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Polling systems with a patient server and state-dependent setup timesYAVUZ GUNALAY a & DIWAKAR GUPTA ba Industrial Engineering Department , Eastern Mediterranean University , Gazi-Magusa,Mersin, 10, Turkeyb Michael G. DeGroote School of Business, McMaster University , 1280 Main Street West,Hamilton, Ontario, L8S 4M4, CanadaPublished online: 30 May 2007.

To cite this article: YAVUZ GUNALAY & DIWAKAR GUPTA (1997) Polling systems with a patient server and state-dependentsetup times, IIE Transactions, 29:6, 469-480, DOI: 10.1080/07408179708966353

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//E Transactions (1997) 29, 469-480

Polling systems with a patient server and state-dependentsetup times

YAVUZ aONALAyl and DIWAKAR GUPTA2•

I Industrial Engineering Department, Eastern Mediterranean University, Gazi-Magusa, Mersin 10, Turkey2Michael G. DeGroote School of Business, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada US 4M4

Received September 1995 and accepted November 1996

We analyze a class of cyclic service (polling) systems with multiple customer classes (stations) in which the server stops cycling uponfinding the entire system empty and initiates a setup only when the polled station has at least one customer in the queue. Interest insuch systems is fueled by applications in design and performance analysis of manufacturing as well as telecommunication systems.We develop a discrete Fourier transform (DFT)-based near-exact numerical technique and an approximate method for systemswith any number of stations. The DFT-based algorithm is accurate but computationally demanding when either the number ofstations is large or server utilization is high. In these cases, the approximate method appears to work well in a large number ofnumerical tests.

1. Introduction

Polling models are frequently used to study problemsarising in the design and performance analysis of com­puter and telecommunication networks as well as pro­duction and transportation systems. Typically, suchsystems have a single server, which is either a token (inlocal area networks using token passing protocol tocontrol access), or a machining center, or a robot, orperhaps an automated guided vehicle (AGV). This serverattends to several types of customer (stations) in a cyclicfashion. Customers are either jobs or messages that queueup for service (machining, transmission) at either geo­graphically or logically distinct locations in the system.There is a vast literature dealing with various applicationsand analyses of polling models. Comprehensive surveysof previous work can be found in two review articles byTakagi [I, 2].

The server behavior can be specified in terms of pos­sible actions following instances when it either 'polls'(checks queue status) or 'completes' a tour of service at astation. Such protocols simplify management and offerdecentralized control. A polling instant occurs immedi­ately after the server moves to a station. The server re­quires a setup before it can commence service at the newlyarrived station.

There are two possible courses of action at each stationcompletion instant: either the server moves to the next

• To whom correspondence should be addressed. ·

0740-817X © 1997 "lIE"

station, irrespective of system state, or it idles until thesystem occupies some desired state. Similarly, two typesof server action may be specified at each polling instant.These determine whether or not the server sets up forservice, and how many customers it serves before regis­tering a station completion instant. Nearly all of the ex­isting literature assumes that the server never idles andthat it sets up irrespective of whether there are anywaiting customers at the polled queue or not. Gupta andSrinivasan (3] describe such models as having a continu­ously roving server with state-independent setups. Com­monly employed regimes that govern the number ofcustomers served at each server visit include exhaustive,gated, and timer-limited.

The focus of attention of the present study is the classof models that can be described as having a patient serverand state-dependent setups. In our models, the server re­mains idle at the station at which it registers a stationcompletion instant if the entire system happens to beempty at that instant. It is restarted next by a new cus­tomer arrival at any station in the system. Similarly,setups are incurred only when the polled station queue isnot empty. Otherwise, the polled station is skipped andthe server moves on to the next station. We present de­tailed analysis assuming exhaustive service strategy ateach station. However, our analysis extends in astraightforward fashion to include gated regime as wen.

Polling models with a patient server and state-depen­dent setups are particularly suitable for representingsystems for which avoiding unnecessary switching andsetups is desirable [3]. For example, they mimic the

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operation of many computer communication networks [4,5) and production systems [3, 6]. Furthermore, modelsthat have either continuously roving server or state-in­dependent setups or both can be studied as special in­stances of the patient server, state-dependent setupsmodel (see Section 3). Therefore, it is a fundamentallyimportant class of models for carrying out detailedanalysis and performance evaluation. Unfortunately, the.analysis of such models is not as well developed as itought to be. The only papers that consider state-depen­dent setups appear to be those by Ferguson [5], Bradlowand Byrd [4], and Gupta and Srinivasan [3]. Of these,Ferguson [5] and Gupta and Srinivasan [3] study con­tinuously roving server systems, and only Bradlow andByrd [4] consider a patient server model.

With the exception of the exact analysis for the two­station model in Gupta and Srinivasan [3], other analysespresented in all of the studies mentioned above are ap­proximate analyses. As such, the problem is widelyregarded as difficult to analyze. Ferguson [5] first observesthe difficulty in 'solving' the functional equations thatrelate intervisit times at a station to the server sojourntime at that station. So, he develops some bounds andapproximations for the mean waiting times. Gupta andSrinivasan [3J present an example in which all of Fer­guson's upper bounds for the mean waiting times of atwo-station system are lower than their correspondingexact values. Gupta and Srinivasan [3] also develop anapproximation procedure that takes the average of twoseparate ways of approximating the system behavior. Theaverage value of the station mean waiting times obtainedfrom the two approximations have been found to betypically within 5°,4 of the simulated values in a largenumber of test cases. This makes their approximation themost accurate among those reported in literature.

Bradlow and Byrd [4] set out to analyze a patient servermodel with zero switchover times and state-dependentsetups. However, in their approximation scheme theyassume that the probability of having the system empty iszero. This assumption effectively reduces their model tothe continuously roving server model studied by Fer­guson and by Gupta and Srinivasan.

In this paper we show first that the two-station systemwith a patient server and state-dependent setups can beanalyzed by using the computationally efficient descen­dant sets technique (see, for example, [7] for details of thedescendant sets method). The same two-station model,but in which setup times are called changeover times, wasanalyzed earlier by Eisenberg [8]. Next, we develop anumerical procedure, utilizing discrete Fourier trans­forms, for finding the joint queue length probabilities atpolling instants for systems with arbitrary number ofstations. These probabilities immediately yield the desiredperformance measures. The procedure assumes thatthe" size of the buffer at each station is finite. Althoughthis makes the technique mathematically approximate,

Gunalay and Gupta

numerical tests show it to be highly accurate when thelimiting buffer sizes are chosen carefully. From a practicalstandpoint, finite buffers are more realistic.

