rarr 0 we have

limararrinfin

PBT (minusab) = b = limararrinfin

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)= exp(2αbσ2)

The left-hand side becomes the probability that the process will reach b somewherealong its path (ie that the maximum of the process exceeds b somewhere alongits path) Therefore we have

PW ge b = exp(2αbσ2) = expminus2|α|bσ2

This example shows us that the supremum of a Brownian motion observed ldquofor-everrdquo (ie on the interval [0infin)) is exponentially distributed with rate 2|α|σ2

4 Heavy Traffic Approximation

We now have all we need to provide a heuristic argument for the approximationof single-servers queue in heavy traffic using Brownian Motion For a rigorousalbeit opaque treatment of this approximation see [8] the seminal work on thissubject by Kingman

Theorem 41 The waiting time of the nth customer in an MM1 queue canbe approximated by the supremum of reflected Brownian motion with negative driftand boundary at 0

18 HARVEY BARNHARD

ProofRecall that an MM1 queue is a process where the number of customer arrivalsfollows a poisson process with rate λ and service times are exponentially distributedwith rate micro From Lindleyrsquos equation in Section 2 we can write the waiting timefor the nth customer as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

where U (n) = S(n)minus I(n) is the difference between the nth service time and the nthinter-arrival time Define the partial sum P

(n)k to be zero when k = 0 and

P(n)k equiv

k983131

i=1

U (nminusi) for k ge 1

We can then rewrite the waiting time of the nth customer as

W (n)q = max

0lekle(nminus1)P

(n)k

The fact that P(n)k is the kth value of a random walk follows immediately from

U (1) U (nminus1) being iid random variables Let α and σ2 be the expectationand variance of U (i) We calculate α and σ2 by first noting that S(i) and I(i) areexponentially distributed with rates micro and λ Recall that ρ = λmicro is the trafficintensity as defined in the first section of this paper

α = E983147U (i)

983148= E

983147S(i)

983148minus E

983147I(i)

983148=

1

microminus 1

λ=

ρminus 1

λ

σ2 = Var983153U (i)

983154= Var

983153S(i)

983154+Var

983153I(i)

983154=

1

micro2+

1

λ2

We next compute the expectation and variance of P(n)k using the fact that U (i) are

iid random variables

E983147P

(n)k

983148=

k983131

i=1

E983147U (nminusi)

983148= kα

Var983153P

(n)k

983154=

k983131

i=1

Var983153U (nminusi)

983154= kσ2

For a finite n W(n)q is therefore the maximum value of a random walk with n steps

We now show that Pnk can be approximated by Brownian motion with drift

parameter α and variance parameter σ2 when n is sufficiently large We restate theproperties of such a Brownian motion Bt below

(1) B0 = 0(2) Bt has stationary and independent increments(3) Bt sim N(αtσ2t)(4) t 983041rarr Bt is continuous with probability one

The first property is clear as P(n)0 = 0 for all n The second property follows from

the fact that U (i) are iid random variables The third property is a result of thecentral limit theorem for iid sequences the result of which is provided below

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 13

also a Brownian motion We do not include a rigorous treatment of stopping timesor a proof of the strong Markov property

We now state a theorem that places a probabilistic bound on the differencebetween a Brownian motion and its Skorokhod embedding

Theorem 37 Define Θ to be the maximum distance between a Brownian motionand its Skorokhod embedding

Θ(BRT ) = max0letleT

|Bt minusRt|

Note that Θ is equivalent to the distance between Bt and Rt in the space ofcontinuous functions on the time interval [0 T ] equipped with the supremum normThere exist c a isin [0infin) such that for all r le n14 and all integers n ge 3

P983153Θ(BRT ) ge rn14

983155log n

983154le ceminusar

ProofIt will suffice to prove the theorem for r ge 9c2 where c is the constant such that

P983153osc(B δ T ) gt r

983155δ log(1δ)

983154le cT δ(rc)

2

andosc(B δ T ) = sup |Bt minusBs| s t isin D s t isin [0 T ] |sminus t| le δ

is the oscillation of Bt restricted to t isin D the dyadic rationals The proof thatsuch a c exists can be found in [3] page 68 where it is also shown that for n isin N

Θ(BSn) le 1 + osc(B 1 n) + max|Bj minusBτj | j = 1 nNow suppose 9c2 le r le n14 If |Bn minusBτn | is large then either |nminus τn| is large

or the oscillation of B is large Consider the three events

(1)983051Θ(BRn) ge rn14

radiclog n

983052

(2)983051osc(B r

radicn 2n) ge (r3)n14

radiclog n

983052

(3) max1lejlen |τj minus j| ge rradicn

From above we know that event (1) is contained in the union of events (2) and (3)As a consequence we need only prove the result of the theorem for events (2) and(3) We first tackle event (2)

P983153osc(B r

radicn 2n) gt (r3)n(14)

983155log n

983154le 3P

983153osc(B r

radicn n) gt (r3)n(14)

983155log n

983154

= 3P983153osc(B rnminus12) gt (r3)nminus(14)

983155log n

983154

le 3P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log

983043n12r

983044983070

Ifradicr3 ge c and r le n14 we can conclude that there exist c and a such that

P983069osc(B rnminus12) gt (

radicr3)

983156rnminus12 log(n12)

983070le ceminusar logn

For event (3) we refer to a proof on page 266 in the appendix of [3] to show thatthere exist c a such that

P983069

max1lejlen

|τj minus j| gt rradicn

983070le ceminusar2

14 HARVEY BARNHARD

333 Strong ApproximationWith preliminaries in order we now show that Brownian motion can be thought ofas a limit of simple random walks a construction known as the strong approximationof Brownian motion Let Bt be a standard Brownian motion with variance

parameter 1 as defined in the probability space (ΩF P) and let B(n)t be the

scaled Brownian motion

B(n)t =

1radicnBnt

From B(n) we use the Skorokhod embedding to derive the simple random walkS(n) And from S(n) we derive the scaled simple random walk R(n)

R(n)t =

1radicnS(n)nt

From Theorem 37 we know that there exists c a isin [0infin) such that for all positiveintegers T

P983069

max0letleTn

983055983055983055S(n)t minusB

(n)t

983055983055983055 ge cr(Tn)14983155log (Tn)

