![Page 1: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/1.jpg)
1
TCOM 501: Networking Theory & Fundamentals
Lectures 4 & 5February 5 and 12, 2003
Prof. Yannis A. Korilis
![Page 2: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/2.jpg)
4&5-2 Topics Markov Chains M/M/1 Queue Poisson Arrivals See Time Averages M/M/* Queues Introduction to Sojourn Times
![Page 3: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/3.jpg)
4&5-3 The M/M/1 Queue Arrival process: Poisson with rate λ Service times: iid, exponential with parameter
µ Service times and interarrival times:
independent Single server Infinite waiting room N(t): Number of customers in system at time t
(state)0 1 n+1n2
![Page 4: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/4.jpg)
4&5-4 Exponential Random Variables X: exponential RV with
parameter λ Y: exponential RV with
parameter μ X, Y: independentThen:1. min{X, Y}: exponential RV
with parameter λ+μ2. P{X<Y} = λ/(λ+μ)
[Exercise 3.12]
Proof:
0 0
0 0
0 0
0
( )
0 0
{ } ( , )
(1 )
( )
1
y
XY
yx y
yy x
y y
y y
P X Y f x y dx dy
e e dx dy
e e dx dy
e e dy
e dy e dy
( )
( )
{min{ , } } { , }{ } { }
{min{ , } } 1
t t t
t
P X Y t P X t Y tP X t P Y te e e
P X Y t e
![Page 5: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/5.jpg)
4&5-5
M/M/1 Queue: Markov Chain Formulation
Jumps of {N(t): t ≥ 0} triggered by arrivals and departures{N(t): t ≥ 0} can jump only between neighboring states
Assume process at time t is in state i: N(t) = i ≥ 1 Xi: time until the next arrival – exponential with parameter λ Yi: time until the next departure – exponential with
parameter μ Ti =min{Xi,Yi}: time process spends at state i Ti : exponential with parameter νi= λ+μ Pi,i+1=P{Xi < Yi}= λ/(λ+μ), Pi,i-1=P{Yi < Xi}= μ/(λ+μ)
P01=1, and T0 is exponential with parameter λ
{N(t): t ≥ 0} is a continuous-time Markov chain with, 1 , 1
, 1 , 1
, 0, 1
0, | | 1
i i i i i
i i i i i
ij
q P iq P iq i j
![Page 6: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/6.jpg)
4&5-6
M/M/1 Queue: Stationary Distribution
Birth-death process → DBE
Normalization constant
Stationary distribution
0 1 n+1n2
1
1 1 0...
n n
nn n n
p p
p p p p
0 00 1
1 1 1 1 , if 1nn
n n
p p p
(1 ), 0,1,...nnp n
![Page 7: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/7.jpg)
4&5-7
The M/M/1 Queue
Average number of customers in system
Little’s Theorem: average time in system
Average waiting time and number of customers in the queue – excluding service
1
0 0 0
2
(1 ) (1 )
1(1 )
(1 ) 1
n nn
n n n
N np n n
N
11NT
1 and
1 2
WNTW Q
![Page 8: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/8.jpg)
4&5-8 The M/M/1 Queue
ρ=λ/μ: utilization factor Long term proportion of
time that server is busy ρ=1-p0: holds for any
M/G/1 queue Stability condition: ρ<1
Arrival rate should be less than the service rate
0
2
4
6
8
10
0 0.2 0.4 0.6 0.8 1
N
![Page 9: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/9.jpg)
4&5-9
M/M/1 Queue: Discrete-Time Approach
Focus on times 0, , 2,… ( arbitrarily small) Study discrete time process Nk = N(k)
Show that transition probabilities are
Discrete time Markov chain, omitting o(
lim { ( ) } lim { }kt k
P N t n P N n
00
, 1
, 1
1 ( )
1 ( ), 1
( ), 0
( ), 0
( ), | | 1
ii
i i
i i
ij
P
P i
P i
P i
P i j
0 1 n+1n2
1 1 1 1
![Page 10: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/10.jpg)
4&5-10
M/M/1 Queue: Discrete-Time Approach
Discrete-time birth-death process → DBE:
Taking the limit →0:
Done!