The DFT-based technique requires considerable com­puter resources. It appears to be practical only when thenumber of stations is small and overall server utilizationis relatively low (no more than 0.5). Therefore we alsopresent an approximate model, which is an adaptation ofan approximation scheme by Gupta and Srinivasan [3], tohandle systems with high server utilization and/or a largenumber of stations. Under such circumstances, theapproximation appears to work well with an average er­ror of about 6%. Thus the problem can be analyzedcompletely and with reasonable accuracy with the help ofthe two complementary approaches we present.

The overall organization of the remainder of this paperis as follows. Section 2 defines the model and our nota­tion. This is followed in Section 3 by the expressions forthe waiting time distribution and the mean waiting timeat each station. The analysis is developed mainly in Sec­tion 4, which is divided into three parts. The first partdeals with an exact analysis of the two-station system.The second presents a DFf-based numerical procedurefor finding the complete probability distribution of jointqueue lengths at polling instants, and the third partcontains an approximate method for finding mean stationwaiting times. Section 5 contains a summary of theexperimental setup and of numerical test results. A vari­ety of different polling models that can be treated usingthe methods presented in this article are discussed inSection 6.

2. Preliminaries and notation

Let stations QI, Q2, ... I QN be indexed in the same orderas they are visited and let the customers served at Qi beknown as type-i customers. Once a tour of service begins,customers at each station are served exhaustively and inthe first-come-first-served (FCFS) fashion. Let Ai denotethe arrival rate of the type-i customers and let the randomvariables Ti,Si, and B, represent the setup time, the servicetime and the busy period at Qi, i = 1, ... ,N, respectively.Whereas arrivals follow independent Poisson patterns,random variables T, and S, have arbitrary distributions.

The overalJ arrival rate is denoted by 1, i.e.,A = L:~l Ai- The long run proportion of type-i arrivals tothe system, Pi, equals 1;/ A; utilization (traffic intensity) atstation i, PI' equals ).jE[Si]; and P = l:~1 Pi denotes theoverall utilization of the server. Note that p < 1 is thenecessary and sufficient condition for the stability of thesystem [9] and this is assumed to be the case. Our aim isto present numerical methods to calculate the steady-stateprobabilities. .

The cumulative distribution function (CDF) of a ran­dom variable A is denoted by A(t) and its Laplace-Stieltjes

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Customer waiting time at Qj, denoted by Wi, can be ob­tained by using intuitive arguments if we imagine that ateach station i completion epoch the server goes on va­cation, which ends at the next station-i station-beginninginstant. Let K;(z) be the PGF of the number of type-i

; customers present at Q; at the end of such a vacation.Then Fuhrmann and Cooper's 'Stochastic Decomposi­tion (SD) Theorem' (see [10] for details) applies andthe LST of the waiting time of type-i customers is theproduct of the following two terms:

(2)

Polling systems and state-dependent setup times

transform (LST), defined as E[e-sA], by A*(s). If the ran­

dom variable is discrete, then its probability generatingfunction (PGF), E[zA], is denoted by A(z). The kth mo­ment of a random variable A is indicated by a(k), wherea(1) ~ E[A] = ii. We use the convention that any emptysum equals zero and that any empty product equals 1.

Four time epochs, at which Markov chains are im­bedded, are of special significance. These are:

• polling instants - time epochs at which the server(physically) arrives at a station;

• station-beginning instants - time epochs at which theserver becomes available to serve a particular type ofcustomer, either after a setup period or after an idleperiod;

• station-completion instants - time epochs at whichthe server completes a tour of service at a station asdictated by the chosen service regime at that station;and

• switch points - time epochs at which the server startsto physically move from one station to the next.

As explained below, every switch point has a corre­sponding station completion instant and similarly eachpolling instant can be associated with a station beginninginstant, but the opposite relationships do not alwayshold. The polling cycle for station i, C;, is defined as thetime interval between two consecutive polling instants ofQi·

At each station-i polling instant the server checks Qiand if it finds any type-i customers waiting, then a setupis performed, Otherwise, -i.e., in the case that Q; isempty, the server simply skips that station and imme­diately switches to the next station. Although the serverdoes not serve any type-i customers during that visit, wemark both a station-i beginning and a station-i com­pletion instant. At each station-i completion instant, theserver looks at the whole system and continues to roveonly if there is at least one station other than i that isnot empty. However, if it observes an empty system, itthen idles at station i and waits for an arrival to occur.A station-r completion epoch is registered, but no switchpoint is marked because the server does not move out ofQ;.

The server is activated next by the first new arrival tothe system. We define the length of time that starts by thisarrival, and ends when the server stops again, as a supercycle. If this arrival happens to be at Q;, the station wherethe server resides, then another station-s beginning instantis registered and service resumes (no setup required).Otherwise, a type-i switch point is registered and theserver leaves Q; immediately, moving in the cyclic se­quence and arriving instantaneously at the desired sta­tion. Thus, during any cycle, there is exactly one polling .instant and one switch point for each station, but theremay be more than one (but an equal number of) station-ibeginning and completion instants, for i = I, ... ,N.

471

Let i (nl, ... ,nN) denote the state of the system at anobservation epoch that is either a polling instant, nor astation completion epoch at station i. The term nj rep­resents the number of customers waiting at queue j. Next,let Ii(n) and g;(s), where n = [nl, ... ,nN], denote the stateprobability at a polling instant and at a station comple­tion epoch of Qj, respectively. These probabilities arecalculated by using the long run proportion of the num­ber of times the system state is i(nl, ... ,nN) at such ob­servation epochs to the total number of stationcompletion events encountered. The PGFs of theseprobability distributions are defined as fi(z) and g;(z). Weuse the upper-case letters to denote the conditionalprobability and PGP of system state. Put differently,F;(n) (Gi(n)) is the joint probability that queue lengthsare (n), ... ,nN) given that a station i has been polled(completed) and Fj(z) (G;(z)) is the corresponding PGF. Itfollows that

F;(z) = /;(z)//;(1), i = I, ... ,N, (1)

where I is a 1 x N vector of ones.

3. Waiting times

• 1 - K;(z) ( (1 - p;) )~ (A.; - AiZ ) = KH I) SiCA; - AjZ) - z .