983070le ceminusar

multiplying by 1radicn this becomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge crT 14nminus14983155log (Tn)

983070le ceminusar

Letting r = c log n where c is sufficiently large for the given T the inequalitybecomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

983070le c

n2

To proceed we require the use of the Borel-Cantelli lemma the result of which isprovided below

Lemma 38 (Borel-Cantelli)If An is a sequence of events in Ω and

983123infini=1 PAn lt infin then

PAn io = 0

where An io = lim supnrarrinfin An is the event that infinitely many An occur

If we set

An = max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

we can then apply the Borel-Cantelli lemma to the above inequality to concludethat

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 le cnminus14 log32 n

with probability one for all n sufficiently large Since

limnrarrinfin

cnminus14 log32 n = 0

we conclude that

R(n)t

asminusminusrarr Bt in C[0 T ]The following theorem condenses the results of this section up to this point

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 15

Theorem 39 (Donskerrsquos Theorem) Let Xi be a sequence of iid random vari-

ables with mean 0 and variance σ2 If R(n)t is a scaled random walk as defined

above then R(n)t rArr Bt where Bt is a standard Brownian motion This result is

also known as the functional central limit theorem

34 Brownian Motion VariantsThe preceding construction of Brownian motion will prove particularly useful in ourapproximation of heavy traffic waiting times However this process requires furthermodification Recall that when traffic intensity ρ is slightly less than 1 customersare being serviced at a slightly faster rate then they are entering the queue Thusgiven enough time traffic will eventually ldquoclear outrdquo due to faster service timesHowever this particular construction of Brownian motion has increments of meanzero

E[Bs+t minusBs] = 0

meaning that the approximate number of customers in line wouldmdashon averagemdashremain unchanged regardless of how long the line is observed Such a propertyis called the martingale property Instead we want a process that exhibits thesupermartingale property

E[Bs+t minusBs] lt 0

Definition 310 Let Fn be an increasing sequence of σ-fields (Fn is called afiltration) Let Xn be a sequence of random variables with finite mean andXn isin Fn for all n Martingales supermartingales and submartingales are thendefined as follows for all n

Martingale E [Xn+1 | Fn] = Xn

Supermartingale E [Xn+1 | Fn] le Xn

Submartingale E [Xn+1 | Fn] ge Xn

In words a sequence that we expect to remain the same over time is a martingalea sequence we expect to increase is a submartingale and a sequence we expect todecrease is a supermartingale To approximate heavy traffic we want a Brownianmotion-like process that exhibits the supermartingale property

Definition 311 A Brownian motion with drift parameter α and variance param-eter σ2 Bα

t is defined by the following properties

(1) Independent increments(2) Increment lengths are normally distributed with mean αt and variance σ2t(3) the path t 983041rarr Bα

t is continuous

Note that these properties are the same as standard Brownian motion exceptincrement lengths now have non-zero means Since we have already shown theexistence of standard Brownian motion the construction of Brownian motion withdrift is now easy

Lemma 312 Let Bt be a standard Brownian motion with variance parameterσ2 then Bα

t = Bt + αt is Brownian motion with drift parameter α and varianceparameter σ2 Moreover Bα

t fulfills the supermartingale property when α lt 0

16 HARVEY BARNHARD

ProofIndependence of increments follows directly from independence of increments forBt and the fact that αt is a constant for a given t To prove that Bα

t has stationaryincrements we have to show that the distribution of increments Bα

s+tminusBαs depends

only the time interval t and not the absolute time Recall from the definition of Bt

that Bs+t minusBt are normal random variables with mean 0 and variance t for s gt 0

E983045Bα

s+t minusBαs

983046= E

983045Bα

s+t

983046minus E [Bα

s ]

= E983045Bα

s+t + α(s+ t)983046minus E [Bs + αs]

= α(s+ t)minus αs

= αt

Var983051Bα

s+t minusBαs

983052= Var Bs+t minusBs + αt= Var Bs+t minusBs= σ2t

And since for fixed t αt is just a constant we use the property of normal randomvariables to conclude that

Bαs+t minusBα

s sim N(αtσ2t) for all s gt 0

Continuity of Bαt directly follows from continuity of Bt We have shown that

Bαt constructed as the sum of standard Brownian motion and drift αt is indeed

Brownian motion with drift

Brownian motion with negative drift provides a better approximation of thewaiting-time of customers in a queue than does standard Brownian motion How-ever the unboundedness of Brownian motion suggests that waiting times couldperhaps be negative In order to circumvent this issue we instead use reflectedBrownian motion with drift Reflected Brownian motion acts much the same asBrownian motion except on a given boundary off of which the Brownian motionismdashunsurprisinglymdashreflected In the one-dimensional case the boundaries may beany interval on the reals that provide a lower and upper bound to the Brownianmotion For our purposes we only use a single boundary zero that gives a lowerbound to Brownian motion That is for all t ge 0 we want Bα

t ge 0

Definition 313 Let Bαt be a standard Brownian motion with drift parameter α

and let Mαt be the running maximum of Bα

t

Mαt = sup

0lesletBα

s

Then reflected Brownian motion with drift parameter minusα and boundary 0 definedon R+ can be constructed as follows

Rα0t equiv max0Mα

t minusBαt

Note that max0 Sαt ge 0 and Mα

t ge Bαt B

α0t ge 0 for all t Most importantly

the local behavior of Bα0t is exactly like Brownian motion with drift parameter minusα

since max0 Sαt is simply a constant when Bα

t ∕= 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 17

We have now provided sufficient modifications of Brownian motion and we nextmove onmdashwhat we will ultimately use to approximate heavy traffic waiting timesThe next example provides us with the distribution of the supremum of a Brownianmotion with negative drift

Example 314Let Bt be a Brownian motion with drift α lt 0 and variance parameter σ2 Define

Mt = sup0leslet

Bs

to be the running maximum then

Minfin sim exp

9830612|α|σ2

983062

that is

PW ge w = eminusminus2|α|σ2 w w ge 0

E [Minfin] =σ2

2|α|

Prooffor constants a b gt 0 Let T (minusa b) be the first time that B minus t hits minusa or b

T (minusa b) = inft Bt = minusa or Bt = bIn [7] it is proved that

P983051BT (minusab)=b

983052=

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)

