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MANAGEMENT SCIENCEVol. 53, No. 6, June 2007, pp. 9911004issn 0025-1909 eissn 1526-5501 07 5306 0991

informs

doi 10.1287/mnsc.1060.0639 2007 INFORMS

Performance Bounds for Flexible

Systems Requiring SetupsMahender P. Singh

MIT Center for Transportation and Logistics, Massachusetts Institute of Technology, Room E40-363,77 Massachusetts Avenue, Cambridge, Massachusetts 02139, [email protected]

Mandyam M. SrinivasanCollege of Business Administration, The University of Tennessee, 611 Stokely Management Center,

Knoxville, Tennessee 37996, [email protected]

Many organizations use product variety as one possible strategy for increasing their competitiveness. Theyhave installed flexible manufacturing systems because these systems offer a powerful means for accom-modating production and assembly of a variety of products. However, increased product variety comes at a

cost. For instance, if the resource requires to be set up each time it switches to operate on a new product, theresulting delays and costs could negate the intended benefits of increased product variety. Analyzing these flex-ible resources for optimal design and operation is therefore very important. To address such issues, we modela flexible resource, serving multiple products, using a queueing modelmore precisely, a polling model. In thismodel, a single server attends to multiple service centers (queues) at which requests arrive and queue up forservice, performing a setup at a polled queue only if that queue is nonempty. This is the state dependent (SD)polling model.

Exact analysis of the SD polling model is inherently very complex. This paper presents a very efficientprocedure to compute a hierarchy of successively improving bounds on the values of performance measuresobtained from the SD polling model with the exact solution as its limit. This procedure can be applied to quicklyestimate performance measures for large SD polling models previously deemed analytically intractable.

Key words : polling models; descendant sets technique; state-dependent service; cyclic serviceHistory : Accepted by Wallace J. Hopp, stochastic models and simulation; received October 4, 2004. This paper

was with the authors 8 12

months for 2 revisions.

1. IntroductionMany organizations are using flexibility as a weaponin an increasingly competitive economy. They haveacquired the capability to offer a large variety ofproducts at a short notice by installing sophisticatedflexible resources. However, unless these resourcesare properly designed and operated, the benefits ofincreased product variety could be offset by increasedcosts and cycle time delays. Models for analyzingthese systems for effective design and operation aretherefore important. Queueing models, more specifi-cally, polling models, enable such analysis. These mod-els are used to analyze a system of multiple servicecenters (queues) attended to by one or more serversthat attend to requests arriving at these centers bypolling (visiting) them in a pre-specified manner.

Polling models have many other real-world appli-cations. They are used to analyze computer networks,telephone switching systems, and manufacturing andmaterial handling systems, to name a few. Real-worldsystems that can be analyzed with polling modelscontinue to grow as new applications and technology

are introduced. For example, token ring networks andnetworks using the fiber distributed data interface(FDDI) protocol are readily analyzed using pollingmodels. The use of radio frequency data communica-tion (RFDC) in warehouse management and materialhandling systems provides a more recent applicationfor polling models because RFDC allows the servers(material handlers) to poll storage locations to deter-mine whether there is any handling needed at theselocations. Consequently, there is a large body of liter-ature on polling models that has continued to growsince the late 1960s when they were first studied; seeTakagi (1990, 1997) for comprehensive surveys.

In this paper, we focus on a special class of pollingmodels known as the state dependent(SD)polling mod-els. In the SD polling model, a single server attends tothe requests waiting at the different queues, perform-ing a setuponlywhen there are customers waiting forservice at the polled queue. Although the SD modelfinds many applications in the real world, it doesnot lend itself to easy analysis. Consequently, eventhough it is a more appropriate model, the SD polling

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Singh and Srinivasan: Performance Bounds for Flexible Systems Requiring Setups992 Management Science 53(6), pp. 9911004, 2007 INFORMS

model is typically approximated by the SI pollingmodel in which the server is assumed to perform asetup (or, alternately, never performs a setup) at apolled queue regardless of its state. Such an approx-imation is acceptable when the system is so highlyloaded that the server will always find the polled

queue nonempty. However, from a practical stand-point, it is usually not desirable to utilize the serverat such high levels because customers will experienceunusually large waiting times.

The inherent complexity in the SD polling modelarises from the fact that the analysis requires addi-tional information to be maintained on the state ofall queues in the system at all times. Most of theresearch on polling models has thus been confined tothestate independent(SI) polling model. To our knowl-edge, there are very few exact procedures availablefor obtaining the moments of the waiting times andthe number of customers for the SD polling model.

Gupta and Srinivasan (1996) obtained these measuresfor the special case of a 2-queue model. Eisenberg(1999) presented an exact analysis for the general Nqueue model. Singh and Srinivasan (2001) and Srini-vasan and Singh (2006) present detailed algorithmsto compute these performance measures for the gen-eralNqueue model. Although exact procedures existfor analyzing the SD model, these procedures are notcomputationally efficient. The computational effortgrows exponentially with the number of queues in thesystem.

A number of papers provide approximate analy-sis of the SD model to estimate the expected waiting

times and these include the work by Altman et al.(1994), Bradlow and Byrd (1987), Ferguson (1986),Gnalay and Gupta (1997), Lennon (1995), Olsen(2001), and Yehia (1998). These papers do not provideboundson the expected waiting times and so it is dif-ficult to determine the accuracy of these estimates.Our intent in this paper is to promote the use ofSD polling models for analyzing real-world systems

by presenting a fast and computationally efficientmethod that determines tight bounds on the values ofperformance measures critical to the design and eval-uation of production systems. This method, the perfor-mance bound hierarchies technique, develops a hierarchy

of successively more accurate bounds on the wait-ing time moments with the exact solution as its limit.We motivate the discussion by presenting examplesof manufacturing systems that are more appropriatelyanalyzed using the SD polling model rather than withthe simpler SI polling model.

2. Manufacturing ApplicationsWe present two different scenarios that can be mod-eled and analyzed with the SD model. The first sce-nario is discussed in greater detail.

Figure 1 Schematic Representation of the RFDC-Enabled Material

Handling System

Store 1

Line 1

Line 3

Line 2

Production Zone I

Cell 1

S3

S2

S1

R1 R2

R3

Cell 2

Store 2

Store 3

Cell 3

2.1. A Material Handling SystemConsider a material handling systempossibly amanual forklift, a person, or an automated guidedvehiclethat moves semifinished products from

three production lines in a production center to stor-age locations positioned close to various other pro-duction cells for further processing. The schematic ofthe material handling system is provided in Figure 1.