1
1 0
[ ( )]π [ ( )]π
( ) ( )π π π
( ) ( )
n n
n
n n
0 00 0 0
( )lim π lim lim π
( )
n n
n np p
0 1 n+1n2
1 1 1 1
![Page 11: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/11.jpg)
4&5-11 Transition Probabilities? Ak: number of customers that arrive in Ik=(k, (k+1)]
Dk: number of customers that depart in Ik=(k, (k+1)] Transition probabilities Pij depend on conditional
probabilities: Q(a,d | n) = P{Ak=a, Dk=d | Nk-1=n} Calculate Q(a,d | n) using arrival and departure statistics Use Taylor expansion e-=1-+o(), e-=1-+o(), to
express as a function of Poisson arrivals: P{Ak ≥ 2}=o( Probability there are more than 1 arrivals in Ik is o() Show: probability of more than one event (arrival or
departure) in Ik is o() See details in textbook
![Page 12: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/12.jpg)
4&5-12 Example: Slowing Down
M/M/1 system: slow down the arrival and service rates by the same factor m
Utilization factors are the same ⇒stationary distributions the same, average number in the system the same
Delay in the slower system is m times higher Average number in queue is the same, but in the 1st
system the customers move out faster
/
1 1 /
/
( / )
/ /
N N
N mT mT
m
mW mW
m m
/ m/ m
/
1 1 /
1
/
N
NT
W
![Page 13: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/13.jpg)
4&5-13
Example: Statistical MUX-ing vs. TDM
m identical Poisson streams with rate λ/m; link with capacity 1; packet lengths iid, exponential with mean 1/μ
Alternative: split the link to m channels with capacity 1/m each, and dedicate one channel to each traffic stream
Delay in each “queue” becomes m times higher Statistical multiplexing vs. TDM or FDMWhen is TDM or FDM preferred over statistical multiplexing?
mT mT
/ m/ m
/ m/ m
m
![Page 14: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/14.jpg)
4&5-14 “PASTA” Theorem Markov chain: “stationary” or “in steady-state:”
Process started at the stationary distribution, or Process runs for an infinite time t→∞
Probability that at any time t, process is in state i is equal to the stationary probability
Question: For an M/M/1 queue: given t is an arrival time, what is the probability that N(t)=i?Answer: Poisson Arrivals See Time Averages!
( )lim { ( ) } lim i
it t
T tp P N t i
t
![Page 15: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/15.jpg)
4&5-15 PASTA Theorem Steady-state probabilities:
Steady-state probabilities upon arrival:
Lack of Anticipation Assumption (LAA): Future inter-arrival times and service times of previously arrived customers are independent
Theorem: In a queueing system satisfying LAA:1. If the arrival process is Poisson:
2. Poisson is the only process with this property (necessary and sufficient condition)
lim { ( ) }nt
p P N t n
lim { ( ) | arrival at }nt
a P N nt t
, 0,1,...n na p n
![Page 16: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/16.jpg)
4&5-16 PASTA Theorem
Doesn’t PASTA apply for all arrival processes? Deterministic arrivals every 10 sec Deterministic service times 9 sec
Upon arrival: system is always empty a1=0
Average time with one customer in system: p1=0.9
“Customer” averages need not be time averages Randomization does not help, unless Poisson!
10 109 20 30 30 39
![Page 17: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/17.jpg)
4&5-17 PASTA Theorem: Proof Define A(t,t+δ), the event that an arrival occurs in [t, t+
δ) Given that a customer arrives at t, probability of finding
the system in state n:
A(t,t+δ) is independent of the state before time t, N(t-) N(t-) determined by arrival times <t, and corresponding
service times A(t,t+δ) independent of arrivals <t [Poisson] A(t,t+δ) independent of service times of customers arrived
<t [LAA]0 0
0
{ ( ) , ( , )}( ) lim { ( ) | ( , )} lim
{ ( , )}
{ ( ) } { ( , )}lim { ( ) }
{ ( , )}
n
P N t n A t ta t P N t n A t t
P A t t
P N t n P A t tP N t n
P A t t
lim ( ) lim { ( ) }n n nt ta a t P N t n p
0{ ( ) | arrival at t} lim { ( ) | ( , )}P N t n P N t n A t t
![Page 18: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/18.jpg)
4&5-18 PASTA Theorem: Intuitive Proof ta and tr: randomly selected arrival and observation
times, respectively The arrival processes prior to ta and tr respectively are
stochastically identical The probability distributions of the time to the first arrival
before ta and tr are both exponentially distributed with parameter
Extending this to the 2nd, 3rd, etc. arrivals before ta and tr establishes the result