Since α lt 0 exp(2αaσ2) rarr 0 we have

limararrinfin

PBT (minusab) = b = limararrinfin

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)= exp(2αbσ2)

The left-hand side becomes the probability that the process will reach b somewherealong its path (ie that the maximum of the process exceeds b somewhere alongits path) Therefore we have

PW ge b = exp(2αbσ2) = expminus2|α|bσ2

This example shows us that the supremum of a Brownian motion observed ldquofor-everrdquo (ie on the interval [0infin)) is exponentially distributed with rate 2|α|σ2

4 Heavy Traffic Approximation

We now have all we need to provide a heuristic argument for the approximationof single-servers queue in heavy traffic using Brownian Motion For a rigorousalbeit opaque treatment of this approximation see [8] the seminal work on thissubject by Kingman

Theorem 41 The waiting time of the nth customer in an MM1 queue canbe approximated by the supremum of reflected Brownian motion with negative driftand boundary at 0

18 HARVEY BARNHARD

ProofRecall that an MM1 queue is a process where the number of customer arrivalsfollows a poisson process with rate λ and service times are exponentially distributedwith rate micro From Lindleyrsquos equation in Section 2 we can write the waiting timefor the nth customer as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

where U (n) = S(n)minus I(n) is the difference between the nth service time and the nthinter-arrival time Define the partial sum P

(n)k to be zero when k = 0 and

P(n)k equiv

k983131

i=1

U (nminusi) for k ge 1

We can then rewrite the waiting time of the nth customer as

W (n)q = max

0lekle(nminus1)P

(n)k

The fact that P(n)k is the kth value of a random walk follows immediately from

U (1) U (nminus1) being iid random variables Let α and σ2 be the expectationand variance of U (i) We calculate α and σ2 by first noting that S(i) and I(i) areexponentially distributed with rates micro and λ Recall that ρ = λmicro is the trafficintensity as defined in the first section of this paper

α = E983147U (i)

983148= E

983147S(i)

983148minus E

983147I(i)

983148=

1

microminus 1

λ=

ρminus 1

λ

σ2 = Var983153U (i)

983154= Var

983153S(i)

983154+Var

983153I(i)

983154=

1

micro2+

1

λ2

We next compute the expectation and variance of P(n)k using the fact that U (i) are

iid random variables

E983147P

(n)k

983148=

k983131

i=1

E983147U (nminusi)

983148= kα

Var983153P

(n)k

983154=

k983131

i=1

Var983153U (nminusi)

983154= kσ2

For a finite n W(n)q is therefore the maximum value of a random walk with n steps

We now show that Pnk can be approximated by Brownian motion with drift

parameter α and variance parameter σ2 when n is sufficiently large We restate theproperties of such a Brownian motion Bt below

(1) B0 = 0(2) Bt has stationary and independent increments(3) Bt sim N(αtσ2t)(4) t 983041rarr Bt is continuous with probability one

The first property is clear as P(n)0 = 0 for all n The second property follows from

the fact that U (i) are iid random variables The third property is a result of thecentral limit theorem for iid sequences the result of which is provided below

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

14 HARVEY BARNHARD

333 Strong ApproximationWith preliminaries in order we now show that Brownian motion can be thought ofas a limit of simple random walks a construction known as the strong approximationof Brownian motion Let Bt be a standard Brownian motion with variance

parameter 1 as defined in the probability space (ΩF P) and let B(n)t be the

scaled Brownian motion

B(n)t =

1radicnBnt

From B(n) we use the Skorokhod embedding to derive the simple random walkS(n) And from S(n) we derive the scaled simple random walk R(n)

R(n)t =

1radicnS(n)nt

From Theorem 37 we know that there exists c a isin [0infin) such that for all positiveintegers T

P983069

max0letleTn

983055983055983055S(n)t minusB

(n)t

983055983055983055 ge cr(Tn)14983155log (Tn)

983070le ceminusar

multiplying by 1radicn this becomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge crT 14nminus14983155log (Tn)

983070le ceminusar

Letting r = c log n where c is sufficiently large for the given T the inequalitybecomes

P983069

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

983070le c

n2

To proceed we require the use of the Borel-Cantelli lemma the result of which isprovided below

Lemma 38 (Borel-Cantelli)If An is a sequence of events in Ω and

983123infini=1 PAn lt infin then

PAn io = 0

where An io = lim supnrarrinfin An is the event that infinitely many An occur

If we set

An = max0letleT

983055983055983055R(n)t minusBt

983055983055983055 ge cnminus14 log32 n

we can then apply the Borel-Cantelli lemma to the above inequality to concludethat

max0letleT

983055983055983055R(n)t minusBt

983055983055983055 le cnminus14 log32 n

with probability one for all n sufficiently large Since

limnrarrinfin

cnminus14 log32 n = 0

we conclude that

R(n)t

asminusminusrarr Bt in C[0 T ]The following theorem condenses the results of this section up to this point

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 15

Theorem 39 (Donskerrsquos Theorem) Let Xi be a sequence of iid random vari-

ables with mean 0 and variance σ2 If R(n)t is a scaled random walk as defined

above then R(n)t rArr Bt where Bt is a standard Brownian motion This result is

also known as the functional central limit theorem

34 Brownian Motion VariantsThe preceding construction of Brownian motion will prove particularly useful in ourapproximation of heavy traffic waiting times However this process requires furthermodification Recall that when traffic intensity ρ is slightly less than 1 customersare being serviced at a slightly faster rate then they are entering the queue Thusgiven enough time traffic will eventually ldquoclear outrdquo due to faster service timesHowever this particular construction of Brownian motion has increments of meanzero

E[Bs+t minusBs] = 0

meaning that the approximate number of customers in line wouldmdashon averagemdashremain unchanged regardless of how long the line is observed Such a propertyis called the martingale property Instead we want a process that exhibits thesupermartingale property

E[Bs+t minusBs] lt 0

Definition 310 Let Fn be an increasing sequence of σ-fields (Fn is called afiltration) Let Xn be a sequence of random variables with finite mean andXn isin Fn for all n Martingales supermartingales and submartingales are thendefined as follows for all n

Martingale E [Xn+1 | Fn] = Xn

Supermartingale E [Xn+1 | Fn] le Xn

Submartingale E [Xn+1 | Fn] ge Xn

In words a sequence that we expect to remain the same over time is a martingalea sequence we expect to increase is a submartingale and a sequence we expect todecrease is a supermartingale To approximate heavy traffic we want a Brownianmotion-like process that exhibits the supermartingale property

