This article was downloaded by: [University of New Mexico]On: 30 November 2014, At: 11:07Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK

American Journal ofMathematical andManagement SciencesPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/umms20

Statistical Inference in M/M/1Queues: A Bayesian ApproachAmit Choudhurya & Arun C. Borthakurb

a Department of Statistics, Gauhati UniversityGuwahati - 781014, Assam, India e-mail:b Department of Statistics, Gauhati UniversityGuwahati - 781014, Assam, IndiaPublished online: 14 Aug 2013.

To cite this article: Amit Choudhury & Arun C. Borthakur (2007)Statistical Inference in M/M/1 Queues: A Bayesian Approach, AmericanJournal of Mathematical and Management Sciences, 27:1-2, 25-41, DOI:10.1080/01966324.2007.10737686

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AIIII!IUCAN .IOURIIAL OF IIIIATHII!IIAnCAL AIIIID IIANAGI!MI!NT SCIUCI!S Copyright © 2007 by American Sciences Press, Inc.

STATISTICAL INFERENCE IN MIMI 1 QUEUES:

ABA YESIAN APPROACH

Amit Choudhury Arun C. Borthakur

Department of Statistics, Gauhati University Guwahati -781014, Assam, India. e-mail: achoudhury@sanchamet.in

SYNOPTIC ABSTRACT

Applications of queueing models for analyzing real-life scenarios are based

on the fundamental premise that parameter values are known. This is a rarity

and more often than not, such parameters require to be estimated. Any analyst

with field level experience will corroborate this. How are such estimates to be

obtained? What are the procedures to be followed? How stable are those?

These and a few other related questions of fundamental practical interest are

dealt with in this paper. The queue we intend to deal with is the M/M/1

queueing model. In other words this paper deals with statistical inference with

regard to the classical single server Markovian queueing model. Inference is

discussed from Bayesian perspective as it has certain advantages. Point

estimates of performance measures are obtained. Predictive distributions of

number of customers in system are derived along with their moments. Interval

estimates and tests of hypotheses are also presented.

Key Words and Phrases: Bayesian analysis, Bayes factor, Credible regions, M/M/1 queue, Performance measures, Posterior predictive distribution, Traffic intensity.

2007, VOL. 27, NOS. 1 & 2, 025-041 0196-6324/07/010025-17 $22.00

25

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26 CHOUDHURY & BORTHAKUR

1. INTRODUCTION. Queueing theory has seen tremendous growth since it started its

journey in 1909 when A. K. Erlang, a Danish engineer, published his

fundamental paper relating to the study of congestion in telephone traffic. A

wide variety of models has since been studied. Almost all of these models

have not only been intimately analyzed but also extended in many almoat

inconceivable ways so as to accommodate the innumerable variations found in

practice.

A practitioner however has a somewhat different agenda. When

assigned to analyze a real life queueing system, he usually begins by

identifying a suitable queuing model fitting the characteristics and description

of the model in hand. Given the wide choice of models available in literature,

this is not too difficult a task. However it is from this point that problems of

the practitioner begin. Having identified a suitable model, most field level

analysis would aim at determination of one or more performance measures.

Traffic intensity, mean number of customers in the system or in the queue etc

are popular examples. These measures in turn are expressed as some function

of parameters of the queue model. Clearly therefore, determination of

performance measures require estimates of such parameters. To the

practitioner, lack of knowledge of numerical values of the parameters presents

a formidable hurdle. How is he to obtain such estimates? What are the

methods available? What is the type of data he would require for obtaining

such estimates? How should such data be collected? These are indeed major

challenges.

This aspect of statistical inference in queue models is the subject of

this paper. Needless to add, without adequately developed inference

procedures, the large body of queueing models would remain confined to the

pages of hard bound journals thereby defeating the very purpose for which

they were proposed in the first place. It is a little disquieting that in spite of

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BAYES IAN INFERENCE IN MIM/1 QUEUE 27

such importance, this branch of knowledge has not yet developed to the

desired extent and we are still in early stages of its growth.