In our model, a station-beginning epoch occurs eitherafter a polling instant is registered at station i and setup iscompleted, or after a server idle period at station i that isfollowed by a new type-i arrival. If Kj(z) = k;(z)/kj{l),then the previous statement permits us to write

ki (z) == [fi (1, ... , 1,Z, 1, ... , 1) - f; (I i)]r;* (A.i - A;Z)

+/;(1;) + g;(O)p;z, (3)

where gi(O) is the empty system probability fori == 1, ... ,N, and I; = [1, ... , 1,0,1, ... ,I]. Substitutingfrom the above into (2) and simplifying, we obtain The­orem 1, which is presented below without proof.

Theorem 1. The LST of the waiting time distribution oftype-i customers in a polling model with a patient serverand state-dependent setups is

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Hj+ (Ai _ liZ) = (I - F; ~1, ... , 1,z, 1, ... , 1)Tt (}~i - liZ)F; (1) + (1 - j;(O)))w,t, +d.p,

_ (I - ~. ()~I .- )~i=))h (0) - diP" (I - =))F, (1) + (1 - fiCO) )}~It, +d.p,

X(I-Pi)

(4)St ().., - ,{,=) - z '

and the mean waiting time is

/,(2) + 11_-t_I"(I)..L II _ r_(0)))2t (2) ).EiS2]E[W;] = I .....,/~, Jj, I \. J I ~, i ~j t,I (I) - + 2(1 _ .)'2l,(/; + (1 - j;(O)))~lt, + d,p;) P,

(5)

where 1,-(0) ~ F,-( 1, ... 1 1,0, 1, ... , 1) is the empty station. . l:1 - - (I) l:1

probability, d, = g,(Oljf, (1), I. = (8j8=)F,(1, , 1,=, 1,... , I) 1%= I, and /;(2) = (82 ja:?)F; (1, ... , 1,Z, 1, 1 1)lz=1.

Next we observe that several of the previously studiedpolling models can be looked upon as special cases of ourmodel and hence their customer waiting time distribu­tions can be obtained from Theorem 1.

Observation J. A patient server polling model with state­independent setups is obtained by setting the empty sta­tion probabilities, .Ii(O), to zero in (4) and (5). The re­sulting expressions yield equations (21) and (30) of [6]after appropriate notational changes and simplification.

Observation 2. The zero switchover time model, firststudied by Cooper and Murray [J l], is obtained when weset T; = 0, i = 1, ... IN. This means we replace 1';* (-) by 1in (5), to obtain the equivalent of equation (33) in [11].

Observation 3. Continuously roving server polling modelsare also a special case ofour model. By setting d, = 0 in (4)and (5) we obtain the continuously roving server systemwith state-dependent setups (see equations (13) and (16)of [3]). Furthermore, setting fiCO) = d, = 0, i = I, ... ,N,reduces our model to the basic polling model havingcontinuously roving server and state-independent setups(see [2] for a variety of methods dealing with this model.).

From Theorem I, it is clear that the moments of stationwaiting time depend on .Fi(], ... , 1,z, 1, ... , 1) throughthe latter's moments. Hence we devote the next section tofinding moments of queue lengths at polling instants. Torna ke the exposition simple, and without any loss ofgenerality, we restrict our attention to station ] only.Similar results for other stations can be obtained simplyby changing the station indices.

4. Queue lengths at polling instants

Consider an arbitrary polling instant of Ql, which weshall can the reference point. The main idea of the de-

Giinalay and Gupta

scendant sets method is to express the queue length at Qlat the reference point as the sum of 'contributions' fromaU previous customers until the reference point. To fa­cilitate this process, we index cycles backward in time anddevelop a recursive method for finding the descendants ofeach customer untiJ the reference point.

Recall that a polling cycle is the length of time thatelapses between any 'two consecutive polling instants at astation. Cycles are indexed such that the cycle that fin­ishes just before the reference point is c = 0, the cyclebefore it is c = 1 and so on. Thus, the reference point isthe start of a cycle, which is indexed c = -1. Let Ci,edenote a type-i customer who gets service during cycle c.The descendant set of Ci,c consists of itself, all customerswho arrive during its service time (children), children ofthose customers (grandchildren), their grandchildren, andso on, until the reference point. Thus the descendant setof a customer is a proper subset of its parent's descendantset, if it has one. Because we can count the size of thedescendant set of each customer, we only need to countthe number of customers from -00 to the reference pointthat do not have a parent. Such customers are those thateither start a super cycle or arrive during setup periods.They were called original customers in [7].

Using these ideas, the PGF of the size of the descen­dant set of Ci,c, which we denote by L"c(z), can be cal­culated recursively by the following expression:

Li,c(z) ~ B; C~I [Aj - AjLj,c(=)] +~ [}'j - }'jLj,C_1 (=)]).

(6)

Boundary values are set as follows: £1,_1 (z) = Z and£,,-1 (z) = 1 so long as i > 1. Similarly, T;,e(z) is the PGFor the contributions of all original customers who arriveduring a setup that occurs c cycles before the referencepoint. This can be calculated as \

Ti,c(=) ~ 1;' (t[},j - )'jLj,c(=)] + ~[}'j - AjLj,C_I(Z)]).

(7)

Finally, the station 1 queue length at reference point canbe written in terms of the sum of contributions of allcustomers without a parent. The resultant relationship is

00 N

F. (z, 1, ... ,1) = IT IT T;,c(z)c=Oi=1

00 N

+ L L.rJ,c(z)(1 - T;,c(z))u;tc(z)c=o ;=1

00 N

- L L d,~,c(z)l1"c(=), (8)c=o ;=1

where for i = 1, ... ,N,

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Polling systems and state-dependent setup times 473

fori= l , ... ,N.

N i-I

Ji,C(Z) = LPj - pjLj,c(z) + LPj - pjLj,c-l (z) and (9)j=i j=l

and if we sum this relation for all i, we obtainNT) = 1 - L~l Pigi(O). The empty system probabilities,gi(O), can be found by solving a system of N Jinearequations as shown by Srinivasan and Gupta. Theseequations have the following form (for details see [6],section 3.3):

However, for N > 2 the Pl(z) terms remain unknown.Notice that these terms represent PGFs and not simpleprobabilities, which makes the problem particularly un­wieldy. In the following two subsections we present twodifferent ways that can be used to overcome this difficulty.