Definition 311 A Brownian motion with drift parameter α and variance param-eter σ2 Bα

t is defined by the following properties

(1) Independent increments(2) Increment lengths are normally distributed with mean αt and variance σ2t(3) the path t 983041rarr Bα

t is continuous

Note that these properties are the same as standard Brownian motion exceptincrement lengths now have non-zero means Since we have already shown theexistence of standard Brownian motion the construction of Brownian motion withdrift is now easy

Lemma 312 Let Bt be a standard Brownian motion with variance parameterσ2 then Bα

t = Bt + αt is Brownian motion with drift parameter α and varianceparameter σ2 Moreover Bα

t fulfills the supermartingale property when α lt 0

16 HARVEY BARNHARD

ProofIndependence of increments follows directly from independence of increments forBt and the fact that αt is a constant for a given t To prove that Bα

t has stationaryincrements we have to show that the distribution of increments Bα

s+tminusBαs depends

only the time interval t and not the absolute time Recall from the definition of Bt

that Bs+t minusBt are normal random variables with mean 0 and variance t for s gt 0

E983045Bα

s+t minusBαs

983046= E

983045Bα

s+t

983046minus E [Bα

s ]

= E983045Bα

s+t + α(s+ t)983046minus E [Bs + αs]

= α(s+ t)minus αs

= αt

Var983051Bα

s+t minusBαs

983052= Var Bs+t minusBs + αt= Var Bs+t minusBs= σ2t

And since for fixed t αt is just a constant we use the property of normal randomvariables to conclude that

Bαs+t minusBα

s sim N(αtσ2t) for all s gt 0

Continuity of Bαt directly follows from continuity of Bt We have shown that

Bαt constructed as the sum of standard Brownian motion and drift αt is indeed

Brownian motion with drift

Brownian motion with negative drift provides a better approximation of thewaiting-time of customers in a queue than does standard Brownian motion How-ever the unboundedness of Brownian motion suggests that waiting times couldperhaps be negative In order to circumvent this issue we instead use reflectedBrownian motion with drift Reflected Brownian motion acts much the same asBrownian motion except on a given boundary off of which the Brownian motionismdashunsurprisinglymdashreflected In the one-dimensional case the boundaries may beany interval on the reals that provide a lower and upper bound to the Brownianmotion For our purposes we only use a single boundary zero that gives a lowerbound to Brownian motion That is for all t ge 0 we want Bα

t ge 0

Definition 313 Let Bαt be a standard Brownian motion with drift parameter α

and let Mαt be the running maximum of Bα

t

Mαt = sup

0lesletBα

s

Then reflected Brownian motion with drift parameter minusα and boundary 0 definedon R+ can be constructed as follows

Rα0t equiv max0Mα

t minusBαt

Note that max0 Sαt ge 0 and Mα

t ge Bαt B

α0t ge 0 for all t Most importantly

the local behavior of Bα0t is exactly like Brownian motion with drift parameter minusα

since max0 Sαt is simply a constant when Bα

t ∕= 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 17

We have now provided sufficient modifications of Brownian motion and we nextmove onmdashwhat we will ultimately use to approximate heavy traffic waiting timesThe next example provides us with the distribution of the supremum of a Brownianmotion with negative drift

Example 314Let Bt be a Brownian motion with drift α lt 0 and variance parameter σ2 Define

Mt = sup0leslet

Bs

to be the running maximum then

Minfin sim exp

9830612|α|σ2

983062

that is

PW ge w = eminusminus2|α|σ2 w w ge 0

E [Minfin] =σ2

2|α|

Prooffor constants a b gt 0 Let T (minusa b) be the first time that B minus t hits minusa or b

T (minusa b) = inft Bt = minusa or Bt = bIn [7] it is proved that

P983051BT (minusab)=b

983052=

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)

Since α lt 0 exp(2αaσ2) rarr 0 we have

limararrinfin

PBT (minusab) = b = limararrinfin

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)= exp(2αbσ2)

The left-hand side becomes the probability that the process will reach b somewherealong its path (ie that the maximum of the process exceeds b somewhere alongits path) Therefore we have

PW ge b = exp(2αbσ2) = expminus2|α|bσ2

This example shows us that the supremum of a Brownian motion observed ldquofor-everrdquo (ie on the interval [0infin)) is exponentially distributed with rate 2|α|σ2

4 Heavy Traffic Approximation

We now have all we need to provide a heuristic argument for the approximationof single-servers queue in heavy traffic using Brownian Motion For a rigorousalbeit opaque treatment of this approximation see [8] the seminal work on thissubject by Kingman

Theorem 41 The waiting time of the nth customer in an MM1 queue canbe approximated by the supremum of reflected Brownian motion with negative driftand boundary at 0

18 HARVEY BARNHARD

ProofRecall that an MM1 queue is a process where the number of customer arrivalsfollows a poisson process with rate λ and service times are exponentially distributedwith rate micro From Lindleyrsquos equation in Section 2 we can write the waiting timefor the nth customer as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

where U (n) = S(n)minus I(n) is the difference between the nth service time and the nthinter-arrival time Define the partial sum P

(n)k to be zero when k = 0 and

P(n)k equiv

k983131

i=1

U (nminusi) for k ge 1

We can then rewrite the waiting time of the nth customer as

W (n)q = max

0lekle(nminus1)P

(n)k

The fact that P(n)k is the kth value of a random walk follows immediately from

U (1) U (nminus1) being iid random variables Let α and σ2 be the expectationand variance of U (i) We calculate α and σ2 by first noting that S(i) and I(i) areexponentially distributed with rates micro and λ Recall that ρ = λmicro is the trafficintensity as defined in the first section of this paper

α = E983147U (i)

983148= E

983147S(i)

983148minus E

983147I(i)

983148=

1

microminus 1

λ=

ρminus 1

λ

σ2 = Var983153U (i)

983154= Var

983153S(i)

983154+Var

983153I(i)

983154=

1

micro2+

1

λ2

We next compute the expectation and variance of P(n)k using the fact that U (i) are

iid random variables

E983147P

(n)k

983148=

k983131

i=1

E983147U (nminusi)