The model we have chosen to analyze is the classical single server

Markovian model. Our choice of this model has been dictated by two

considerations. First it is a popular model with vast applications which in turn

means that the necessity of inference is greater here. Additionally while we are

aware that some work in this direction has already been carried out, a few

drawbacks have drawn our attention. Devising a way out of these drawbacks

and suggesting improved methods formed our second consideration. We

propose to take up the same using Bayesian approach.

This paper is organized into six sections of which this is the first.

Section 2 contains a brief overview of important results of M/M/1 queuing

model. Section 3 reviews literature. Section 4 justifies our choice of Bayesian

methodology. Section 5 is the core section of this paper containing Bayesian

analysis. An example is considered in section 6. Section 7 concludes the

paper.

2. A FEW WORDS ON M/M/1 OUEUES.

2.1 The Model Assumptions. Typical assumptions associated with this model

are:

(i) Inter arrival time of customers arriving into the system follow exp(A.).

(ii) Customers arrive into the system one by one.

(iii) There is only one server.

(iv) Time required to serve each customer follows exp(Jl).

(v) Customers are served on first come first served basis.

(vi) There is no restriction on waiting space.

(vii) Calling population is infinite.

There are many real life queuing systems which satisfy these

assumptions. Common examples are one man barber shops, doctors clinics

etc.

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28 CHOUDHURY & BORTHAKUR

2.2 A Few Notable Results. Steady state behavior is often assumed. It

implies that traffic intensity (p) which is defined as the ratio of arrival rate to

service rate is less than one. Mathematically, it implies that A. < f.J which is a

very natural thing to happen for stability of the queuing system. Even in case

this assumption is not initially met, operational manager(s) can be trusted upon

to act on the system so that the assumption is ultimately met. The system

would otherwise 'explode' as queue size would continue to increase.

Under this steady state assumption, a well known result of queuing

theory is the Pollaczek-Khinchine (P-K) formulae (Medhi [2003], page 260).

Derived in the context of an M/G/1 queue, this formulae gives the probability

generating function (p.g.f) of a non negative integer valued random variable

(N) representing number of customers in the system at the instant a customer

departs from the system. Also popularly known as the P-K formulae, this

generating function is given by

V(s) = {(1 - p)(I-s)s*(A.-A.s) }/ {s*(A.-A.s)-s}

where B\s) is the Laplace transform of B(t}, the arbitrary (general)

distribution function of service time. Mean of this service time distribution is

assumed to be 1111. For the particular case of an M/M/1 queuing system, the P­

K formulae reduces to

V(s) =(1-p)/(1-ps)

The distribution of number of customers in the system at the instant a

customer departs from the system (N) can be obtained from the above p.g.f

and is given by

P(N = m) = {< 1 - p ) p m , m = 0,1,2 ... 0 , otherwise

(I)

Mean system size and mean queue size can be shown to be L= p/(1-p)

and Lq = p2/(l-p) respectively.

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BAYES IAN INFERENCE IN MIM/1 QUEUE 29

3. LITERATURE REVIEW.

An early attempt at statistical inference in MIMI! queues was made by

Clark ( 1957). Based upon a likelihood function constructed out of data on

initial queue size, the time T up to the m1h departure from the system and the

total number of arrivals up to time T, he presented M.L. estimates of the

arrival rate and service rate viz. A, and J.l . Basawa and Prabhu ( 1988) in their

landmark paper constructed four likelihood functions for four different data

collection experiments. However, they presented only asymptotic results.

Schruben and Kulkarni (1982) and thereafter Zheng and Seila (2000) showed

that under frequentist approach, performance measures like L , Lq . W and

W q all have undesirable properties in the sense that expected values and

standard errors of estimators of these performance measures do not exist.

Zheng and Seila (2000) have presented a way out of this imbroglio by

restricting the upper bound of p to some value Po (<I) .

McGrath et al (1987a, 1987b) in their seminal two part paper put

forward the concept of Bayesian approach to the statistical inference in

queues. This approach has since been significantly developed by Armero and

Bayarri (1994a, 1994b, 2000). They have worked with a likelihood function

constructed by observing some independent inter arrival and service times. In

their 1994a paper, moments of the predictive distribution were shown not to

exist. In their l994b paper, they circumvented this undesirable situation by

proposing a new distribution for the prior viz. the Gauss Hypergeometric

distribution. However, the number of hyper parameters had to be increased. In

general, lower the number of hyper parameters the better. This is because

those too would require to be estimated except of course in the rare case when

all the hyper parameters are known as part of prior information.