. 4.2. The method of discrete Fourier transforms

DFTs have been used in several previous studies to obtainqueue length distributions from PGFs (see [12] and ref­erences therein). Our technique, which computes jointqueue lengths' distributions at polling instants, is fash­ioned after a numerical approximation used by Le­ung [13] to analyze polling models with a continuouslyroving server, state-independent setups and probabilisticservice strategy. Under the probabilistic service regime, ofwhich K -limi ted service is a special case, the server pro­cesses at most j customers at its visit to station. i withprobabilitya{. Different service strategies can be obtainedby specifying the probability distribution of at for each i.Our approach is a numerical approximation as it requiresthe maximum station queue lengths to be finite. However,this limitation does not seem to have a significant effecton the accuracy of our calculations, so long as the lim­iting queue lengths are chosen carefully. Note that whilecomputing queue length moments for even the mostsimple polling models, the accuracy of calculations islimited either by the machine accuracy or by a user­specified tolerance level or both. Therefore, when limitingqueue sizes are chosen carefully, our method yields nearlyexact values of the mean station waiting times. Themaximum buffer size at Q;, which we denote by Li,i = 1, ... ,N, is chosen such that P(number atstation Qi ~ Li) is a small quantity (e.g. not more than10-8) .

Another important aspect of the algorithm is that themaximum number of service completions at each visit toQi is assumed to be known in advance. Even for the ex­haustive service discipline, we know that in steady statethere exists a finite K, such that

P(number of customers served at each

visit to Qi ::; JCi) = 1.

We defer the discussion of how to set JCi to Section 5 andfor the time being simply assume that this number isknown for each station.

Before describing the algorithm, let us derive thefunctional relationship between the PGF of queue lengthsat polling instants. First, we define v;J(z) as the PGF ofthe queue lengths immediately after the jth service com­pletion at Qi, for j = 0, ... , K: Note that the end of theOth service actually corresponds to the station beginningepoch at Qi. Suppose Qi becomes empty after Y servicecompletions, Y ::; JCi . Then we still perform the x, - }'fictitious service completions; however, the system statedoes not change as a result of these service completions.

(12)

Observation 4. By setting d, = 0 in (8) we obtain the PGFof the queue length at Ql polJing instants for the con­tinuously roving server model with state-dependent set­ups (see equation (23) of [3]). Similarly, after eliminatingexpressions containing the Pl(z) terms from (8), the re­sulting PGF corresponds to a patient server model withstate-independent setups (see equation (12) of [6]).

Let V denote h(I), the probability of polling station i,Note that this probability is independent of the stationindex because each station is polled exactly once in eachcycle. Each station-(i + 1) polling instant is preceded by astation-i completion epoch, except when the latter hap­pens to be an empty system epoch to which the first ar­rival is of type i. Therefore

4.1. Two-station models

When N = 2, notice that Pl(z) = 0, i = 1,2, as thesecorrespond to the polling instants of Qi when the systemis empty, and that never happens. Upon eliminating theseterms the resulting mathematical model reduces to a pa­tient server model with state-independent setups. Such amodel is analyzed in detail by Srinivasan and Gupta [6].Using their technique one can calculate the empty systemprobabilities, gj(O), i = 1,2, by solving a set of linearequations like (13) above.

N c-l N

O"i,c(Z) = IT Tj,c(z) IT IT TkJ'(z). (10)j=i+l p=O k=1

r:c(z) is the PGF of the sum of contributions from allcustomers present in the system at a polling instant of Qi,i = 1, ... ,N, when Qi is empty. An empty queue does not'contribute' anything to station 1 queue at referencepoint. Thus

~c(z) ~ fi(L 1,c- 1(z), .. . ,Li-1,c-l[z}, 1,Li+l,c(Z), .. . ,LNtc(z)).

(11 )

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474 Gunalay and Gupta

Similarly, the DFT of queue lengths at the jth servicecompJetion epoch when Qi is empty, v?J(ki ) can be cal-culated by taking the average of v?J(k) with respect to the

(23)

(22)

(21)

(20)

(19)

to the

(N )... -. k,

5, (k) = S, ~ )_j - )_/w/ ')=1

i; (k) = r,.(t ).j - IjW~j) .)=1

Let .ff(k;) denote the Off which correspondsPGFs, /;(Zi), i = 1, ... ,N. Then,

{,I-I £/_1- 1 {,;+I-I {,N-)

.ff(k;) ~ L ... L L' .. L111=0 ni_I==O 11;+1=0 "N=O

f i(n- .) kill t wk,-In,-I Wki+llIl+l JcN lIN, Wi . .. i_I j+) ... Wi.; .

Notice that, .ff(.) has N - 1 independent variables. Toindicate this, we define the domain set as 5 i = {(kl,... , L;, ... ,kN ) : kj =.0, ... , L j - 1, j = 1, ... ,N andj =1= i}. Therefore, 5' is the projection of 5 into jN-),

where I denotes the set of natural numbers. Finally, byusing the initial value property of OFT [15], we obtain

- 1 {,j-I ... -

.ff(k;) = L.l:h(k)., k;=O

Even though the nested mappings described in Theo­rem 3 ought, in principle, to yield the steady-state PGF ofqueue lengths at polling instants, steps described in(14K17) require knowledge of PGFs f;(Zj) , which cannotbe derived from other known relationships. Therefore weuse the DFT of the state probabilities for which theanalogous unknown DFTs can be obtained. Details ofthe method are described next.

For a finite discrete random variable, one can view theDFT of this random variable as a finite sampling of itsP~F on.the unit circle and obtain .it sim2!Y by replacing Z

with e-)W for -1t ~ (J) ~ 1t and} = v'-1. Then for anN-dimensional PGF, such as Ji(z), after defining themaximum (possible) queue length as L;, i = 1, ... ,N, itsDFT equivalent is ([15], section 10.7)

where Wi = e-21tj / {,; and k E 5 = {(kl , ... ,kN ) : k, = 0,... , L, - 1, for each i = l, ... ,N}.

Similarly we define the DFTs of gi(ii) and tfJ(ii) forj = 0, ... , ICi and i = 1, ... ,N, and denote them as gj(k)and viJ(k), k E 5 respectively. Note that just like the PGFgi(=,), the DFf g,(k) also does not depend on the ithparameter, kj , because gi(ii) = 0, unless n, = O. Further­more, the DFTs of arrivals during the service time andthe setup time are denoted by S,* (k) and i; (k) for k E 5and i = 1, ... ,N, respectively, and given as follows:

Then. starting with an arbitrary initial value for /;(z) andrepeatedly applying .0;[.] infinitely many times. yields thesteady state /;(z).A detailed account of the convergence of such proceduresas the number of times .0; is applied approaches infinitycan be found in [14]. It is also reported there that the rateof convergence is fast for reasonable p values.