983148= kα

Var983153P

(n)k

983154=

k983131

i=1

Var983153U (nminusi)

983154= kσ2

For a finite n W(n)q is therefore the maximum value of a random walk with n steps

We now show that Pnk can be approximated by Brownian motion with drift

parameter α and variance parameter σ2 when n is sufficiently large We restate theproperties of such a Brownian motion Bt below

(1) B0 = 0(2) Bt has stationary and independent increments(3) Bt sim N(αtσ2t)(4) t 983041rarr Bt is continuous with probability one

The first property is clear as P(n)0 = 0 for all n The second property follows from

the fact that U (i) are iid random variables The third property is a result of thecentral limit theorem for iid sequences the result of which is provided below

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 15

Theorem 39 (Donskerrsquos Theorem) Let Xi be a sequence of iid random vari-

ables with mean 0 and variance σ2 If R(n)t is a scaled random walk as defined

above then R(n)t rArr Bt where Bt is a standard Brownian motion This result is

also known as the functional central limit theorem

34 Brownian Motion VariantsThe preceding construction of Brownian motion will prove particularly useful in ourapproximation of heavy traffic waiting times However this process requires furthermodification Recall that when traffic intensity ρ is slightly less than 1 customersare being serviced at a slightly faster rate then they are entering the queue Thusgiven enough time traffic will eventually ldquoclear outrdquo due to faster service timesHowever this particular construction of Brownian motion has increments of meanzero

E[Bs+t minusBs] = 0

meaning that the approximate number of customers in line wouldmdashon averagemdashremain unchanged regardless of how long the line is observed Such a propertyis called the martingale property Instead we want a process that exhibits thesupermartingale property

E[Bs+t minusBs] lt 0

Definition 310 Let Fn be an increasing sequence of σ-fields (Fn is called afiltration) Let Xn be a sequence of random variables with finite mean andXn isin Fn for all n Martingales supermartingales and submartingales are thendefined as follows for all n

Martingale E [Xn+1 | Fn] = Xn

Supermartingale E [Xn+1 | Fn] le Xn

Submartingale E [Xn+1 | Fn] ge Xn

In words a sequence that we expect to remain the same over time is a martingalea sequence we expect to increase is a submartingale and a sequence we expect todecrease is a supermartingale To approximate heavy traffic we want a Brownianmotion-like process that exhibits the supermartingale property

Definition 311 A Brownian motion with drift parameter α and variance param-eter σ2 Bα

t is defined by the following properties

(1) Independent increments(2) Increment lengths are normally distributed with mean αt and variance σ2t(3) the path t 983041rarr Bα

t is continuous

Note that these properties are the same as standard Brownian motion exceptincrement lengths now have non-zero means Since we have already shown theexistence of standard Brownian motion the construction of Brownian motion withdrift is now easy

Lemma 312 Let Bt be a standard Brownian motion with variance parameterσ2 then Bα

t = Bt + αt is Brownian motion with drift parameter α and varianceparameter σ2 Moreover Bα

t fulfills the supermartingale property when α lt 0

16 HARVEY BARNHARD

ProofIndependence of increments follows directly from independence of increments forBt and the fact that αt is a constant for a given t To prove that Bα

t has stationaryincrements we have to show that the distribution of increments Bα

s+tminusBαs depends

only the time interval t and not the absolute time Recall from the definition of Bt

that Bs+t minusBt are normal random variables with mean 0 and variance t for s gt 0

E983045Bα

s+t minusBαs

983046= E

983045Bα

s+t

983046minus E [Bα

s ]

= E983045Bα

s+t + α(s+ t)983046minus E [Bs + αs]

= α(s+ t)minus αs

= αt

Var983051Bα

s+t minusBαs

983052= Var Bs+t minusBs + αt= Var Bs+t minusBs= σ2t

And since for fixed t αt is just a constant we use the property of normal randomvariables to conclude that

Bαs+t minusBα

s sim N(αtσ2t) for all s gt 0

Continuity of Bαt directly follows from continuity of Bt We have shown that

Bαt constructed as the sum of standard Brownian motion and drift αt is indeed

Brownian motion with drift

Brownian motion with negative drift provides a better approximation of thewaiting-time of customers in a queue than does standard Brownian motion How-ever the unboundedness of Brownian motion suggests that waiting times couldperhaps be negative In order to circumvent this issue we instead use reflectedBrownian motion with drift Reflected Brownian motion acts much the same asBrownian motion except on a given boundary off of which the Brownian motionismdashunsurprisinglymdashreflected In the one-dimensional case the boundaries may beany interval on the reals that provide a lower and upper bound to the Brownianmotion For our purposes we only use a single boundary zero that gives a lowerbound to Brownian motion That is for all t ge 0 we want Bα

t ge 0

Definition 313 Let Bαt be a standard Brownian motion with drift parameter α

and let Mαt be the running maximum of Bα

t

Mαt = sup

0lesletBα

s

Then reflected Brownian motion with drift parameter minusα and boundary 0 definedon R+ can be constructed as follows

Rα0t equiv max0Mα

t minusBαt

Note that max0 Sαt ge 0 and Mα

t ge Bαt B

α0t ge 0 for all t Most importantly

the local behavior of Bα0t is exactly like Brownian motion with drift parameter minusα

since max0 Sαt is simply a constant when Bα

t ∕= 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 17

We have now provided sufficient modifications of Brownian motion and we nextmove onmdashwhat we will ultimately use to approximate heavy traffic waiting timesThe next example provides us with the distribution of the supremum of a Brownianmotion with negative drift

Example 314Let Bt be a Brownian motion with drift α lt 0 and variance parameter σ2 Define

Mt = sup0leslet

Bs

to be the running maximum then

Minfin sim exp

9830612|α|σ2

983062

that is

PW ge w = eminusminus2|α|σ2 w w ge 0

E [Minfin] =σ2

2|α|

Prooffor constants a b gt 0 Let T (minusa b) be the first time that B minus t hits minusa or b

T (minusa b) = inft Bt = minusa or Bt = bIn [7] it is proved that

P983051BT (minusab)=b

983052=

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)

Since α lt 0 exp(2αaσ2) rarr 0 we have

limararrinfin

PBT (minusab) = b = limararrinfin

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)= exp(2αbσ2)