Sharma and Kumar ( 1999) addressed the issue of statistical inference

both from frequentist as well as from Bayesian perspective. A unique feature

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30 CHOUDHURY & BORTHAKUR

of their work was construction of UMP critical region for testing hypotheses

on performance measures using randomized testing procedure.

4. WHY BAYESIAN.

In any inference exercise, construction of estimators is not the only

exercise. Derivation of appropriate sampling distribution, confidence intervals,

methods for testing hypothesis are always addressed. In queuing systems, the

frequentist approach has been found to be wanting in these areas except

possibly in asymptotic cases. An unpleasant fact is that none of the frequentist

approaches reviewed in section 3 has provided derivation of the sampling

distribution, sampling variance or confidence intervals of estimators of typical

performance measures. These matters are indeed very involved. Add to this

evidence the work by Zheng and Seila (2000) who have provided an elegant

proof establishing non-existence of expectations and standard errors of

common estimators of performance measures under frequentist approach. This

approach therefore is not without roadblocks.

In contrast, statistical inference under Bayesian set up provides

techniques which do not get weighed down by these difficulties. Standard

errors of estimators of performance parameters 'are most naturally given by

the standard deviation of those (posterior) distributions' (Armero and Bayarri

[2000]). The posterior distributions are sampling distributions. Consequently,

construction of credible intervals (the Bayesian equivalent of frequentist

confidence intervals) can easily be accomplished. Testing of hypothesis using

'Bayes Factor' could also be carried out with simple interpretation unlike the

randomised testing procedure outlined by Sharma and Kumar (1999).

Additionally, 'there is plenty of information a priori about the queue especially

if it is assumed to be in equilibrium (which implies at the very least that the

system has been working for an extended period of time)' (Armero and

Bayarri [2000]). This prior knowledge is presented in the form of prior

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BAYES IAN INFERENCE IN MIM/1 QUEUE 31

distribution. In conjunction with the likelihood, it contributes greatly in

improving quality of the probability model.

5. BAYES IAN INFERENCE.

5.1. The likelihood function. In order to generate data, it will be necessary to

observe the queuing system at epochs of departing customers. One has to note

the number of customers left behind in the system at the departure epoch of

customers departing from it after receiving service. In this way, the number of

customers left behind by n such departing customers has to be recorded. Let

these be x1.x2, ... ,X0 where x; denotes number left behind by the i1h such

customer on whose departure, the number left behind in the system is

recorded. Such a scheme will ensure that the data generating process is

consistent with probability distribution (1). Having thus obtained a sample

x1.x2, • • • ,x0 from this distribution, the likelihood function can be constructed

and is given by

n

L= (1-p)" P ;:t;

= (1-p)" pY, n

where Y= I:x;. i=l

(2)

It is pertinent to note here that this data generating scheme ensures

independence of sample observations provided they are well spaced, meaning

departing customers for whom number left behind in the system is noted

should be sufficiently spaced. Thus data on number left behind in the system

should not be collected for consecutive departing customer or even for nearly

consecutive customers. This fact this will ensure independence of sample

observations follows from the ergodicity property of the Markov chain

considered in the derivation of P-K formulae (Medhi[2003), page 259-60).

Briefly, if Xm denotes number in the system at the departure epoch of the mth

customer and Pii = P(Xm+,=j I Xm=i), then the ergodicity property says that

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32 CHOUDHURY & BORTHAKUR

provided p < 1, the limiting probabilities v j = lim pfkl, j = 0,1 ,2, ... all exist k-><Xl J

and are independent of the initial state i.

5.2 The Prior.

Bayesian prior distributions belong to two broad categories viz.

subjective and objective. As the name suggests, subjective priors are those

which are based upon prior belief or prior knowledge on random behavior of

the parameter of interest. Such priors are therefore dependent upon depth and

extent of knowledge available on random behavior of the parameter. Objective

or non informative priors on the other hand are those that are constructed out

of a state of ignorance on behavior of the parameter. Both have their critics

and votaries.