Theorem 2. The relationship, h+l C=) :::: ..A.1, {f;(=)}. betweenthe PGFs of the queue length distributions at consecutivepolling instants of Qj and Q;+1 is obtained from the fol­lowing equations:

v"u(z) = If;(z) - j; (=,)]1;+(t )'J - )'JZJ))=1

+ };(=,) + g,(O)PJ=" (14)

V;J+i (z) =v'J(=') + [vJJ(z) - ViJ(Z;)]

x s;(t Aj - AJZJ) /z" j = 0, ... ,1C" (15)

g;(z;) = »o: (Zj) , (16)N

/;+1 (=) = [g,(Zi) - 9;(0)] + g,(O) l:P)=)l (17)j=tjf:.j

where Zj = [ZI, ... ,Z;_I, O,Zj+I,.·· ,ZN].

The following arguments lead to relationships (14)-( 17)of Theorem 2.

• The number of customers present in the system at astation-i beginning epoch are either those that werealready in the system at the polling instant of Q; orthose that arrive during the setup period that follows,Furthermore. if the system gets empty at a station-icompletion epoch, then with probability p; anotherstation-i beginning with onJy one customer in Qj canbe observed. Putting these together yields (l4).

• There are exactly IC j services at each server visit to Q;and the system state changes only when Qi is notempty. Equation (l5) accounts for the change insystem state during these service completions.

• By definition, the ICith service completion exhaustsQit yielding (16).

• The polling instant of Qi+l matches with the station-scompletion epoch unless the server finds the systemempty. In that case the polling instant coincides withthe arrival epoch of the type-j customer, where j =I- i.The resulting relationship is (17).

D;[Ji(z)] = M,-I {M,-2{ . . .M J+ ) {M,{h(z)}} ...}}.(18)

Theorem 3. Let 0; be the functional relationship betweenthe PGF of the queue lengths at two consecutive pollinginstants of Q;, defined as

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Polling systems and state-dependent setup times 475

The folJowing iterative algorithm is based on Theorem 4,and assumes that the critical values, L, and JC;,i = 1, ... ,N, are known.

The algorithm can be adapted to solve several variationsof this base model. For example, continuous roving, non­zero switchover times and gated service are readily han­dled. We discuss these extensions in Section 6.

variable ki for each j = I, .K: Next the empty systemprobabilities, gi(O), i = 1, ,N, can be evaluated bytaking the average of 9i(k) with respect to all N variables,i.e.,

might provide the foundation of a good approximationbecause (1 - h(O))E[1j] is indeed the long run averageamount of setup time that the server spends at Qj in eachcycle in the state-dependent setups model. This method ofscaling setup times was suggested by Gupta and Srin­ivasan [3] for approximating continuously roving servermodels with state-dependent setups.

Notice that scaling reduces not only the mean setuptime per cycle, but also its variance (if the scaled setuptime in iteration k is denoted as r;(k) = (1 - /;(k-I) (O))1i,for k ~ 1, then Var(Tl) = (1 - .t:(k-l)(0))2Var (1i) , whichis less than or equal to Var(1i)). In contrast, linking set­ups with system state actually increases the variance ofthe setup time incurred at station i, for now we have somecycles with zero setup time. Therefore, we approximatethe distribution of setup time at station i as follows. In thekth iteration, we make the setup time to be 0 withprobability /;(k-I) (0), and T; otherwise. Next, assumingr h h r(k-l) (0) . ktor the rnoment that j, ,t=l, ... ,N,are Down,(7) leads to

T~~)(z) = /;(k-l) (0) + (1 - /;(k-I)(O))

x ~* (tlAj -AjLj,c(z)] +I:[Aj -AjLj,c-1(Z)]) .J=r J=O

(26)

Once the T/:) terms have been calculated, we solve apatient server model with state-independent setups to findthe next set of empty station polling probabilities,/;(k) (O)s. For example, these can be calculated by settingz = 0 in equation (12) of [6], i.e.,

Numerical experiments are performed to gain a betterunderstanding of the DFf-based approach as well as theiterative approximation scheme of Section 4.3. Becausetesting involved a large number of data sets, we start thissection by describing the test data. Later, we report theresults of the analysis in the form of three tables. Accu­racy of the approximation is tested by comparing its re­sults with those obtained from a computer simulation of

5. Numerical tests

00 NooN

.t:(k)(O) = rrIIT)~(O) - L:L:djk)Jjlc(O)CT)~J(O), (27)c=O j=1 c=O j=1

where aYJ (0) is calculated using LST of the scaled setupdistribution at iteration k.

We start the iterative procedure with /;(0) (0) = 0)i = 1, ... ,N, and stop when all /;(k) (0) values, i = 1, ... ,N, converge within a specified tolerance. This typicallyhappens in 20 or fewer iterations in our numerical ex­periments. Finally, the expected waiting times are calcu­lated by substituting g}") (0), /;(.) (0), moments of FP (z),/;(1) and /;(2) obtained from the final iteration, into (5).

(25)'t k E S.

Algorithm:Step O. Calculate S;* (k) and i; (k) Vk E S, i = I, ... ,N.

Set the initial state (arbitrarily) to idle systemwith server resting at station 1, i.e., gl (k) = 1,Yj(k) = 0, j > I, andAfl(~) = 0, V ~ ~ S. _

Step 1. ForJ == 1 to N do: h+l(k) = M;Cfi(k)) V k E S.Step 2. If fl (k) converges V k E S, then continue. Else

go to step 1.Step 3. For i = 1 to N do:

(i) Invert i(k) to get the joint probabilities ofqueue lengths at polling instants, fi(nLi = I, ... ,N and ii E S.

(ii) Calculate the marginal probabilities ofqueue length at polling instants, /;(ni),i = 1, ... ,N, which also includes the emptystation probability, /;(0). Then calculate thefactorial moments of the queue length atpolling instant of Qi.

(iii) Calculate the empty system probability, gi(O)using (24). If (i < N), then generate ./;+1 (k)utilizing the mapping M j •

Step 4. Calculate the expected customer waiting times bysubstituting the values from step 3 in (5).

1 £1- 1 .cN-l _

9t(O) = L) ... LNt; ...~ gj(k). (24)

We are now ready to state the following key Theorem,concerning the PGF-style mappings for the DFTs.

Theorem 4. The DFT of the steady-state probabilities ofqueue lengths at polling instants, /;(k), satisfies the map­ping, 0i, defined for PGFs in Theorem 3 with the PGFsreplaced by the appropriate DETs, i.e.,

4.3. The approximate method

Suppose we make setups state-independent, but scaletheir duration to (1 - ji(O))Ti. We can now solve the re­sulting state-independent setups model with a patientserver using a mathematically exact algorithm (see, forexample, [6]), provided we can find h(O)s. Such a method

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476

the corresponding system. We also report on the limita­tions of each method.