The left-hand side becomes the probability that the process will reach b somewherealong its path (ie that the maximum of the process exceeds b somewhere alongits path) Therefore we have

PW ge b = exp(2αbσ2) = expminus2|α|bσ2

This example shows us that the supremum of a Brownian motion observed ldquofor-everrdquo (ie on the interval [0infin)) is exponentially distributed with rate 2|α|σ2

4 Heavy Traffic Approximation

We now have all we need to provide a heuristic argument for the approximationof single-servers queue in heavy traffic using Brownian Motion For a rigorousalbeit opaque treatment of this approximation see [8] the seminal work on thissubject by Kingman

Theorem 41 The waiting time of the nth customer in an MM1 queue canbe approximated by the supremum of reflected Brownian motion with negative driftand boundary at 0

18 HARVEY BARNHARD

ProofRecall that an MM1 queue is a process where the number of customer arrivalsfollows a poisson process with rate λ and service times are exponentially distributedwith rate micro From Lindleyrsquos equation in Section 2 we can write the waiting timefor the nth customer as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

where U (n) = S(n)minus I(n) is the difference between the nth service time and the nthinter-arrival time Define the partial sum P

(n)k to be zero when k = 0 and

P(n)k equiv

k983131

i=1

U (nminusi) for k ge 1

We can then rewrite the waiting time of the nth customer as

W (n)q = max

0lekle(nminus1)P

(n)k

The fact that P(n)k is the kth value of a random walk follows immediately from

U (1) U (nminus1) being iid random variables Let α and σ2 be the expectationand variance of U (i) We calculate α and σ2 by first noting that S(i) and I(i) areexponentially distributed with rates micro and λ Recall that ρ = λmicro is the trafficintensity as defined in the first section of this paper

α = E983147U (i)

983148= E

983147S(i)

983148minus E

983147I(i)

983148=

1

microminus 1

λ=

ρminus 1

λ

σ2 = Var983153U (i)

983154= Var

983153S(i)

983154+Var

983153I(i)

983154=

1

micro2+

1

λ2

We next compute the expectation and variance of P(n)k using the fact that U (i) are

iid random variables

E983147P

(n)k

983148=

k983131

i=1

E983147U (nminusi)

983148= kα

Var983153P

(n)k

983154=

k983131

i=1

Var983153U (nminusi)

983154= kσ2

For a finite n W(n)q is therefore the maximum value of a random walk with n steps

We now show that Pnk can be approximated by Brownian motion with drift

parameter α and variance parameter σ2 when n is sufficiently large We restate theproperties of such a Brownian motion Bt below

(1) B0 = 0(2) Bt has stationary and independent increments(3) Bt sim N(αtσ2t)(4) t 983041rarr Bt is continuous with probability one

The first property is clear as P(n)0 = 0 for all n The second property follows from

the fact that U (i) are iid random variables The third property is a result of thecentral limit theorem for iid sequences the result of which is provided below

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

16 HARVEY BARNHARD

ProofIndependence of increments follows directly from independence of increments forBt and the fact that αt is a constant for a given t To prove that Bα

t has stationaryincrements we have to show that the distribution of increments Bα

s+tminusBαs depends

only the time interval t and not the absolute time Recall from the definition of Bt

that Bs+t minusBt are normal random variables with mean 0 and variance t for s gt 0

E983045Bα

s+t minusBαs

983046= E

983045Bα

s+t

983046minus E [Bα

s ]

= E983045Bα

s+t + α(s+ t)983046minus E [Bs + αs]

= α(s+ t)minus αs

= αt

Var983051Bα

s+t minusBαs

983052= Var Bs+t minusBs + αt= Var Bs+t minusBs= σ2t

And since for fixed t αt is just a constant we use the property of normal randomvariables to conclude that

Bαs+t minusBα

s sim N(αtσ2t) for all s gt 0

Continuity of Bαt directly follows from continuity of Bt We have shown that

Bαt constructed as the sum of standard Brownian motion and drift αt is indeed

Brownian motion with drift

Brownian motion with negative drift provides a better approximation of thewaiting-time of customers in a queue than does standard Brownian motion How-ever the unboundedness of Brownian motion suggests that waiting times couldperhaps be negative In order to circumvent this issue we instead use reflectedBrownian motion with drift Reflected Brownian motion acts much the same asBrownian motion except on a given boundary off of which the Brownian motionismdashunsurprisinglymdashreflected In the one-dimensional case the boundaries may beany interval on the reals that provide a lower and upper bound to the Brownianmotion For our purposes we only use a single boundary zero that gives a lowerbound to Brownian motion That is for all t ge 0 we want Bα

t ge 0

Definition 313 Let Bαt be a standard Brownian motion with drift parameter α

and let Mαt be the running maximum of Bα

t

Mαt = sup

0lesletBα

s

Then reflected Brownian motion with drift parameter minusα and boundary 0 definedon R+ can be constructed as follows

Rα0t equiv max0Mα

t minusBαt

Note that max0 Sαt ge 0 and Mα

t ge Bαt B

α0t ge 0 for all t Most importantly

the local behavior of Bα0t is exactly like Brownian motion with drift parameter minusα

since max0 Sαt is simply a constant when Bα

t ∕= 0

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 17

We have now provided sufficient modifications of Brownian motion and we nextmove onmdashwhat we will ultimately use to approximate heavy traffic waiting timesThe next example provides us with the distribution of the supremum of a Brownianmotion with negative drift

Example 314Let Bt be a Brownian motion with drift α lt 0 and variance parameter σ2 Define

Mt = sup0leslet

Bs

to be the running maximum then

Minfin sim exp

9830612|α|σ2

983062

that is

PW ge w = eminusminus2|α|σ2 w w ge 0

E [Minfin] =σ2

2|α|

Prooffor constants a b gt 0 Let T (minusa b) be the first time that B minus t hits minusa or b

T (minusa b) = inft Bt = minusa or Bt = bIn [7] it is proved that

P983051BT (minusab)=b

983052=

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)

Since α lt 0 exp(2αaσ2) rarr 0 we have

limararrinfin

PBT (minusab) = b = limararrinfin

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)= exp(2αbσ2)

The left-hand side becomes the probability that the process will reach b somewherealong its path (ie that the maximum of the process exceeds b somewhere alongits path) Therefore we have