In this scenario, the server is the material handlingsystem and the customers are the semifinished prod-ucts that wait in their respective queues for the server.The server visits the stationsproduction linesin acyclic manner and polls each one to ascertain the stateof that station. If there are customers waiting at thepolled station, the server performs service as follows:

stops in front of the polled station, picks up semifinished parts, one at a time, from

the queue, scans the part to update necessary records main-

tained in a database, and places the part in an empty container available

at that location.This process is repeated until there are no more

parts at the station. After serving (picking up, scan-ning, and placing in container) all the customers(semifinished parts), including customers that arriveduring the service period (this is the exhaustive servicediscipline), the server departs from the polled stationto the storage area designated for the production cell.It is assumed that the container is large enough to

store all the semifinished parts present at the station.After depositing the full container at the store, theserver picks up an empty container from that loca-tion and takes it back to the station. The server nowplaces the empty container at the station and proceedsto the next station. Parts waiting at the station whenthe server returns are transported in the next cycle.If the polled station is empty, the server directly pro-ceeds to the next station (in an RFDC enabled system,the server knows the state of the system immediatelyon arriving at the station, so there is no need for the

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Singh and Srinivasan: Performance Bounds for Flexible Systems Requiring SetupsManagement Science 53(6), pp. 9911004, 2007 INFORMS 993

server to come to a halt to establish the state of thatstation).

Let us examine the servers activities in more detailto understand the modeling nuances for this system.The time required by the server to move from sta-tion i to station i + 1 is the switchover time, denoted

by Ri, i = 1 2 3, in Figure 1. The time spent mov-ing the filled container to the respective store and tobring back an empty container is treated as the setuptime, denoted by Si in Figure 1. This setup activityis different from the typical setup in the sense thatthe server executes it after completing service at thestation instead of performing a setup activity beforestarting service. Indeed, for the SD polling model weanalyze in this paper, a setup is assumed to be com-pleted before the service period begins. However, ourmethodology accommodates the alternate setup activ-ity assumed for the model of the above material han-dling system simply by modifying one expression in

the standard SD polling model (see Remark 1 in 4).

2.2. A Flexible Manufacturing SystemNow consider a flexible manufacturing system thatprocesses multiple products. A workcenter in thisflexible manufacturing system must have the flexi-

bility to cater to multiple product types, each oneof which might require the workcenter to performa setup to prepare itself for service. Following thedescription of the model in Figure 1, it is easy tosee that the exhaustive service SD model presentsa very effective modeling framework for analyzingsuch flexible systems for their desired measures of

performance such as capacity analysis, waiting timedistributions, and so on. The exhaustive service strat-egy may, for instance, be the appropriate model touse when switchover and/or setup times are high,

but it should be pointed out that more general servicestrategies can be analyzed within the framework ofthis paper. Choosing an optimal service strategy is,however, far from trivial. Therefore, fast algorithmsthat can allow us to search more easily across differ-ent strategies would be of value in this regard.

3. The Model and PreliminariesThe classical polling model considers a system inwhich multiple queues are served by a single servertraveling from queue to queue in a prescribed se-quence. Note that, in isolation, each queue is an ordi-nary M/G/1 queue, but because they share a singleserver they interact in complex ways, and their perfor-mance measures are strongly dependent on each other.We assume that customers arrive at various queuesfollowing Poisson processes and the server adoptsthe exhaustive service discipline (the server contin-ues to serve customers until the queue is empty).We only consider the exhaustive service discipline

in this paper, but our model can accommodate anyservice discipline that allows a multitype branchingprocess interpretation (Resing 1993). We assume thatthe server serves the queues in a cyclic order. Themethodology described below can be adapted to ana-lyze models where the server uses a polling table

(Eisenberg 1972) to switch between queues, switchesbetween queues in a random manner (Boxma andWeststrate 1989, Kleinrock and Levy 1988, Srinivasan1991), or uses fractional service policies (see, forinstance, Levy 1991).

Although we explicitly include switchover times,our model allows for zero switchover times so longas at least one queue has a nonzero setup time. Alter-nately, the model can accommodate systems withzero setup times and at least one nonzero switchovertime (this is the simple SI model). We note, inpassing, that most real-world systems have nonzeroswitchover times. Certainly, the material handling

system described in the 2 has fairly significantswitchover times, and if the polling model is usedto analyze a production facility processing multipleproducts (see, for instance, Kreig and Kuhn 2004),then again the facility will need a finite amount oftime to switch between products. Even computercommunication systems, which arguably motivatedthe development of polling models, have switchovertimes that can be nontrivial relative to service times.We also note that Levy and Kleinrock (1991) modeland analyze the zero switchover time model as a lim-iting case of the nonzero switchover time model.

3.1. NotationA single server serves a sequence of N infinite-capacity queues in a cyclic order. The arrival processat queuei is Poisson with ratei. The time required toserve a customer at queue i is denoted by Bi and the

busy period at queue i is i. Theswitchover timefromqueuei(the time required for the server to travel fromqueuei to queuei + 1) is denoted byRi. Thesetup timeat queuei(the preparation time at a queue prior to the

beginning of a service period) is denoted by Si, andit is incurred if one or more customers are present atqueue i at the instant this queue is polled. The arrivalprocesses, service times, setup times, and switchover

times are all mutually independent. The traffic inten-sity at queueiis i iEBi, and the server utilizationis

Ni=1 i. For the polling system to be stable,

must be less than one, and this is assumed to be thecase. The waiting time at queuei (the time a customerwaits to receive service) is denoted by Wi.

Unless otherwise stated, the Laplace-Stieltjes trans-form (LST) of a nonnegative random variable A, de-fined asEexpsA, is denoted by As; and whenAis discrete, its probability generating function (PGF),defined as EzA, is denoted by Az. Multivariate

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LSTs and PGFs are defined similarly. It is assumedthat the index used for a summation over the queuesis (a) reset to one if it increases to N+ 1, and (b) resetto N if it drops to zero.

We will refer to the time at which the serverpolls queue i as an i-polling epoch. Let X

ji denote

the number of customers present at queue j at ani-polling epoch, and let Fiz1 zN Fiz denotethe (multivariate) PGF of X1i X

Ni . In particular,

Fi1 1 zi 1 1 denotes the (univariate) PGFof Xii . Let fi0 denote the probability that queue iis empty at an i-polling epoch. Then, the LST of thewaiting time for customers at queue i is (Gupta andSrinivasan 1996)

Wii izi= 1

iECBii izizi

1Fi11zi11Sii izi

fi01Sii izi (1)

where C is the cycle time, which is the time requiredby the server to complete a scan of all the queues. Theexpression for the expected cycle time is

EC =ER +

Ni=11 fi0ESi

1 (2)

where ER is the expected total switchover time re-sulting from a complete scan of all the queues.