State of the system at a given time t depends only on the arrivals (and associated service times) before t Since the arrival processes before arrival times and random times are identical, so is the state of the system they see
![Page 19: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/19.jpg)
4&5-19
Arrivals that Do not See Time-Averages
Example 1: Non-Poisson arrivals IID inter-arrival times, uniformly distributed between in 2
and 4 sec Service times deterministic 1 sec
Upon arrival: system is always emptyλ=1/3, T=1 → N=T/λ=1/3 → p1=1/3
Example 2: LAA violated Poisson arrivals Service time of customer i: Si= Ti+1, 1
Upon arrival: system is always emptyAverage time the system has 1 customer: p1=
![Page 20: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/20.jpg)
4&5-20 Distribution after Departure Steady-state probabilities after departure:
Under very general assumptions: N(t) changes in unit increments
limits an and exist dn
an = dn, n=0,1,…
In steady-state, system appears stochastically identical to an arriving and departing customer
Poisson arrivals + LAA: an arriving and a departing customer see a system that is stochastically to the one seen by an observer looking at an arbitrary time
| departure at lim { ( ) }nt
d P X nt t
![Page 21: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/21.jpg)
4&5-21 M/M/* Queues Poisson arrival process
Interarrival times: iid, exponential Service times: iid, exponential Service times and interarrival times: independent N(t): Number of customers in system at time t
(state){N(t): t ≥ 0} can be modeled as a continuous-time Markov chainTransition rates depend on the characteristics of the systemPASTA Theorem always holds
![Page 22: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/22.jpg)
4&5-22 M/M/1/K Queue
M/M/1 with finite waiting room At most K customers in the system Customer that upon arrival finds K customers in system is
dropped Stationary distribution
Stability condition: always stable – even if ρ≥1 Probability of loss – using PASTA theorem:
0 1 KK-12
0
0 1
, 1,2,...,1
1
nn
K
p p n K
p
1
(1 ){loss} { ( ) }
1
K
KP P N t K
![Page 23: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/23.jpg)
4&5-23 M/M/1/K Queue (proof)
Exactly as in the M/M/1 queue:
Normalization constant:
Generalize: Truncating a Markov chain
0 1 KK-12
1
0 00 1
0 1
11 1 1
11
1
KK Kn
nn n
K
p p p
p
0 , 1,2,...,nnp p n K
![Page 24: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/24.jpg)
4&5-24 Truncating a Markov Chain
{X(t): t ≥ 0} continuous-time Markov chain with stationary distribution {pi: i=0,1,…}
S a subset of {0,1,…}: set of states; Observe process only in S
Eliminate all states not in S Set
{Y(t): t ≥ 0}: resulting truncated process; If irreducible: Continuous-time Markov chain Stationary distribution
Under certain conditions – need to verify depending on the system
if
0 if
j
ij
i S
pj S
pp
j S
0, ,ji ijq q j S i S
![Page 25: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/25.jpg)
4&5-25
Truncating a Markov Chain (cont.) Possible sufficient condition
Verify that distribution of truncated process1. Satisfies the GBE
2. Satisfies the probability conservation law:
Another – even better – sufficient condition: DBE!Relates to “reversibility”Holds for multidimensional chains
( ) ( ),
j ij ji i ij j ji i ij ji ij
i i i S i S i S i S
j ji i ij j ji i iji S i S i S i S
p pp q p q p q p q q q
p S p Sp q p q p q p q j S
,j ji i iji S i S
p q p q j S
( )1, ( )
( ) ( )i
i ii S i S i S
p p Sp p S p
p S p S
![Page 26: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/26.jpg)
4&5-26
M/M/1 Queue with State-Dependent Rates
Interarrival times: independent, exponential, with parameter λn when at state n
Service times: independent, exponential, with parameter μn when at state n
Service times and interarrival times: independent{N(t): t ≥ 0} is a birth-death processStationary distribution:
0 1 n+1n2
0
1
n
1n
1
2
1n
n
11 1
0 00 1 01 1
, 1 1n n
i in
i n ii i
p p n p
![Page 27: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/27.jpg)
4&5-27 M/M/c Queue
Poisson arrivals with rate λ Exponential service times with parameter μ c servers Arriving customer finds n customers in system
n < c: it is routed to any idle server n ≥ c: it joins the waiting queue – all servers are busy
Birth-death process with state-dependent death rates
[Time spent at state n before jumping to n -1 is the minimum of Bn= min{n,c} exponentials with parameter μ]
, 1
,n
n n c
c n c
0 1 c+1c2
c
2
c
![Page 28: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/28.jpg)
4&5-28 M/M/c Queue
0 1 c+1c2
c
2
c
Detailed balance equations
Normalizing
1 0 0 0
0 0 0
1 ( )1 : ,
( 1) ! !