PW ge b = exp(2αbσ2) = expminus2|α|bσ2

This example shows us that the supremum of a Brownian motion observed ldquofor-everrdquo (ie on the interval [0infin)) is exponentially distributed with rate 2|α|σ2

4 Heavy Traffic Approximation

We now have all we need to provide a heuristic argument for the approximationof single-servers queue in heavy traffic using Brownian Motion For a rigorousalbeit opaque treatment of this approximation see [8] the seminal work on thissubject by Kingman

Theorem 41 The waiting time of the nth customer in an MM1 queue canbe approximated by the supremum of reflected Brownian motion with negative driftand boundary at 0

18 HARVEY BARNHARD

ProofRecall that an MM1 queue is a process where the number of customer arrivalsfollows a poisson process with rate λ and service times are exponentially distributedwith rate micro From Lindleyrsquos equation in Section 2 we can write the waiting timefor the nth customer as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

where U (n) = S(n)minus I(n) is the difference between the nth service time and the nthinter-arrival time Define the partial sum P

(n)k to be zero when k = 0 and

P(n)k equiv

k983131

i=1

U (nminusi) for k ge 1

We can then rewrite the waiting time of the nth customer as

W (n)q = max

0lekle(nminus1)P

(n)k

The fact that P(n)k is the kth value of a random walk follows immediately from

U (1) U (nminus1) being iid random variables Let α and σ2 be the expectationand variance of U (i) We calculate α and σ2 by first noting that S(i) and I(i) areexponentially distributed with rates micro and λ Recall that ρ = λmicro is the trafficintensity as defined in the first section of this paper

α = E983147U (i)

983148= E

983147S(i)

983148minus E

983147I(i)

983148=

1

microminus 1

λ=

ρminus 1

λ

σ2 = Var983153U (i)

983154= Var

983153S(i)

983154+Var

983153I(i)

983154=

1

micro2+

1

λ2

We next compute the expectation and variance of P(n)k using the fact that U (i) are

iid random variables

E983147P

(n)k

983148=

k983131

i=1

E983147U (nminusi)

983148= kα

Var983153P

(n)k

983154=

k983131

i=1

Var983153U (nminusi)

983154= kσ2

For a finite n W(n)q is therefore the maximum value of a random walk with n steps

We now show that Pnk can be approximated by Brownian motion with drift

parameter α and variance parameter σ2 when n is sufficiently large We restate theproperties of such a Brownian motion Bt below

(1) B0 = 0(2) Bt has stationary and independent increments(3) Bt sim N(αtσ2t)(4) t 983041rarr Bt is continuous with probability one

The first property is clear as P(n)0 = 0 for all n The second property follows from

the fact that U (i) are iid random variables The third property is a result of thecentral limit theorem for iid sequences the result of which is provided below

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 17

We have now provided sufficient modifications of Brownian motion and we nextmove onmdashwhat we will ultimately use to approximate heavy traffic waiting timesThe next example provides us with the distribution of the supremum of a Brownianmotion with negative drift

Example 314Let Bt be a Brownian motion with drift α lt 0 and variance parameter σ2 Define

Mt = sup0leslet

Bs

to be the running maximum then

Minfin sim exp

9830612|α|σ2

983062

that is

PW ge w = eminusminus2|α|σ2 w w ge 0

E [Minfin] =σ2

2|α|

Prooffor constants a b gt 0 Let T (minusa b) be the first time that B minus t hits minusa or b

T (minusa b) = inft Bt = minusa or Bt = bIn [7] it is proved that

P983051BT (minusab)=b

983052=

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)

Since α lt 0 exp(2αaσ2) rarr 0 we have

limararrinfin

PBT (minusab) = b = limararrinfin

exp(2αaσ2)minus 1

exp(2αaσ2)minus exp(minus2αbσ2)= exp(2αbσ2)

The left-hand side becomes the probability that the process will reach b somewherealong its path (ie that the maximum of the process exceeds b somewhere alongits path) Therefore we have

PW ge b = exp(2αbσ2) = expminus2|α|bσ2

This example shows us that the supremum of a Brownian motion observed ldquofor-everrdquo (ie on the interval [0infin)) is exponentially distributed with rate 2|α|σ2

4 Heavy Traffic Approximation

We now have all we need to provide a heuristic argument for the approximationof single-servers queue in heavy traffic using Brownian Motion For a rigorousalbeit opaque treatment of this approximation see [8] the seminal work on thissubject by Kingman

Theorem 41 The waiting time of the nth customer in an MM1 queue canbe approximated by the supremum of reflected Brownian motion with negative driftand boundary at 0

18 HARVEY BARNHARD

ProofRecall that an MM1 queue is a process where the number of customer arrivalsfollows a poisson process with rate λ and service times are exponentially distributedwith rate micro From Lindleyrsquos equation in Section 2 we can write the waiting timefor the nth customer as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

where U (n) = S(n)minus I(n) is the difference between the nth service time and the nthinter-arrival time Define the partial sum P

(n)k to be zero when k = 0 and

P(n)k equiv

k983131

i=1

U (nminusi) for k ge 1

We can then rewrite the waiting time of the nth customer as

W (n)q = max

0lekle(nminus1)P

(n)k

The fact that P(n)k is the kth value of a random walk follows immediately from

U (1) U (nminus1) being iid random variables Let α and σ2 be the expectationand variance of U (i) We calculate α and σ2 by first noting that S(i) and I(i) areexponentially distributed with rates micro and λ Recall that ρ = λmicro is the trafficintensity as defined in the first section of this paper

α = E983147U (i)

983148= E

983147S(i)

983148minus E

983147I(i)

983148=

1

microminus 1

λ=

ρminus 1

λ

σ2 = Var983153U (i)

983154= Var

983153S(i)

983154+Var

983153I(i)

983154=

1

micro2+

1

λ2

We next compute the expectation and variance of P(n)k using the fact that U (i) are

iid random variables

E983147P

(n)k

983148=

k983131

i=1

E983147U (nminusi)

983148= kα

Var983153P

(n)k

983154=

k983131

i=1

Var983153U (nminusi)

983154= kσ2

For a finite n W(n)q is therefore the maximum value of a random walk with n steps

We now show that Pnk can be approximated by Brownian motion with drift

parameter α and variance parameter σ2 when n is sufficiently large We restate theproperties of such a Brownian motion Bt below