3.2. The Descendent Sets Technique ApproachKonheim et al. (1994) proposed the descendant sets

technique that provides an expression for the (uni-variate) PGF of the queue-length distribution at apre-specified queue at various polling instants. Whilethis technique can analyze a wide variety of pollingmodels, it cannot be used to analyze models such asthe SD model that require computation of the jointqueue-length distribution at polling instants. We willrefer to this technique as the univariate descendant setsUDS technique. To address the shortcoming of theUDS technique, its multidimensional version, themul-tidimensional descendant sets (MDS) technique, is intro-duced in Singh and Srinivasan (2001).

A key difference between the UDS technique and

the MDS technique is the availability of the com-plete state description of the whole system at all timesin the MDS technique. A pure application of theMDS technique is computationally intense and veryslow. Srinivasan and Singh (2006) present a hybridapproach that utilizes both the MDS and the UDStechniques. The hybrid procedure results in a much

better performance and provides a typical improve-ment of three times or more over the pure proce-dure. However, even the hybrid procedure is limitedin its ability to analyze models of larger real-world

systems rapidly. Thus, while the hybrid procedure canbe readily used to analyze descriptive models with arelatively small number of stations, it would not beas useful for analyzing models with a large numberof stations, as it would require a very large amount ofmemory and computational effort. Furthermore, the

hybrid procedure would be impractical to use withinthe context of a prescriptive algorithm that couldrequire the evaluation of a number of alternate modelconfigurations. In such a situation, a bounding tech-nique that provides upper and lower bounds on thevalues of performance measures would be extremelyvaluable.

Thus, although exact procedures typically result ina better understanding of the model and the system

being modeled, the increased computational effortoften negates their benefits. In most cases, a proce-dure that quickly provides reasonably tight boundson the exact values may be adequate. To that end, we

develop a new iterative procedure to generate hier-archies of successively improving (tighter) bounds onthe waiting time moments for the SD model withN > 2 queues. The new bounding method offers anefficient way to analyze large, complicated pollingmodels. To motivate the discussion, we first providea brief review of the UDS and MDS techniques below.

3.3. The Univariate Descendant Sets (UDS)Technique

The UDS technique (Konheim et al. 1994) starts by(re)numbering the N queues in the given pollingmodel, starting with the queue under consideration,

labeled as queue 1. Define a cycle time as the elapsedtime between two successive 1-polling epochs. Atqueue 1, define a reference point by considering a ran-domly selected 1-polling epoch that corresponds tothe initiation of a cycle with cycle indexc = 1. Then,the number of customers at queue 1 at the referencepoint is X11 .

The UDS technique expresses each customer pres-ent at a polling epoch in terms of contributionsfrom customers that arrived in all previous cycles.Following this approach, to determine all the cus-tomers represented by X11 , we look back in time andconsider the cth cycle prior to the reference point,

wherec = 0 1 Denote a type-i customer, who isserved during the cth cycle, as a i c customer, anddefine the immediate offspring of a customer as theset of all customers that arrive to the system (at allqueues) during its service period. Then, the descendantset of a i c customer is defined as the set consistingof this i c customer, its offspring (if any), and thedescendants of its offspring.

Let Li c be the number of waiting customers atqueue 1 at the reference point that are in the descen-dant set of a i c customer, and let Li cz denote

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its PGF. Li c can be interpreted as the contributionto X11 from a i c customer. Every waiting customer(if any) at queue 1 at the reference point contributes acount of exactly 1 to X11 ; none of the customers wait-ing at the other queues at this instant contribute to X11 .Thus, L1 1 = 1, Li 1 = 0 for i > 1, and so L1 1z = z,

withLi 1z = 1 for i > 1. For c 0, the PGF ofL i c isnow recursively expressed in terms of the PGF of thecontributions to X11 made by customers who arriveduring the service to a i c customer. Customers thatarrive at a queue j > i while a i c customer is beingserved are, by definition, j c customers. Customersthat arrive at a queue j < iduring the service to a i ccustomer are, by definition, j c1 customers becausethey are served in the next cycle, which is cycle c 1.Therefore, it follows that

Li cz1= i

Nj=i+1

j1Lj cz1+i1j=1

j1Ljc 1z1

i = 1 N c 0 (3)

The index j on the right-hand side of (3) does notcover queue i because the contribution from theith queue is implicitly accounted for by the busyperiod i.

3.4. The multidimensional Descendant Sets(MDS) Technique

The MDS technique (Singh and Srinivasan 2001,Srinivasan and Singh 2006) is based on the descendentsets technique proposed in Konheim et al. (1994) inthe sense that it expresses F1zsolely in terms ofF1

at previous 1-polling epochs. In principle, it followsthe basic idea of the UDS technique and goes back intime, relatingX11 to customers present at each of the1-polling epochs in previous cycles, until the contri-

butions from customers present at a 1-polling epochbecomes negligible.

There is, however, one key difference between theUDS and the MDS techniques. To highlight this dif-ference, denote the presence or absence of a setupat a polled queue as an outcome, and use the termsample path to denote the sequence of possible out-comes experienced by the server as it continues to pollthe queues. The UDS technique works with a single

sample path. More precisely, it relates the i c cus-tomers contribution to X11 using a single expressionforL i cz1. However, for the SD polling model, thereare multiple sample paths depending on the outcomesat each queue at each cycle. Each sample path resultsin a unique expression for the contribution from a i ccustomer (to X11 ). In this respect, the MDS techniqueis reminiscent of the buffer occupancy approach(Eisenberg 1972), which requires the moments ofX

ji .

To address the issue of multiple sample paths, theMDS technique uses a multidimensional PGF,Mi cz.

In its generic form, the expression for Mi czis simi-lar to the expression for Li cz1:

Mi cz= i

Nj=i+1

j1Mj cz+i1j=1

j1Mj c1z

i = 1 N c 0 (4)

The Mi c terms are key to the execution of theMDS technique because they are used to carry thesample path information. This is explicated in Singhand Srinivasan (2001), but to clarify the point to thereader of this paper, consider a model with N =3queues and consider the sequence of outcomes as theserver moves from a 1-polling epoch in cycle c+ 1to a 1-polling epoch in cycle c (with the descendentsets approach, as we move forward in time, eachsucceeding cycle has a lower cycle index). Denotethe presence or absence of a setup at each queue

by a one or a zero, respectively. When we relate a

1-polling epoch in cycle c + 1 to a 1-polling epoch incyclec, there are 23 = 8 possible outcomes to consider,1 1 1 0 1 1 1 0 1 0 0 0 (the rightmostposition changes the slowest). In general, it is conve-nient to capture these outcomes by a vector, Ip, p=

1 2N, where for each p, Ip is an N 1 vector ofN binary switches. Let Ipj denote the jth element

of this vector. For instance, with three queues, ifIp=1 0 1, then Ip1 = Ip3 = 1 and Ip2 = 0.