1 :
! ! !
n n
n n
n c c n c nc c n
n c
cn c p p p p p
n n n n n c
c cn c p p p p p
c c c c c c
1 11 1
00 1 0
( ) ( ) ( ) ( ) 11 1
! ! ! ! 1
k c k cc ck c
nn k k c k
c c c cp p
k c k c
![Page 29: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/29.jpg)
4&5-29 M/M/c Queue
Probability of queueing – arriving customer finds all servers busy
Erlang-C Formula: used in telephony and circuit-switching Call requests arrive with rate λ; holding time of a call
exponential with mean 1/μ c available circuits on a transmission line A call that finds all c circuits busy, continuously attempts to
find a free circuit – “remains in queue” M/M/c/c Queue: c-server loss system
A call that finds all c circuits busy is blocked Erlang-B Formula: popular in telephony
0 0
( ) ( ) 1{queueing}
! ! 1
c cn c
Q nn c n c
c cP P p p p
c c
![Page 30: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/30.jpg)
4&5-30 M/M/c Queue
Expected number of customers waiting in queue – not in service
Average waiting time (in queue)
Average time in system (queued + serviced)
Expected number of customers in system
0 0 2
2
( ) ( )( ) ( )
! ! (1 )
(1 )(1 ) 1
c cn c
Q nn c n c
Q Q
c cN n c p p n c p
c c
P P
(1 )Q
Q
NW P
1 1
(1 )QT W P
(1 )QN T P c
![Page 31: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/31.jpg)
4&5-31 M/M/∞ Queue: Infinite-Server System
Infinite number of servers – no queueing Stationary distribution:
Poisson with rate λ/μ Average number of customers & average delay:
The results hold for an M/G/∞ queue
0 1 n+1n2
( 1)n
2
n
/( / ), 0,1,...
!
n
np e nn
1,
NN T
![Page 32: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/32.jpg)
4&5-32
M/M/c/c Queue: c-Server Loss System
c servers, no waiting room An arriving customer that finds all servers busy is blocked Stationary distribution:
Probability of blocking (using PASTA):
Erlang-B Formula: used in telephony and circuit-switchingResults hold for an M/G/c/c queue
0 1 c2
2
c
1
0
( / ) ( / ), 0,1,...,
! !
n kc
nk
p n cn k
1
0
( / ) ( / )
! !
c kc
ck
pc k
![Page 33: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/33.jpg)
4&5-33
M/M/∞ and M/M/c/c Queues (proof)
DBE:
Normalizing:
0 1 n+1n2
( 1)n
2
n
1 1 2 0
0
( )( 1) ( 1)
( / ), 0,1,...
!
n n n n n
n
n
n p p p p p pn n n n n
p p nn
1
00
( / ), for M/M/c/c
!
kc
k
pk
1
/0
0
( / ), for M/M/c/c
!
k
k
p ek
![Page 34: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/34.jpg)
4&5-34 Sum of IID Exponential RV’s X1, X2,…, Xn: iid, exponential with parameter λ T = X1+ X2+…+ Xn
The probability density function of T is:
[Gamma distribution with parameters (n, λ)]
If Xi is the time between arrivals i -1 and i of a certain type of events, then T is the time until the nth event occurs For arbitrarily small :
Cummulative distribution function:
1( )( ) , 0
( 1)!
nt
T
tf t e t
n
1( ){ arrival occurs in [ , )} ( )
( 1)!
nth t
T
tP n t t f t e
n
1
0
( ){ } 1 { arrival occurs after }
( 1)!
nt s thn
sP t t e ds P n t
n
![Page 35: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/35.jpg)
4&5-35 Sum of IID Exponential RV’s
Example 1: Poisson arrivals with rate λ 1: time until arrival of 1st customer i: ith interarrival time 1, 2,…, n: iid exponential with parameter λ tn= 1+ 2+…,+n: arrival time of customer n
tn follows Gamma with parameters (n, λ).
For arbitrarily small :
1 1
0
( ) ( )( ) , 0; { }
( 1)! ( 1)!
n ntt tn
t tf t e t P t t e dt
n n
1( ){ arrival occurs in [ , )} ( )
( 1)!
nth t
T
tP n t t f t e
n
![Page 36: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/36.jpg)
4&5-36 Sojourn Times in a M/M/1 Queue
M/M/1 Queue – FCFS Ti: time spent in system (queueing + service)
by customer i Ti: exponentially distributed with parameter μ-λ Example of a sojourn time of a customer:
describes the evolution of the queue together with the specific customer
Proof:1. Direct calculation of probability distribution function2. Moment generating functions3. Intuitive: Exercise 3.11(b)
![Page 37: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/37.jpg)
4&5-37 M/M/1 Queue: Sojourn Times (proof)P roof 1: Let ti be the arrival time of customer i, and N i = N (t¡
i ), thenumber of customers in the system right before the ith arrival.