(1) B0 = 0(2) Bt has stationary and independent increments(3) Bt sim N(αtσ2t)(4) t 983041rarr Bt is continuous with probability one

The first property is clear as P(n)0 = 0 for all n The second property follows from

the fact that U (i) are iid random variables The third property is a result of thecentral limit theorem for iid sequences the result of which is provided below

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

18 HARVEY BARNHARD

ProofRecall that an MM1 queue is a process where the number of customer arrivalsfollows a poisson process with rate λ and service times are exponentially distributedwith rate micro From Lindleyrsquos equation in Section 2 we can write the waiting timefor the nth customer as

W (n)q = max

983153U (1) + middot middot middot+ U (nminus1) U (2) + middot middot middot+ U (nminus1) U (nminus1) 0

983154

where U (n) = S(n)minus I(n) is the difference between the nth service time and the nthinter-arrival time Define the partial sum P

(n)k to be zero when k = 0 and

P(n)k equiv

k983131

i=1

U (nminusi) for k ge 1

We can then rewrite the waiting time of the nth customer as

W (n)q = max

0lekle(nminus1)P

(n)k

The fact that P(n)k is the kth value of a random walk follows immediately from

U (1) U (nminus1) being iid random variables Let α and σ2 be the expectationand variance of U (i) We calculate α and σ2 by first noting that S(i) and I(i) areexponentially distributed with rates micro and λ Recall that ρ = λmicro is the trafficintensity as defined in the first section of this paper

α = E983147U (i)

983148= E

983147S(i)

983148minus E

983147I(i)

983148=

1

microminus 1

λ=

ρminus 1

λ

σ2 = Var983153U (i)

983154= Var

983153S(i)

983154+Var

983153I(i)

983154=

1

micro2+

1

λ2

We next compute the expectation and variance of P(n)k using the fact that U (i) are

iid random variables

E983147P

(n)k

983148=

k983131

i=1

E983147U (nminusi)

983148= kα

Var983153P

(n)k

983154=

k983131

i=1

Var983153U (nminusi)

983154= kσ2

For a finite n W(n)q is therefore the maximum value of a random walk with n steps

We now show that Pnk can be approximated by Brownian motion with drift

parameter α and variance parameter σ2 when n is sufficiently large We restate theproperties of such a Brownian motion Bt below

(1) B0 = 0(2) Bt has stationary and independent increments(3) Bt sim N(αtσ2t)(4) t 983041rarr Bt is continuous with probability one

The first property is clear as P(n)0 = 0 for all n The second property follows from

the fact that U (i) are iid random variables The third property is a result of thecentral limit theorem for iid sequences the result of which is provided below

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

APPROXIMATING HEAVY TRAFFIC WITH BROWNIAN MOTION 19

Theorem 42 Let Xiinfini=1 be a sequence of iid random variables with E [Xi] = microand VarXi = σ2 If Sn = X1 + middot middot middotXn then

radicnSn

d=rArr N(microσ2)

In words as n rarr infinradicnSn converges in distribution to a normal random variable

with mean micro and variance σ2

When k is sufficiently large the central limit theorem implies that

Pinfink equiv lim

nrarrinfinP

(n)k

is approximately normally distributed with mean kα and variance kσ2 In practicetaking n rarr infin represents a queue that has been operating continuously with a largeamount of customers

Property (4) is where we make use of the heavy traffic assumption Since ρ is

very close to one |α| is small so increments of P (n)k are on average small Small

increments ensure that continuity approximately holds Therefore Pinfink (approxi-

mately) fulfills properties (1)-(4) above so the waiting time of the nth customercan be approximated as the supremum of Brownian motion with negative drift

Before concluding with a computation of the expected waiting time we take noteof the conditions under which the above approximation is accurate

(1) Service and inter-arrival times are exponentially distributed with knownconstant rates

(2) ρ = λmicro is very close to but does not exceed one(3) Many other customers preceded the nth customer in line the customer

whose waiting time we wish to approximate In common usage the resultof the central limit theorem is used when n ge 30

We conclude with a computation of the expected waiting time for large n Ex-ample 314 proved that the supremum of Brownian motion with negative drift is

exponentially distributed so we conclude that W(n)q can be approximated by an

exponential distribution with rate 2|α|σ2 when n is at least greater than 30

E983147W (n)

q

983148asymp σ2

2|α| =1

2

1micro2 + 1

λ2983055983055ρminus1λ

983055983055

Acknowledgements

I would like to extend my sincerest gratitude to Peter May for providing me andmy peers with the opportunity to participate in the University of Chicago MathREU My interest in pure math now heightened and my mathematical maturitydeveloped I consider myself extremely fortunate to have taken part in this programI must also thank Gregory Lawler whose lectures and texts have proved invaluablein the writing of this paper Finally I am forever grateful to my mentor YuchengDeng for guiding me throughout the entire writing process I attribute much ofthe fruit borne by my exploration of Brownian motion and queues to Yuchengrsquosmotivation and insight

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu

20 HARVEY BARNHARD

References

1 Gross Donald and Harris Carl M Fundamentals of Queueing Theory John Wiley and SonsInc 2008

2 Lawler Gregory F An Introduction to Stochastic Processes Chapman and Hall 20063 Lawler Gregory F and Limic Vlada Random Walk A Modern Introduction

httpswwwmathuchicagoedusimlawlersrwbookpdf4 Durrett Richard Probability Theory and Examples Cambridge University Press 20105 Billingsley Patrick Convergence of Probability Measures John Wiley and Sons Inc 20096 Peskir Goran ldquoOn Reflecting Brownian Motion with Driftrdquo Proc Symp Stoch Syst (Os-

aka 2005) ISCIE Kyoto 2006 (1-5) Research Report No 3 2005 Probab Statist GroupManchester (8 pp)

7 Huang Yibi Lecture notes for STAT 253317 ldquoIntroduction to Probability Modelsrdquo Univer-sity of Chicago 2014 httpsgaltonuchicagoedusimyibiteachingstat3172014

8 Kingman J F C rdquoOn Queues in Heavy Trafficrdquo Journal of the Royal Statistical SocietyAccessed on JSTOR 2984229 1962

Department of Mathematics University of ChicagoEmail address harveybarnharduchicagoedu