The test data vary in terms of number of stations, totaltraffic intensity, service and setup time distributions, de­gree of symmetry in the system, and relative magnitude ofthe setup times with respect to the service times. Weconsider five- and ten-station problems for testing theapproximation. However, for the DFT-based method, Nis chosen to be either 2 or 5. For all problem sets thearrival rate of each customer type is kept constant at 1.0and the load is varied by changing the mean service timeat stations. We test the problems for low (L), medium (M)and high (H) total work load, which are 0.1, 0.5 and 0.9respectively. Either constant or exponential setup andservice time distributions are used.

Our data-labeling scheme consists of a seven-characteralphanumeric code. The first two digits of the code denotethe number of stations, N. The third digit is used for thetotal work load of the system (L, M or H). The fourthdigit indicates whether or not the system consists ofsymmetric sta tions. A system is called symmetric, denotedby S, if all the stations are identical. In contrast, anasymmetric system, denoted by A, has PI == 0.9p andPj = 0.1 p/(N - I), for j == 2, ... 1 N. The relative magni­tude of the mean setup time to the mean service time isindicated by the fifth digit: the fifth-digit values of 0, Iand 2 correspond to the problem instants in which themean setup time is one tenth of, equal to and ten timesthe mean service time, respectively. The last two digits arereserved for service time and setup time distributions; 0(constant) or I (exponential). This way we obtain 24different problem sets for each problem size, i.e., for each(N, p) couple.

To evaluate the performance of the approximation andof the DFT-based approach, we have also simulatedsystem operation using an event-driven simulation pro­gram written in Ansi C. The simulation program is run ona 486 PC with a 60 MHz clock frequency and it takesanywhere from 0.25 hours to 4 hours depending on thesize of the data set, N, and the total traffic intensity, p.Computer codes of the approximation and the numericalmethod are also written in Ansi C. For matrix and DFTinversion required in our methods, we used standardsubroutines from [16].

5.1. Test results for the DFT-hased algorithm

The D FT-based method yields near-exact values of sys­tem performance measures, if L; and JC j , i = 1, ... ,N, arechosen carefully. Of these, the choice of suitable 12; valuesis extremely important. On the one hand, if the chosen L;values are too small, then large errors can result fromhaving an excessively truncated state space. On the otherhand, if the chosen L, values are too large, the use ofcomputer resources as well as the chance of havingcomputational errors increase. The latter happens be-

Gunalay and Gupta

cause the algorithm makes operations on very smallnumbers that correspond to the probabilities of the sys­tem having large queue lengths at all stations.

We use the following procedure for obtaining initialestimates of L;s. First, the mean customer waiting timesare estimated from the approximation of Section 4.3.From this, the expected station queue lengths are foundusing Little's Law. We equate E[L;], i = I, ... ,N, to theseexpected queue lengths. Next, the server utilization, Pi'for an equivalent M / M /1 queue which has the same ex­pected queue length is calculated, i.e., Pi is chosen suchthat E[LJ ] == Pi/ (1 ~ Pi)' Finally, we set the L, value suchthat the probability of observing a queue length greaterthan or equal to L, in the M / M /1 model is less than €

(which is set to 10- 8) . Put differently, the chosen L, sat­isfies pfi < c.

The goodness of our initial estimates for L, can heascertained by checking the marginal queue lengthprobabilities obtained from the DFT-based algorithm.We expect these probabilities to reduce to € or less as thequeue size approaches L, - 1. If this does not happen, thealgorithm is rerun, after increasing Li values. In our ex­periments, we doubled the initial E, estimate for suchcases.

In the implementation of the DFT-based method, in­stead of performing exactly JC; services at each server visitto Qi, we let the server continue serving type-i customersuntil Qi is empty. In other words, we repeatedly evaluate(15) until vlJ(k) converges for all k E S.

The CPU time and computer memory requirementsnecessary to implement the algorithm grow exponentiallyin N. These requirements also grow in P because we haveto increase L, accordingly. Therefore, as Nand/or p getslarger the algorithm becomes inefficient. Table 1 presentsthe results for N = 2 and N == 5. The mean waiting timescalculated by the exact method (for N == 2) and the sim­ulation (for N = 5) are also included in the table in pa­rentheses. The same data code is used with a minorchange; for data sets with N == 5, 'M' as the third char­acter of the code represents p = 0.3, instead of p == 0.5.We report expected waiting times as (E[W'i], (E{Hj])),where {E[HjD represents the average of the mean waitingtimes of type-j customers for j == 2, ... ,N.

As seen in Table 1, when N = 2, choosing L; values ashigh as 64 does not create excessive requirements for ei­ther CPU time or memory. However, this changes whenN == 5. Large CPU times are necessary whenever Li > 8.Because of the memory limitations of PCs, the DFf­based algorithm was run on a Sun Spare 2 workstationwith a clock speed of 80 MHz.

As a final remark about the DFf-based algorithm wepoint out that for the purpose of carrying out numericaltests, the computer implementation of the algorithm wasnot optimized and no special data structure was used. Thealgorithm is also suitable for parallel computing. All thesemeasures and the availability of greater computing power

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Polling systems and state-dependent setup times 477

Table 1. A sample of mean waiting times obtained from the OFT-based algorithm. Values shown in parenthesis are exact resultsfor N = 2 and the simulation results for N = 5. Experiments were performed on a Sun Spare 2 workstation with 80 MHz clockspeed.

Code (LI,···,LN) (E[WI], (E[ijj)))

05LSOll (8,8,8,8,8) 0.0040,0.0040(0.0040,0.0040)

05LAIOI (16,8,8,8,8) 0.0235,0.0265(0.0238,0.0252)

05MSOll (8,8,8,8,8) 0.1204,0.1204(0.1211,0.1211)

02MSOl1 (32,32) 0.2762,0.2762(0.2760,0.2760)

02MA201 (64,64) 4,0694,6.6296(4,0690,6.6328)

CPU time (minutes)

17

50

24

11

Memory (Kbytes)

7788

14400

7788

450

1800

can substantially improve the performance of the DFTbased algorithm, making it practical even for systemswith many stations and large p.