In this paper, we shall assume a popular objective prior for the

parameter of interest p. As noted earlier, we assume steady state meaning

thereby p < 1. Further p cannot be negative. Consequently, we shall assume

the uniform prior for p given by

0(p) oc 1' O<p<l, (3)

This prior basically says that beyond the natural limits, nothing more is

known about the random behavior of p . The density curve is therefore flat in

the interval [0,1).

5.3 The Posterior.

Having collected data and zeroed in on a well defined prior, we next

proceed to construct the posterior of p which by definition is proportional to

the product of prior and the likelihood. This posterior distribution reflects

random behavior of p after the prior information has been updated by data

generated from the system (in the form oflikelihood).

Using prior (3) and likelihood (2), posterior distribution of p is given by

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BAYES IAN INFERENCE IN MIM/1 QUEUE 33

nCp l data) oc p Y (1-p) n, O< p < 1

= {{1/B(y+1,n+1)} p Y (1-p) n, O<p<l (4)

0 , otherwise

where B(.,.) is the beta function. Using it, we can proceed to obtain

point estimates, interval estimates as well as carry out tests of hypotheses.

5.4 Point Estimators of Traffic Intensity ( p ).

Point estimator in Bayesian framework is tenned as Bayes estimator. By

definition, it is mean of the posterior distribution. Bayes estimator of p using

posterior (4) is given by

p= E( pI data)

= (y+ I) I (y+n+ 2).

Associated to any estimator is a measure of its variability. In Bayesian

framework where the parameter is a variable, the natural standard error of the

estimator is standard deviation of the posterior. For Bayes estimator of p,

standard error is given by

S.E. (pNc) = 'I'Var(pldata)

= ~(y+l) (n+l) /(y+n+2Y (y+n+3)

5.5 Predictive Distribution of Number in the System.

Given sufficient infonnation about the past and present behavior of an

event or an objective, an important objective in scientific investigation is to

predict the nature of its future behavior' [Sinha ( 1998), page 103]. Predictive

distributions are designed to perfonn this role.

Let D = {x1.x2, ••. ,x0 } be a random sample from some distribution with

p.m. f. f(x I 8), e E n. With a prior g(S) and the corresponding posterior me I

D), the posterior predictive distribution of a future observation x is defined as

the posterior expectation of f(x I 8), e En and is given by

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34 CHOUDHURY & BORTHAKUR

h(x I D) = I f(x I 9) me I D) d9 n

Using this definition, the predictive distribution of number in the

system immediately on the departure of a customer (N) will be given by

1 P(N=m I D)= J(l-p)p m O(p I D) dp

0

{B(y+1+m,n+2)/B(y+1, n+1) , m=0,1,2, ...

0 , otherwise

Moments of this predictive would exist subject to extremely mild

conditions. The factorial moment of order r is

E{N(N-l) .. . (N-r+l) I D} = Ep/data [E{N(N-l) .. . (N-r+l)ID,p}

f (r!){p/ 1-pf {pY(1-p)Il/B(y+1,n+l)}dp 0

{(r!)B(y+r+l,n-r+l) }/B(y+l,n+1) ,

r=1,23, ...

which will exist provided n > r-1. Condition for existence of the first four

moments is n>3. It is difficult to visualize a scenario where sample size

does not exceed 3. The first four moments would thus exist for all practical

purposes (higher order moments are rarely of interest).

5.6 Point estimator of L and Lq.

Mean system size (L) and mean queue size (Lq) are two popular performance

measures. Here we shall present their Bayes estimators along with their

standard errors. We obtain these from the posterior (4)

L =E(LID)= B(y+2, n)/B(y+1,n+1),

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BAYES IAN INFERENCE IN MIM/1 QUEUE

Lq =E(Lq/D) = B{y+3,n)/B{y+1,n+1),

SE ~ )= ~B{y + 3,n -1) /B{y + 1, n + 1)- { B( y+2,n) /B{y+1,n+1) }2 ,

SE ~q)= ~B{y + 5, n -1) /B{y + 1,n + 1)- { B{y+3,n)/B{y+1,n+1)}2 .