P f Ti > tg =1X
k=0
P f Ti > tjN i = kgP f N i = kg
=1X
k=0
P f D(ti + t) ¡ D(ti ) · kgpk (1)
=1X
k=0
kX
n=0
e¡ ¹ t (¹ t)n
n!¢(1¡ ½)½k (2)
= e¡ ¹ t1X
n=0
(¹ t)n
n!
1X
k=n
(1¡ ½)½k (3)
= e¡ ¹ t1X
n=0
(¹ t)n
n!¢½n = e¡ ¹ t
1X
n=0
(¸t)n
n!(4)
= e¡ ¹ te¸ t = e¡ (¹ ¡ ¸ )t
![Page 38: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/38.jpg)
4&5-38 M/M/1 Queue: Sojourn Times (proof)
P roof 1: Note that:
² Time customer i stays in the system is greater than t, giventhat it ¯nds k customers in the system, i®the number ofdepartures in interval (ti ; ti + t) are less than k + 1. The serveris always busy during that interval, thus times betweendepartures are iid, exponential with parameter ¹ . Then:
P fD(ti + t) ¡ D(ti ) = ng= e¡ ¹ t (¹ t)n
n!; 0 · n · k
² P fNi = kg= pk, by PASTA theorem.
² Eq. (3) follows by changing order of summation.
² Eq. (4) uses:
1X
k=n
½k =1X
k=0
½k ¡n¡ 1X
k=0
½k =1
1¡ ½¡
1¡ ½n
1¡ ½=
½n
1¡ ½
![Page 39: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/39.jpg)
4&5-39 M/M/1 Queue: Sojourn Times (proof)
P roof 2:
² N i : number of customers in system upon arrival of customer i
² T (k)i : sojourn time of customer i when it ¯nds k customers in system
T (k)i = Si + Si ¡ 1 + ::: + Si ¡ k+1 + R i ¡ k
Sj is service time of customer j , and R i ¡ k the residual service timeof the customer in service.
² Si ; : : : ;Si ¡ k+1: iid, exponential with parameter ¹
² R i ¡ k : exponential with parameter ¹ , independent of Si ; : : : ;Si ¡ k+1
² T (k)i is the sum of k iid exponential RV's
² Ti = T (N i )i is the sum of a random number of iid exponential RV's
² Use moment generating functions
![Page 40: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/40.jpg)
4&5-40 M/M/1 Queue: Sojourn Times (proof)
P roof 2:
M T i (t) = E [etT i ] =X 1
k=0E [etT i jN i = k]P f N i = kg
=X 1
k=0E [etT ( k )
i ]pk =X 1
k=0M T ( k )
i(t)pk
A RV is exponential with parameter ¹ if and only if its momentgenerating function is ¹ =(¹ ¡ t).
T (k)i sum of k + 1 independent RV's:
M T ( k )i
(t) = MS i (t)MS i ¡ 1 (t) : : :MS i ¡ k + 1 (t)MR i ¡ k (t) =
µ¹
¹ ¡ t
¶ k+1
Then:
M T i (t) =1X
k=0
µ¹
¹ ¡ t
¶k+1
(1¡ ½)½k = (1¡ ½)¹
¹ ¡ t
1X
k=0
µ¸
¹ ¡ t
¶k
=¹ ¡ ¸¹ ¡ t
11¡ ¸
¹ ¡ t
=¹ ¡ ¸
(¹ ¡ ¸) ¡ t
![Page 41: 1 TCOM 501: Networking Theory & Fundamentals Lectures 4 & 5 February 5 and 12, 2003 Prof. Yannis A. Korilis](https://reader036.vdocuments.net/reader036/viewer/2022062409/56649f4c5503460f94c6cd3b/html5/thumbnails/41.jpg)
4&5-41 Moment Generating Function1. De nition: for any t 2 IR:
MX (t) = E [etX ] =
8>><
>>:
Z 1
¡ 1etx f X (x) dx; X continuous
X
j
etx j P fX = xj g; X discrete
2. If the moment generating function MX (t) of X exists and is ¯nite insome neighborhood of t = 0, it determines the distribution of Xuniquely.
3. Fundamental Properties: for any n 2 IN :
(i)dn
dtn MX (t) = E [X n etX ]
(ii)dn
dtn MX (0) = E [X n ]
4. Moment Generating Functions and Independence:
X ;Y : independent ) MX +Y (t) = MX (t)M Y (t)
The opposite is not true.