5.2. Test results for the approximate method

The approximate method is tested over 144different datasets. Because no work conservation relationships areavailable for state-dependent setup models, simulation isused as a benchmark for comparison. We let these sim­ulations go on until the 95% confidence interval on themean waiting times is in the ±O.5 % neighborhood of itspoint estimate. In contrast, both Ferguson [5] andBradlow and Byrd [4] use time-limited and/or event-lim­ited stopping criteria. Our computational experience hasshown that such criteria do not necessarily yield thesteady-state performance measures. Major differencesbetween our experiments and those reported in previousstudies (4, 5] include the facts that the latter consider only'heavily loaded' systems and do not report empty stationand/or empty system probabilities. In contrast, we ex­amine systems with a range of values of overall serverutilization and report empty station and empty systemprobabilities. We believe that in addition to having a di­rect impact on the accuracy of the estimate of the meanwaiting time, these probabilities are important perfor­mance measures. For example, if the empty systemprobabilities are very small, then we know that the systembehaves like a continuously roving server model (seeObservation 3) and can be approximated as such.

In Table 2, we report the mean, the standard deviation,and the worst-case values of the absolute error of ourestimate of the expected waiting time at station 1. Thesame three statistics are also reported for the average ofthe mean waiting times at all other stations, which wedenote by (E[fJj]). These descriptive statistics are calcu­lated after first representing each absolute error as apercentage of the corresponding value obtained fromsimulation. For the empty station and empty systemprobabilities, similar statistics are calculated over their

absolute errors, because these values, rather than per­centage errors, better reflect their effect on the accuracy ofthe mean waiting time estimates. The terms (jj(O) and(gj(O) denote respectively the average of the empty sta­tion and empty system probabilities when the server is atQj, j = 2) ... ,N. Recall that even for the 'asymmetric'problem sets, stations Q2" . QN are identical and there­fore little information is lost by such averaging.

The absolute percentage error in E[Wi] over all datasets has a mean of 3.58%, and a standard deviation of3.78%. In the worst case, the absolute percentage errorcan be as high as 16.850/0. However, such instances arerare and happen when low overall server utilization iscombined with very large setup times, i.e., when the meansetup time is ten times as large as the mean service time atthe corresponding station. Note that such instances havenot been studied before. For example, Bradlow and Byrd[4] consider cases in which the setup times are at most ofthe same magnitude as the service times. Still, they report10-150/0 error in their estimates of the mean waiting timesfor low load (p = 0.3) systems. If such extreme cases areexcluded from our data set and we then consider only lowload systems, i.e., when p = 0.1, the absolute percentageerror in E[Wi] has a mean of 1.160/0 and a standard de­viation of 1.70% (for (E[ijj]) these statistics are 1.44%and 1.490/0 respectively).

Table 2 also shows that the iterative approximation isvery effective in estimating the empty station and theempty system probabilities. In most cases these proba­bilities can be estimated to lie within two-digit accuracy.The absolute error in fi(O) has a mean of 0.012 and astandard deviation of 0.019 and the absolute error in g;(O)has a mean of 0.024 and a standard deviation of 0.018.

Figs 1 and 2 show the station mean waiting times ob­tained from the approximate and simulation proceduresfor some problem instants with N = 5. The solid lineconnects simulation results, and the corresponding ap­proximate values are shown by the asterisk. Notice thatthe error in estimating E[Wi] and (E[Wj]) is small for allproblem instants and that the approximation is not bi-

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478 Giinalay and Gupta

Table 2. Performance statistics of the approximate method. Statistics are calculated on the absolute percentage error for meanwaiting times and on the absolute error for empty station and empty system probabilities.

N p E[Wl] (E[~·]) 11(0) (fj(0) } gl (0) (gj(O)}

5 L Mean 1.95 2.01 . 0.010 0.011 0.070 0.093Std. 2.25 2.41 0.011 0.011 0.025 0.025Worst 6.76 7.34 0.031 0.033 0.104 0.118

M Mean 2.12 2.49 0.008 0.010 0.015 0.038Std. 2.32 2.46 0.008 0.008 0.017 0.034Worst 7.23 7.64 0.030 0.027 0.051 0.096

H Avg, 3.96 4.94 0.007 0.012 0.003 0.013Std. 2.83 3.63 0.008 0.015 0.003 0.022Worst 10.23 17.42 0.030 0.050 0.011 0.065

10 L Mean 4.22 4.19 0.015 0.016 0.019 0.030Std. 4.71 5.17 0.017 0.019 0.010 0.005Worst 14.05 14.02 0.044 0.047 0.032 0.036

M Mean 4.26 5.06 0.008 0.011 0.004 0.011Std. 5.57 5.80 0.009 0.013 0.005 0.010Worst 16.85 16.98 0.026 0.035 0.014 0.029

H Mean 4.96 6.23 0.017 0.024 0.002 0.007Std. 3.76 3.68 0.031 0.044 0.002 0.009Worst 12.78 13.33 0.094 0.128 0.006 0.023

Overall Mean 3.58 4.15 0.011 0.014 0.019 0.032Std. 3.78 4.06 0.016 0.022 0.013 0.020Worst 16.85 17.42 0.094 0.128 0.104 0.118

ased in the sense of estimating mean waiting times moreaccurately for some stations.

The approximate algorithm utilizes a descendant-sets­based recursive algorithm with the result that it is veryfast when compared with either the simulation or theDFT-based method. It takes less than 4 minutes to run 24

data sets on a 486 PC, whereas the simulation of a singledata set could take between 15 minutes and 4 hours onthe same PC. However, if the problem of interest has lowload and large setup times in relation to service times, werecommend that the DFT-based algorithm be used. Al­though this would require considerable computer re-

so

40

rJ30.!3

~CJl

.~20.;

~

5 10~

0

·10

~

''1./

..................._ ....................--x-....... )...U ...............r""\..,.:x/rM lrx~

o~o~o~o~g~a~O~O~O~D"O~O~O~O~O"o~o~o~o"o~a~o~o~o~o~a"o"O~O~O"D~O~O~O~O~O~

§53s5;55§3§s3~§~333333§3~;iiiiiiiiiiiiiiiiiiiiiiiiiiiiii~~iiii~i~~~i~iiiData Sets

Fig. l. Mean waiting times at station 1: simulation and approximation results for a five-station system.

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Polling systems and state-dependent setup times

80

70

60

-ioo~o~o~a~a~o~o~O"O~O~D~D~O~C~O~O~O~O~O"O_D~C"O~O~O~O~O~O~O~O-D~O~O"O~O~O~

;§§;5s55SS§S33~~333~3333ii~iiiiiiiiiiliiiiiiiiii~~~~iiiiii~5~ii~i~ii~i~~Data Sets

Fig. 2. Mean waiting times at stations 2, 3, 4, and 5: simulation and approximation results for a five-station system.