5.7 Credible Region of Traffic Intensity.

35

There is a school of thought which argues that a region to which the

parameter of interest can belong to with high certainty is perhaps more

meaningful than simply proposing a point estimate. In the classical or

frequentist setup, this is known as confidence interval whereas in Bayesian

setup, the equivalent term coined for the purpose is credible region.

In general, any region I c R 1 such that

f 0(9/D)dS= l- a I

(5)

is called the posterior I 00(1 - a) % credible region of (}. Given data (i.e. the

sample), this region r is the region where the true value of 8 would lie with

probability (l - a). There could be more than one credible region. One

advantage with credible regions is the straight forward interpretation they lend

themselves to as compared to the somewhat roundabout interpretation that

confidence intervals have.

Let I= [I, l+h], h > 0. We need to determine I and h such that I is the

l 00(1 - a) %credible region of p. Here (5) can equivalently be stated as

P(l S p s I + h) = l - a ( 6)

=> P(p-< 1)+ P(p :--l+h)= a

=> P(p-< 1) =a} (7)

and P(p :--I+ h)= a2 with a 1 + a2 =a, O<a1, a2< I.

Determination of! and h would depend upon choice of a 1 and a 2 . With

posterior (4), (7) is equivalent to

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36 CHOUDHURY & BORTHAKUR

I 1 f TI (piD) dp = a) and f TI (piD) dp = a2 0 l+h

:::> B(l, y + 1, n + 1 )/B(y + 1, n +I)= a)

and B(l + h, y +I, n +I )/B(y + 1, n + 1) = 1- a2 .

Here B(.,., .) is the incomplete beta function. For determination of 1 and

h, one can use standard packages. Often one would be interested in the 1 00(1-

a)% shortest credible interval of p. Our posterior (4) being unimodal, the

condition which ensures that h is minimum is (Zellner [ 1971 ), page 27)

TI{I+h I D)= TIOI D), (8)

TI(. I D), being posterior (4). Determination ofl and h so that I= [I, l+h], h>O

is the shortest credible interval thus implies numerically solving equations (6)

and (8).

Another popular notion is that of Highest Posterior Density or HPD

interval. Such an interval I is defined to be one which in addition to being the

shortest possible is also such that posterior density at every point inside I is

greater than posterior density at every other point outside I. It is known that if

the posterior density is unimodal but not necessarily symmetric, the shortest

credible interval and HPD interval are one and same (Sinha [1998], page 143).

5.8 Testing of Hypothesis.

One could be interested in three types of hypothesis. These are:

HorP~Po vs H11: P>-Po

H 02 : L ~ L0 vs H 12 : L >- Lo

(9)

(10)

(11)

Testing of hypothesis under Bayesian frame work involves computation

of Bayes factor B 10 which is the ratio of the posterior odds ratio in favor of H 1

over the prior odds ratio in favor of H1. Symbolically,

P(H1 I data) I P(Ho I data)= B10 P(H1) I P(Ho) .

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BAYESIAN INFERENCE IN M/M/1 QUEUE 37

Bw is a summary of the evidence provided by the data in favor of one

scientific theory (H1), as opposed to another (H0). {Kass and Raftery (1995)} .

Once Bw has been computed, decisions in favor or against H1 can be taken on

the basis of the following rule by Kass and Raftery (1995).

Table 1 : Decision rules using Bayes Factor

2loge(BJO) Evidence against Ho

0 to 2 Not worth more than a bare mention

2 to 6 Positive

6 to 10 Strong

>10 Decisive

Using uniform prior, appropriate expressions for testing hypothesis (9) are

P(Ho I data)= B(po, y+l, n+l) I B(y+l, n+l), P(Ho) =Po,

P(H1 I data)= 1- P(Ho I data), P(HI) = 1-P(Ho).