To explicitly capture each outcome sequence, letM

pi c denote the contribution of a i c customer

(just like Li cz in the UDS technique) resulting fromthepth binary switch configuration,p = 1 2N. Thecase p= 2N describes a situation where Ipj= 0 forall j, that is, the server found each queue empty dur-ing that cycle. For this case, there are no such con-tributions, and so we simply set M2

N

i c z= 0. Thus,using (4), M

pi czis evaluated as follows:

Mpi cz =

0 Ipi = 0

i

Nj=i+1

j1 wpj cz

+i1

j=1j1 w

pj c1z

otherwise

(5)

where, for any c ,

wpi cz =

0 ifIpi = 0

Mpi cz ifIpi = 1

A fundamental property of Mpi cz (Theorem 1 in

Singh and Srinivasan 2001) is that if we initializethe iteration with, say, w

pi cz= 0, and c= 1, and

continue to iteratively express Mpi cz using (5), then

it converges monotonically to a unique, nonnegative

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value, pi 1. That is, limc M

pi cz=

pi 1, inde-

pendent ofz. In particular, for the case p = 1, whereIpj= 1 for all j, we have limc M

1i cz

1i =1

for all i. Note that, by definition, pi =0 whenever

Ipi = 0, andpi = 0 for all i whenp = 2

N.

4. Performance Bound HierarchiesThe performance bound hierarchies (PBH) techniquedevelops a sequence of bounds on the waiting timemoments by developing a corresponding sequence ofupper and lower bounds onFiz, with the exact solu-tion as its limit. In this section, we first present theexpression for Fizobtained in Singh and Srinivasan(2001) and show how the bounds on this expressionare obtained.

Let cz = i cz, wpc+1z = w

pi c+1z, x

pc+1zk =

xpi c+1zk, and y

pc+1zk= y

pi c+1zk, i = 1

N, denote N 1 vectors. The expression for F1 in

cyclec is (Singh and Srinivasan 2001)

F1cz=2N

p=1

F1 w

pc+1z

N

k=1

k xpc+1zkk y

pc+1zk

(6)

where

kypc+1zk =

k y

pc+1zk ifIpk = 1

1 kypc+1zk ifIpk = 0

kypc+1zk =Sk N

j=1

j jypj c+1zk

kxpc+1zk =

Rk

Nj=1

j jxpj c+1zk

and wpi c+1z, y

pi c+1zk, and x

pi c+1zk are defined

fori = 1 N as follows:

wpi c+1z =

0 ifIpi = 0

Mpi c+1

cz ifIpi = 1

xpi c+1zk =

0 ifIpi=

0 and i >k

i cz ifi k

Mpi c+1

cz otherwise

and

ypi c+1zk =

0 ifIpi = 0 and i >k

i cz ifi k

i cz ifi k

Mpi c+1

cz otherwise

4.1. Basic Concept for the PBH TechniqueThe PBH technique starts with the user specifying a

bound level, n (the level-n bound), where a highernindicates a higher degree of accuracy. It then applies(6) iteratively, n 1 times. The first application pro-duces an expression for F1z with 2

N terms on theright-hand side of the expression. Each term consistsof an unknown, F1, with an associated weight

involving the coefficients kand k. One of theseunknowns is F10, which represents the case whereevery queue was empty at the 1-polling epoch. Theremaining 2N 1 unknowns have at least one oftheir arguments nonzero and we refer to these as thenonzero-argument terms. The arguments for these termsare either 0 or M

pi c+1 for p= 1 2

N. In particu-lar, one of these terms isF1M

11 c+1z M

1N c+1z

F1M1c+1z, which represents the situation where

every queue is nonempty at the 1-polling instant. Werefer to such a term as apure term.

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If n > 1, then (6) is applied to express the 2N 1nonzero-argument F1 terms iteratively, in termsof new F1 terms. A second application of (6) onthe nonzero-argument terms results in a total of2N2N 1 + 1 terms, which now include 2N 1 newterms involving F10 in addition to the F10 term

in the original equation. Thus, there are 2N

1 + 1terms with F10, and 2N 12 nonzero-argument

F1 terms. After n such applications of (6), therewill be 2N 1n nonzero-argument F1 terms andn1

j=0 2N 1j terms involvingF10.

The PBH technique now obtains an upper boundon this expression for F1zby replacing the 2

N 1n

nonzero-argument F1 terms with a pure term.For the lower bound, the PBH technique replacesall nonzero-argument F1 terms except the pureterm with F10. Thus following the replacement, thedesired bound on F1 is expressed using just a pureterm and F10. Theorem 1 shows that these replace-

ments provide the desired bounds.

Theorem 1. For any cycle c 0,

F1cz

F1M1c+1z

2N1p=1

Nk=1

kxpc+1zkky

pc+1zk

+ F10N

k=1

k

x2

N

c+1zkk

y2

N

c+1zk

and (7)

F1 cz

F1M1c+1z

Nk=1

kx1c+1zkk y1c+1zk

+ F102N

p=2

Nk=1

kxpc+1zkky

pc+1zk (8)

Proof. The expression forF1czin Equation (6),reproduced below, is

F1cz

=2N

p=1

F1 w

pc+1z

Nk=1

kxpc+1zkky

pc+1zk

Recall that the superscript p in the expression forw

pi c+1z corresponds to the configuration of binary

switches. In particular, when p = 1, we have I1i = 1for all i (all binary switches are set equal to one) andwhen p= 2N, we have I2Ni = 0 for all i. Thus, toprove Theorem 1, we need to show that F1 w

1c+1z

F1 wpc+1z F1 w

2N

c+1z, p = 2 3 2N 1.

We will prove the first inequality for p= 2, whichcorresponds to the configuration in which the first

binary switch is a zero with all other binary switchesset to one. To prove that the inequality holds, it is

enough to show that F1z1 zN F10 z2 zN.By definition,

F1z =

n1=0

nN=0

zn11 znNN P1n1 nN (9)

Equation (9) can be rewritten as

F1z=

n1=1

n2=0

nN=0

zn11 z

n22 z

nNN P1n1 n2 nN

+ F10 z2 zN (10)

Because

n1=1

n2=0

nN=0

zn11 z

n22 z

nNN P1n1 n2 nN

is a partial PGF, it is a nonnegative number, and so,F1z F10 z2 zN. The same reasoning appliesto show that the first inequality holds for all other

values ofp. The proof of the second inequality followsdirectly because F10 = F10 0 = P10 0. The PBH technique now goes back in time, itera-

tively expressing the upper or lower bound for F1zusing just the pure term and F10, for increasing val-ues ofc. Unlike the hybrid technique which generates2N 1 new unknowns at each succeeding iteration,the PBH technique generates exactly two unknownterms.

In the limit, as c , it can be shown (simi-lar to Cooper and Murray 1969, VI) that M1i cz M1i z = 1. (Intuitively, the contribution from a cus-tomer servedc cycles ago approaches 0 as c .) At

this point in time, all F1M

1

c z F11 1 = 1, andthe upper (lower) bound for F1z is now expressedentirely as a sum of a large number of k andk terms and F10. The technique now obtains atight upper (lower) bound for F10 to obtain thedesired bound on F1z.