479

sources, these would still be less than what simulationwould need. Furthermore, the DFT-based algorithmprovides not only the mean waiting times but also thehigher moments of waiting times with negligible extraeffort.

6. Extensions

Both the DFT-based method and the iterative approxi­mation can be used to analyze a variety of differentmodels, involving nonzero switchover times, continu­ously roving server, state-independent setups and forsystems with some or all stations using the gated serviceregime. Before describing these extensions, we first in­troduce notation for nonzero switchover times.

Let Ri, i = 1, ... )N, denote the random switching timeneeded by the server to travel from station i to station(i + 1). The patient server polling system with bothswitchover and setup times has two types of stationoverhead: (1) state dependent setup times and (2) stateindependent switchover times. Because the switchovertimes are state independent, according to Srinivasan, Niuand Cooper [17], their presence in the system affects thecustomer-waiting times in an additive fashion. Therefore,the analysis of such systems is similar to what we presentin Section 3.

The DFT-based method can be modified to handle thecontinuously roving (CR) server polling system withstate-dependent setups. This is done by setting the emptysystem probabilities to zero in Theorem 2 and intro­ducing nonzero switchover times. Next, we show thenecessary changes needed to handle the gated service

strategy. We assume that a gate closes behind all type-icustomers present at station i at its station beginninginstant. The server processes only those customers thatare in front of the gate before registering a stationcompletion instant. Thus, server vacation from station iends either by the end of a setup period or by a type-iarrival to the system when the server is idling at Q;, andthe relationship in (3) is still valid. The waiting timedistribution for customers at station i is found by ap­plying Fuhrmann and Cooper's stochastic decompositiontheorem (see example 3 in [10]):

~*(Ai _ AiZ

) = Ki(S;(A.; - ,J...;z)) - K;(z) ( I ).K;(I) St(A; - A;Z) - Z

(28)

Upon taking the derivative of the LST above and thensetting s = 0, we obtain the mean waiting time of type-icustomers as

E[~] = (1 + p;)U;(2) + 2AitJ;(1) + (1 - /;(O))).,;t?))J (1)

2AiC!; + (1 - /;(0) )A;t; + diP;)

(29)

It once again depends on the first and second moments ofqueue lengths at polling instants. To find these momentswe can use either one of the two methods described inSection 4. However, during calculations we need to dothe following:

(1) pap and corresponding DFT mapping relationshave to be changed. For example, (15) and (l6) arereplaced with

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00 00 00

9i(Z) = L'" L'" L vi'O(nl,'" ,ni,"" nN)n, =0 nj=O nN=O

x [st(t A.)- A.)Z))] n'ITz? ' (30)1=1 ~I

J#-iwhere vi,O(fii ) is the joint probabilities of queue lengthsat station beginning instants, and can be calculated byinverting its DFT at each iteration. Note that for thegated service regime g;(z) is no longer independent ofthe ith element, z..(2) In the iterative approximation, the PGF of thecontribution of a type-i customer who is served at cthcycle is redefined as

(

N i )L"c(z) =S; L V~1 - )~1L1,C(=)J +L[J~j - ).JLJ,c-l(=)] .

j=i+l j=l

(31)

For simplicity, the above expression has been shownwithout the iteration index, i.e., superscript k.

Acknowledgements

This research has been supported, in part, by the NaturalSciences and Engineering Research Council of Canadathrough research grant number OGP0045904 awarded toDiwakar Gupta. This research was completed while YavuzGiinalay was a doctoral student at McMaster University.

References

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[2] Takagi, H. (1994) Queueing Analysis of Polling Models: Progressin 1990-93. Institute of Socio-Economic Planning, University ofTsukuba, Japan.

[3] Gupta, D. and Snruvasan, M. M. (1996) Polling systems withstate dependent setup times. Queueing Systems, 22. 403-423.

(4) Bradlow, H. S. and Byrd, H. F. (1987) Mean waiting time eval­uation of packet switches for centrally controlled PBX's. Perfor­mance Evaluation. 7, 309-327.

[5] 'Ferguson, M. J. (1986) Mean waitmg time for token ring with sta­tion dependent overheads, in Local Area and Multiple Access Net­works, Pickholtz, R. L. (ed.), Computer Science Press, Rockville,MD. pp. 43-67.

Gunalay and Gupta

[6] Srinivasan, M. M. and Gupta. D. (1996) When should a rovingserver be patient? Management Science, 42(3),437-451.

[7] Konheim, A. G., Levy, H. and Srinivasan. M. M. (1994) De­scendant set an efficient approach for the analysis of pollingsystems. IEEE Transactions on Communications, 42(2/3/4), 1245­1253.

[8] Eisenberg, M. (1971) Two queues with changeover Times. Oper­ations Research, 19(3), 386-40 I.

[9] Altman, E., Konstantopoulos, P. and Liu, Z. (1992) Stability,monotonicity and invariant quantities In general polling Systems.Queueing Systems, II, 35-57.

[10] Fuhrmann, S. W. and Cooper. R. B. (1985) Stochastic decom­positions in the MjGjl queue with generalized vacations. Opera­tions Research. 33(5), JII7-1129.

[11] Cooper, R. B. and Murray, G. (1969) Queues served in cyclicorder. The Bell System Technical Journal, 19(3), 675-689.

[12] Daigle, J. N. (1989) Queue length distributions from probabilitygenerating functions via discrete Fourier transforms. OperationsResearch Letters, 8(4), 229-236.

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Biographies

Yavuz Giinalay is an Assistant Professor in the Department of In­dustrial Engineering at Eastern Mediterranean University, North Cy­prus. He received a B.Sc. degree in Electrical and ElectronicsEngineering from Middle East Technical University in 1987, an M.Sc.degree in Industrial Engineering from Bilkent University in 1990, and aPh.D. in Business Administration from McMaster University in 1996.His research interests include stochastic modeling, queueing theory,polling systems, and optimization.

Diwakar Gupta is an Associate Professor of Production and Man­agement Science at the Michael G. DeGroote School of Business.McMaster University. He has received a B.Tech. (Mechanical Engi­neering) from the Indian Institute of Technology Delhi, an M.A.Sc.(Industrial Engmeenng) from the University of Windsor, and a Ph.D.(Management SCience) from the University of Waterloo. His researchinterests are in measurement and evaluation of manufacturing flexi­bility, design and operational analysis of manufacturing and healthcare delivery systems. and other stochastic models.

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