For testing hypothesis on L and Lq, we note that for 0 < p < I, these

are increasing functions of p. Consequently

(i) L ::-::; Lo <=> p::<;;p~

and L >- Lo <=> p>- p~

where p~ = L0 /(l + Lo) s

' o-< Po-< I.

(ii) Lq ::-::; Lq0 <=> p::<;;p3

and Lq >- Lq0

<=> p>- PO

where p~ is the positive root of 2 P + Lq0 P - Lq

0 = 0, 0-<p-<1.

Testing hypothesis on Land Lq is thus equivalent to testing hypothesis

on p. The same procedure can now be applied.

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38 CHOUDHURY & BORTHAKUR

6. AN EXAMPLE.

We apply our results to a real life queuing problem. The setting is a

computerized branch of a renowned large bank which has country wide

network in India. During Monday to Friday, banking hours are from 10 A.M.

to 2 P.M. and again from 2.30 P.M. to 4.00 P.M. On Saturday, customer

service is offered for half of the day during 10 A.M. to 12.30 P.M. The branch

is located in a thickly populated urban residential area of a large city

(population 8,08,021 in census 2001 ). Within a radius of 2 kms of the branch,

there is just one more branch of a competing bank so that there is heavy

demand for banking service from neighboring residents. Our focus was on a

particular counter manned by a single clerk. Service offered by the counter

were of three types - cash receipt (non bulk), cash withdrawal and updating

passbook. Customers requiring service form a queue (if the clerk is not idle)

and are served on the basis of FIFO rule.

The assumptions of MIM/1 model were confirmed to hold good. For

Bayesian inference, data was collected at around four time points during

Monday - Friday and two time points during Saturday. The counter was

observed at 10.45 A.M., 12.15 P.M., 1.45 P.M. and 3.30 P.M. during Monday

-Friday. On Saturday, the counter was observed at 10.45 A.M. and 12.15

P.M. Data collected pertained to number of customers left behind in the

counter (i.e. system) at the departure epoch of a customer departing the system

(after receiving service) at around the above mentioned time points. In all,

data was collected during nine full working days and two half days

(Saturday's). Forty observations were thus obtained. These observations were

11 , 7, 3, 3, 1, 1, 14, 17, ~ 11,6, 1, 1, 2, 10, 8,2,6,4,6,4,0,~8,4,8, 11,0,

19, 2, 11 , 0, 13, 12, 1, 0, 0, 10, 15, 13 . Using this sample, the following were

obtained:

i) Estimate of traffic intensity (p) = 0.86054

ii) Standard error of estimate of p = 0.02017

iii) Table 2: Distribution of predictive probabilities

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BAYESIAN INFERENCE IN MIMI I QUEUE 39

No of customers at departure epoch Predictive probability

0 0.139456

I 0.119601

2 0.102631

3 0.088117

4 0.075698

5 0.065065

6 0.055956

7 0.048148

8 0.041452

9 0.035706

10 0.030773

iv) The first four factorial moments of the predictive distribution are 6.325,

82.3871794, 1658.584008, 45902.43308.

7. CONCLUSION.

A Bayesian approach to statistical inference in the single server

Markovian queuing system has been presented. Unlike the frequentist

approach which has been known to possess a number of undesirable properties

with regard to statistical inference of queue models, Bayesian technique lends

itself to lucid analysis. Not only have we presented point estimates, credible

intervals and testing of hypothesis has also been discussed. The technique we

have presented has two notable improvements. Unlike existing results,

moments of our predictive distribution exists subject to very mild restrictions

and without increasing the number of hyper parameters. Secondly, our

sampling scheme is designed for accurate data collection. Methods presented

by other researchers require data on inter arrival times and service times which

are little difficult to measure accurately. In contrast, our scheme requires data

on number of customers left behind in the system at departure epoch. This is

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40 CHOUDHURY & BORTHAKUR

an advantage as data on number of customers can be collected accurately. For

the same reason, viz. differences in type of data required for analysis, it is not

possible to compare our results with those obtained by other approaches.

Lastly, we have presented closed form expressions which are simple enough to

be implemented even in an Ms-Excel sheet.

Acknowledgments. The authors wish to thank the invisible referee whose

careful comments assisted in sharpening a few elements of the paper.

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