Corollary 1.

F1z RS1z + RS10

1 RS20RS2z and (11)

F1z RS3z + RS30

1 RS40RS4z (12)

where RSmz and RSm0, m= 1 4, are constantsinvolving k and k,k = 1 N .

Proof. We will prove the corollary only for theupper bound. The proof for the lower bound followsin an identical manner. Let

RScz =2N1p=1

Nk=1

kxpc+1zkky

pc+1zk and

cz =N

k=1

k x2N

c+1zkk y2N

c+1zk

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Then, from Theorem 1,

F1cz F1M1c+1zRS

cz

+ F10cz (13)

Replacing F1M1c+1z in Equation (13) with F1

M1c+2zpreservesthe upper bound.So, if we iterateon this pure

term for increasing values of c, the resulting expres-sion will provide an upper bound. As we continueto iterate in this fashion, limc M

1i cz= 1 for i =

1 N , and so

limc

F1M1c z= limc

F1M11 cz M

1N cz

=F11 1 = 1

As a result, the upper bound for F1czultimatelyhas the form

F1 cz RS1cz + F10RS2cz (14)

where RS1cz is the sum of all RScz terms,associated with the pure term, over all cycles, andRS2 cz is the sum of all cz terms associ-ated with F10. We ignore the exact specification ofRS1 cz and RS2 cz because they are not rel-evant to the proof, but emphasize that they can becomputed in a relatively straightforward manner.

Set i cz = 0 for i= 1 N in (14). Solving theresulting expression for F10,

F10 RS10

1 RS20 (15)

whereRS10and RS20are the values ofRS1czand RS2cz obtained by setting zi= 0, i= 1 N. From (14) and (15), we get the desired expressionfor the upper bound.

4.2. Bounds for the Waiting Time MomentsAfter expressing F1z using only the k and kterms, waiting time moments are obtained by differ-entiating this expression the appropriate number oftimes. The nth moment of the waiting time requiresFiz to be differentiated n + 1 times. We now showhow to apply the PBH technique to obtain bounds onthe expected waiting time.

4.2.1. Computing Bounds on the Expected Wait-ing Time. Let fi denote the first factorial moment ofthe number of customers present at queue i at an

i-polling epoch, and let fni denote the nth factorialmoment of this number. Let fi0 denote the emptystation i probability at an i-polling epoch. Differenti-ating (1) twice with respect to zi and letting zi 1,we get the expression for the expected waiting timeat queue i :

EWi= f

2i + 2ifiESi +

2i ES

2i1 fi0

2i EC1 i

+ iEBi

21 i (16)

The expressions for f1 and f2

1 , required to computeEW1, are (Srinivasan and Singh 2006)

f1 =

c=0

Ni=1

ERiJi c + ESiJi1 c fi0ESiJi1 c

(17)

and

f2

1 =

c=0

Ni=1

ER2i J

2i c +ES

2iJ

2i1c +2ERiESiJi cJi1c

c=0

Ni=1

fi0ES

2i J

2i1 c + 2f

0i ESiJi1 c

2

c=0

Ni=1

ERiJi c + EGi cESiJi1 c (18)

wheref0

i =N

j=1 EXj

i0, withXj

i 0 denoting the num-ber of customers present at queue j at an i-pollingepoch that found queue i empty. (By definition,Xi

i 0

= 0.) In these equations, E Ri

, E Si

, E R2i

, E S2i

,

Ji1 c,J2

i c , andEGi care constants that are either dataor terms that can be easily computed using the UDStechnique. For instance, Ji1 c and J

2i c are computed

as follows:

Ji c=N

j=i+1

jj c +i1j=1

jj c1 c 0 (19)

J2

i c =N

j=i+1

j2j c +

i1j=1

j2j c1 c 0 (20)

where i c and 2i c are calculated using Equations

(21), (22), and (23):

i c= EBi

1 i

Nj=i+1

jj c +i1j=1

jj c1

c 0 (21)

2

ic = EB2i

1i3

Nj=i+1

j2j c +

i1j=1

j2j c1

c 0 (22)

11 = 211 = 1 j 1 =

2j 1 = 0 2 j N (23)

To compute bounds on the expected waiting times,we partition the terms needed to compute fi andf

2i

into two categories: terms that can be computedexactly using the UDS technique and terms that

require the PBH technique. The only unknowns com-puted using the PBH technique are fi0and f

0i . The

termfi0is computed readily from the expression forF1zsimply by setting zj= 0 for all j= i. To compute

the terms, EXj

i 0, j= 1 N , needed to evaluate f0

i ,we set z i= 0 in the expression for Fiz and differen-tiate it with respect to j. In expressions (17) and (18),the terms containing fi0 and f

0i have a negative

sign in front of them. Hence, to apply the boundsobtained from Corollary 1, an upper (lower) bound onf1 is obtained by using a lower (upper) bound on the

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fi0 terms. A similar observation holds for the boundson f

21 . Corollary 2, presented below without proof,

summarizes these characteristics.

Corollary 2. Differentiating the expression for theupper (lower) bound obtained from Corollary 1 results ina lower (upper) bound on f

1. Similarly, an upper (lower)

bound on f1 is obtained by differentiating the expressionfor the lower (upper) bound obtained from Corollary 1.

4.2.2. Algorithm BOUNDS. Algorithm BOUNDSproduces a hierarchy of successively improving

bounds on the expected waiting time. A similar ap-proach applies for computing higher waiting timemoments. Similar to the hybrid procedure (Srinivasanand Singh 2006), all terms that can be computed usingthe UDS technique (the [S] terms) are computedexactly in Algorithm BOUNDS. For the rest of theterms (the [P] terms), Algorithm BOUNDS com-

putes upper and lower bounds using (11) or (12).The user specifies the bound level as n 1, wherea higher value of n indicates a higher degree ofaccuracy. The algorithm first evaluates the terms,

pi = limc M

pi cz, i = 1 N , p = 1 2

N. Theseterms are determined regardless of the bound level.The algorithm then applies Equation (6) n 1 times

before replacing the nonzero-argument F1 termswith the pure term or F10. To elaborate, for thelevel-n upper bound, the algorithm expresses F1zin terms of 2N 1n nonzero-argument F1 terms.(Recall from 4.1 that these are all the F1 termsexcept F10.) It then replaces each of these nonzero-

argument F1 terms with the pure term. In thesame way, for the level-nlower bound, the algorithmreplaces all the nonzero-argument F1 terms exceptthe pure term with F10. The bounds on F1z arenow determined using Corollary 1, providing the datanecessary to determine bounds on expected waitingtimes. Obviously, the bounds are tighter if we performmore iterations using (6), before making the replace-ment. However, at the same time, the computationaleffort increases significantly if a higher number of iter-ations are first performed using (6), as we show in thefollowing section on complexity analysis.

Algorithm BOUNDS(A) Evaluate the categoryS terms using the UDS

technique.(B) Compute bounds on category P terms using

the PBH technique as follows:(I) Compute the limiting values for the Mi cz

terms by first generating the vectors, Ip, p = 1 2N.

For each p 1 2N, do Steps (1) through (3)below:

(1) Setc to some arbitrary value, say, c = 0, andinitialize M

pj c = 0, j= 1 N .

(2) For each i in the order, N N 1 1, ifIpi = 0, then set M

pi c+1 = 0, else evaluate M

pi c+1

using (4), with the current values of Mpj c, j =

1 N .(3) If M

pi c+1 M

pi c for all i, for some

tolerance level,, then return the limiting values, pi =

M

p

i c. Otherwise, set c= c + 1 and repeat Steps (2)and (3).(II) Determine n, the bound level. Use (6) to

expressF1z iteratively,n times. The resulting expres-sion has 2N 1n nonzero-argumentF1 terms.

(III) Replace all nonzero-argument F1 termsexcept the pure term with either the pure term orF10, depending on whether a lower or an upper

bound is being computed. (Equation (13) presents theform of the resulting expression for the upper bound.)With the resulting expression for F1z, continue theiteration on c until the pure term converges to one.(Equation (14) presents the form of the expression for

the upper bound onF1zat this stage.) Now set zi = 0for all i in the resulting expression to solve for F10.(Equation (15) presents the form of the expression forthe upper bound onF10.) Now use either (11) or (12)to obtain the upper or lower bound on F1z. (SeeCorollary 1.)

(C) After obtaining bounds on all Fiz, i= 1 N, computefi0,i = 1 N , simply by settingzi = 0and zj = 1 for all j = i in the expression for the

bound on F1z. Obtain EXj

i 0 for i= 1 N andj= 1 N , by analytically differentiating the appro-priate bound for Fiz1 zi1 0 zi+1 zN withrespect tozj forj= i.

The procedure for computing the desired boundson F1zis initiated with arbitrary nonzero values forzi< 1. These bounds onF1zare necessary for obtain-ing any waiting time moment bounds.

4.3. Complexity AnalysisWe describe the computational effort required by thePBH algorithm to compute bounds on Fiz for anyqueue, say, queue i = 1. Bounds on waiting timemoments are then obtained in a fairly straightfor-ward manner with relatively little effort. The algo-rithm first applies the UDS technique to computeall terms that can be evaluated with this technique.

This step requires ONlog operations, where isthe utilization of the server and is the tolerance(see Konheim et al. 1994). It next computes

pi , i =

1 N , p= 1 2N. Determining a pi term also

requires ONlog operations. Because there are atotal of N2N1 N such terms, computing all the

pi requires ON

22N log operations. (Note that the

number of pi s to be evaluated is not N2

N because

pi = 0 ifIpi = 0, and

1i = 1 for all i .)

Now consider the recursion on F1z using (6). Forease of exposition, consider first the level-1 upper

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bound. After the first application of (6), initializedwith c = 0, applying the PBH technique results in anupper bound with the following form (see (13)):

F1z F1M11

2N1p=1

Nk=1

kxp1kx

p1

+ F10N

k=1

k

x2

N

1 k

x2

N

1

Computing the k and k terms associatedwithF1M

11 in the above equation requires ON2

Noperations, and computing the corresponding coeffi-cients of F10 requires ON operations. Hence, thisstep requires ON2N operations. The pure term isnow repeatedly expressed using (6) until its argu-ments all converge to one. At each iteration, the newcoefficients generated require ON2N operations toevaluate, just as before. Because the iterations con-

tinue until theM

p

i cterms converge to one, the num-ber of iterations performed is Olog and so therequired computational effort is O N2N log . Thus,for the level-1 bound, the level of effort is deter-mined by the step that computes the

pi s, which is

ON22N log .For the higher-level bounds, the step that deter-

mines the order of effort is the one that first expressesF1z iteratively using (6) n times, before replacingall the nonzero-arguments with pure term or withF10. It can be verified that for the level-n bound,this requires ON2Nn operations. Now, followingthe same logic as used to determine the complexity of

the level-1 bounds, the computational effort requiredfor the level-n bound, n 2, is ON2Nn log .

It is instructive to compare the computational com-plexity of the PBH algorithm with that of the hybridprocedure (Srinivasan and Singh 2006). The hybridprocedure also applies the UDS technique to computeall terms that can be evaluated with the UDS tech-nique ONlog operations), and computes

pi , i =

1 N ,p = 1 2N ON22N log operations). Atthis stage, it could apply (6) on the F1z terms untilthe M

pi c arguments converge to their limiting val-

ues. However, when the arguments converge, they doso to

pi which is not equal to one unlessp = 1. Hence,

each F1 term converges to a F1r, r 1 2N,where r = r1

rN, and so that leaves many

unknowns still to be determined. To overcome thisdifficulty, the hybrid algorithm instead initializes theiterations using (6) with z = r for somer 1 2Nand iterates until it results in an expression forF1

r in terms of F1p, p= 1 2N. That essen-

tially amounts to expressing F1z using (6) Olog

times, requiring ON2Nlog operations to evaluateall coefficients in the resulting expression. Repeatingthis process for every r1 2N results in a set

of 2N equations for the 2N unknowns, F1r. Obtain-

ing these 2N equations thus requires O2NN2Nlogoperations. Solving them involvesO23Noperations.

The hybrid procedure now uses (6) again to expressF1zin terms ofF1terms. It now continues to iterateuntil the F1 terms converge to their limiting F1

pi

values. At this stage, F1z is expressed entirely interms of the F1pi s (now known constants), and the

kand kterms. This step requiresON2Nlog

operations. Thus, the overall level of computationaleffort required by the hybrid procedure to computeF1zis O2

NN2Nlog. We remark that log can bea fairly large number. Note that to determine F1zaccurately, we had to use 105. Note, too, thatlog is very sensitive to , the system utilization.Thus, when = 07, log > 30.

5. Numerical Results

The primary intent of this paper is to provide a hier-archy of improving bounds on F1z that converge tothe exact values. Once the upper and lower boundson F1z are computed, we can compute correspond-ing bounds on the expected waiting time as wellas higher moments in a relatively straightforwardmanner. We were interested, though, in seeing howeffective the bounding scheme was in generating thehierarchy of bounds on the expected waiting time.To this end, numerous experiments were conductedon a number of alternate model configurations. Wefirst wanted to identify how the bound hierarchy pro-gressed toward the exact values of expected waiting

time, and the computational effort required to achievethese bounds. The results presented in Table 1 pro-vide level-1, level-2, and level-4 bounds on expectedwaiting time for models with = 03 and = 07.The mean service, switchover, and setup times wereall set equal to one for every model tested. Differ-ent levels of variation were assumed for the service,switchover, and setup times, with the coefficient ofvariation (CV) ranging from 0 to 9.85. Some of theseCVs, for example, the CV of 9.85, were generatedusing 2-point distributions (see Gupta and Srinivasan1996 for examples). The exact expected waiting timesobtained using the hybrid procedure are provided for

comparison. Table 1 also presents an approximationfor the expected waiting time obtained by averagingthe level-1 upper and lower bounds. All times (in sec-onds) in the table are rounded to the nearest integer.

The results provided in Table 1 suggest that thelevel-1 bounds on expected waiting time are fairlytight, and that a simple average of the upper and lowerlevel-1 bounds produces an estimate of the expectedwaiting time that is generally very close to the exactvalue, at least for the cases considered. Because theSD model is very complex, it is not possible for us

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Table 1 Expected Waiting Time: Bound Hierarchy and Exact Results

Data set E1W1 E2W1 E

4W1 EW 1 E1 W1 CPU

1 CPU2 CPU4 CPU

3SLEVC-UB 5068 5039 5038 5038 5038 0.0 10 10 70-LB 5007 5013 5024

3SHEEE-UB 2946 2757 2751 2741 2744 0.0 00 00 70

-LB 2541 2723 2730

3AHCCE-UB 1607 1602 1601 1601 1601 0.0 70 110 190-LB 1595 1600 1601

5SHEHC-UB 1562 1541 1540 1539 1543 0.0 1140 1520 2540

-LB 1523 1538 1539

5AHCCC-UB 3130 2841 2830 2818 2816 0.0 5240 6390 7000-LB 2502 2789 2801

7SLEEE-UB 251 225 224 223 223 0.0 23900 30330 39780

-LB 196 221 222

Explanation of columns:

Column 1 (#abcde):

#: Number of stations in the model.

a: SSymmetric or AAsymmetric. b: Station utilizationL = 03and H = 07.c/d/e: CV of the service/switchover/setup time distributionsC = 0, E = 10, H = 349, and V = 985.UB: Upper bound. LB: Lower bound.

Columns 210:

EiW1: Level-iupper and lower bounds on the expected waiting time, i= 1 2 4.

EW1: Exact expected waiting time.

E1 W1: Approximate value of EW1 = Average of level-1 upper and lower bounds.

CPUi: CPU time in seconds to computeEiW1,i= 1 2 4 (computingE1W1took less than 0.5 seconds).

CPU: CPU time in seconds to compute E W1.

to determine a priori as to when the bounds wouldbe tight, or how much computational effort wouldbe required to produce upper and lower bounds onthe expected waiting time, that were within a givenpercentage range of the exact value. Therefore, wenext ran many experiments to determine when these

bounds would be tight, when they would not performwell, and when the midpoint approximation wouldnot perform very effectively.

To undertake a systematic study, we restricted theexperiments to models with five queues, at variouslevels of utilizations (= 010307, and 0.9) andonly computed level-1 bounds. We experimented withdifferent CVs for the service times, the switchovertimes, and setup times, using CVs of 0, 1.0, 1.25, 1.5,1.75, 2.0, 3.49, and 9.85. As before, we set the meanservice, switchover, and setup times all equal to one.The number of queues in these experiments rangedfrom N= 357 to 10 queues. We experimented withsymmetric and asymmetric systems. For asymmet-ric systems, while we retained the average service,switchover, and setup times equal to one, the arrivalrates at each queue were different.

These experiments also suggested that the level-1bounds were reasonably tight for many cases and, fur-thermore, the average of the upper and lower level-1

bounds were within 5% of the actual values for amajority of the cases. However, the bounds were rel-atively loose, with a corresponding decrease in theaccuracy of the midpoint approximation when there

was a high degree of asymmetry. They were also rel-atively loose for models with a high switchover timeCV (of 3.49 or higher) and relatively high utilizations = 07, although this was not uniformly, true acrossall such models. No doubt, with real-world appli-cations, it is highly unlikely to find systems with a

switchover time CV of 9.85. We chose models withthis CV because they readily produce some counter-intuitive behavior as discussed next. Table 2 presentssome of these results. The criteria used to decide onwhich experimental results were included in the tableare presented following the discussion on the coun-terintuitive behavior.

Because the SD model does not lend itself to easyanalysis, its performance is often approximated bythe simpler SI model in which the server is assumedto perform a setup (or, alternately, never performsa setup) at a polled queue, regardless of its state.Indeed, common sense dictates that the performanceof the SD model should be at least as good as thatof its SI counterpart in which a setup is assumedto take place at each polled queue regardless of itsstate. Based on this reasoning, the SI model has beenused to provide a conservative bound on the cor-responding SD model (Ferguson 1986). Strangely, thiscommon-sense intuition is not always correct. Asoriginally pointed out by Sarkar and Zangwill (1991)and by numerous other papers subsequently, increas-ing setup times can sometimes reduce the expectedwaiting times.

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Table 2 PBH Technique Performance Evaluation

Data set EW 1 SI E W 1 E1W1 E

W1 Error (%) Perf Imp

SMHC-UB 0.1 855 999 1148 1070 706 14

-LB 887 992

A1MHM-UB 0.1 859 984 1155 1066 835 10

-LB 882 978

A2MCM-UB 0.3 675 543 698 613 1299 55

-LB 347 529

SCCC-UB 0.3 693 586 601 586 000 116

-LB 357 570

SCCV-UB 0.3 3118 1843 1896 1854 060 Inf

-LB 357 1811

SCVC-UB 0.3 3118 5083 5192 4994 175 Inf

-LB 5208 4796

SEVC-UB 0.3 3140 5090 5216 5027 124 Inf

-LB 5230 4837

SEVE-UB 0.3 3165 5103 5222 5039 125 Inf

-LB 5230 4855

SVCV-UB 0.3 5198 3749 3763 3750 003 Inf

-LB 2436 3737SVVC-UB 0.3 5198 7355 7351 7340 020 Inf

-LB 7287 7329

SCCC-UB 0.7 1550 1538 1557 1536 013 96

-LB 833 1515

SCVC-UB 0.7 3976 4381 6005 4921 1233 85

-LB 5685 3836

SEVC-UB 0.7 4092 4628 6186 5104 1029 110

-LB 5801 4021

SEVE-UB 0.7 4117 4718 6213 5149 912 103

-LB 5801 4084

SVCV-UB 0.7 15295 1454 5 1 4612 14540 003 Inf

-LB 12153 14467

SCVV-UB 0.7 6401 6422 7098 6454 050 147

-LB 5685 5809

SCCC-UB 0.9 4550 4550 4552 4550 000 8

-LB 2500 4548

SEGC-UB 0.9 5056 5004 5014 5005 000 7

-LB 2950 4995

SMHC-UB 0.9 4967 5044 4970

-LB 3221 4896


Column 1 (abcd):

a: SSymmetric or AAsymmetric. For asymmetric systems, mean

waiting times are presented only for queue 1.

A1:1= 002,2= 0005, 3= 0063,4= 001,5= 0002. A2:1=01,2 = 005,3 = 006,4 = 001,5 = 008.

b/c/d: CV of the service/switchover/setup time distributionsC = 0, M =

05, E = 10, G = 15, H = 349, and V = 985.Columns 38:

EW1 SI: Upper and lower bounds on the expected waiting time usingSIapproximation.

EW1: Exact expected waiting time using algorithm HYBRID.E1W1: Level-1, upper and lower bounds on the expected waiting time.

E W1: Approximate value ofE W1 = Average of level-1 upper and lowerbounds.

Error (%): % Error betweenEW1andEW1.

Perf Imp: Factor improvement in CPU time = CPUExact/CPULevel-1 (Inf if

CPULevel-1 = 0).Exact waiting time not determined because CPU time was unduly long.

We report a select number of cases in Table 2 whereeither the level-1 bounds were relatively loose, or theupper bound from the SI model exhibits counterintu-itive behavior in the sense that this bound is lowerthan the exact value (often significantly lower), orthe SI bounds were so loose as to have no practical

value. It is noted that not all such cases are reportedin the table. For the upper SI bound, we use an SImodel and assume that a setup takes place at eachpolled queue, regardless of its state. For the lowerSI bound, we use an SI model and assume that nosetups take place at any queue (i.e., the setup timesare zero).

Table 2 shows that with high switchover timevariances, the common-sense bounds are completelyunpredictable. For example, with model SMHC with = 01 and a CV of 3.49, the upper SI bound is sig-nificantly less than the exact value, and is actuallylower than the lower SI bound. Similarly, for SCVC

with = 03, the upper SI bound is much lower thanthe exact value, while the lower SI bound is largerthan the exact value. When the service time or thesetup time has a very high CV, often both boundsare below the exact value. We also note that for somemodels with a low CV for switchover time, with highsetup time variances the SI bounds are so loose as tohave no practical utility.

We note at this point that the SI model producescounterintuitive behavior even with switchover timeCVs as low as 1.75, although we do not report it here

because they were obtained for models with N= 3queues. In general, the experiments we conducted

seem to suggest that for models with a smaller num-ber of queues, such counterintuitive behavior occurswith lower CVs. More detailed sets of experimentsare, however, needed before such a conclusion can

be made.As we have remarked in the introduction section,

a number of papers provide approximation schemesfor the expected times for several variants of theSD model. The model analyzed in this paper is a vari-ant which includes both switchover and setup times.It is possible to modify our model to specialize itto the other variants proposed in the past, and onecan also undertake an extensive numerical compari-son of the accuracy of the approximations. However,

because the focus in this paper is on providing ahierarchy of bounds on the moments of the waitingtime, we did not undertake such a comparison in thispaper.

In general, based on the numerous experiments weconducted, we suggest the following approach. If thelevel-1 upper bound (UB) is fairly close to the level-1lower bound (LB), say if (UB LB)/average < 5%,whereaverageis the simple average, then use the aver-age. Otherwise, obtain the level-2 bound, take the

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Table 3 Expected Waiting Time Level-1 Bounds

Data set E1W1 CPU1 E1 W1

7SHEHC-UB 44.11 20 44.10-LB 44.09 20

7SLEEC-UB 25.23 10 18.26-LB 11.29 10

7AHEEE-UB 16.27 10 16.23-LB 16.19 10

10SLEEC-UB 13.61 160 11.88

-LB 10.15 16010SMCCC-UB 42.21 660 37.54

-LB 32.87 62015SMEEE-UB 27.66 21630 25.13

-LB 22.60 21360


Column 1 (#abcde): See the explanation provided in Table 1.

Columns 24:

E1W1: Level-1, upper and lower bounds on the expected

waiting time.

CPU1: CPU time in seconds to computeE1W1.

E1 W1: Approximate value of EW1 = Average of level-1upper and lower bounds.

average of these bounds, and apply the same rule. Wenote that the level-1 bounds are computed quickly,even for models with a relatively large number ofqueues (as high as 15 queuessee Table 3). Inciden-tally, from a practical perspective, it seems unlikelythat that a single server would attend to such a largenumber of queues, at least in a manufacturing set-ting. On the other hand, the higher-level bounds takeconsiderable computational effort even with sevenqueues, as can be observed for the 7-queue example

in Table 1.In Table 3, we present level-1 bounds for various

large SD models. In general, for models of this size,obtaining the exact values for the expected waitingtimes is either not possible or highly time consum-ing. As before, we also compute an approximate valueE1W1 simply by averaging the level-1 upper andlower bounds. When the level-1 bounds are tight, theexperiments conducted above suggest that the aver-age should present a fairly reliable estimate of theexact value so long as the switchover time CVs arenot extremely high. We remind the reader that if thelevel-1 bounds are not very tight, then the level-2 orhigher level bounds may be applied.

6. ConclusionsDespite its ability to represent real-world systemsmore accurately, the SD polling model is not wellstudied. The inherent complexity of the SD pollingmodel has limited research in this area, and the easier-to-analyze SI polling model is often used as a surro-gate. However, in many instances, using the SI modelin place of the more realistic SD model can result in

questionable values for the desired performance mea-sures, as demonstrated in 5. That provides a moti-vation for developing techniques to compute desiredperformance measures for the SD model more easily.

The descendant sets technique offers a convenientplatform for analyzing the SD model. The technique

not only facilitates the modeling process, it alsoallows us to explain the model in an intuitive manner.For instance, when we discussed two manufactur-ing scenarios, one of which required a new defini-tion of the setup time, the technique allowed us toaccommodate this alternate definition easily, and ina fairly intuitively manner. Tackling this modifica-tion using other mathematical approaches could havepresented additional challenges. The descendant setsapproach also facilitated the development of the PBHtechnique which obtains a hierarchy of increasinglytighter bounds on the waiting time moments for theSD model with increased computational effort.

The PBH technique allows for quick computationof bounds on waiting time moments. In particular, forexpected waiting times, level-1 bounds are computedvery quickly and are usually fairly tight, particularlyfor symmetric models and models where the CV ofthe switchover time is not very high. The memory andcomputational effort required to compute level-1 and,often, even level-2 bounds are relatively minimal andso we can obtain tight bounds for large polling mod-els. The model can be used to quickly determine, forinstance, the affect of adding another product (queue)to the existing product portfolio. The exact analysis ofsuch models using available techniques would have

either taken an extremely long time, or would nothave been possible.

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