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General Open and Closed Queueing Networks withBlocking: A Unified Framework for ApproximationMark Vroblefski, R. Ramesh, Stanley Zionts,

To cite this article:Mark Vroblefski, R. Ramesh, Stanley Zionts, (2000) General Open and Closed Queueing Networks with Blocking: AUnified Framework for Approximation. INFORMS Journal on Computing 12(4):299-316. http://dx.doi.org/10.1287/ijoc.12.4.299.11878

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General Open and Closed Queueing Networks with Blocking:A Unified Framework for Approximation

MARK VROBLEFSKI � Department of Management Science and Information Technology, Virginia Polytechnic Institute and StateUniversity, 1007 Pamplin Hall (0235), Blacksburg, VA 24061, Email: mvro@vt.edu

R. RAMESH and STANLEY ZIONTS � Department of Management Science & Systems, School of Management, State University ofNew York at Buffalo, Buffalo, NY 14260, Email: rramesh@acsu.buffalo.edu, szionts@buffalo.edu

(Received: March 1999; revised: March 2000; accepted: May 2000)

In this paper, we develop a unified framework for approximatingopen and closed queueing networks under any general block-ing protocol by extending and generalizing the approximationalgorithm for open tandem queues under minimal blocking pre-sented in Di Mascolo et al. (1996). The proposed framework isbased on decomposition. We develop decomposition struc-tures and analysis algorithms for any general blocking systemusing the framework. The proposed algorithms have been ex-tensively tested using simulations as a benchmarking device.The results show that the proposed framework yields robust,reliable, and accurate estimates of system characteristics, suchas throughput and Work-In-Process inventory in a wide range ofsystem configurations. The computational load is minimal. Theunified framework presents a highly useful set of tools of anal-ysis for queueing-system designers to use in evaluating theperformance under numerous design alternatives. Directionsfor future research are presented, with a focus on critical appli-cation areas such as packet-switching-network design and cel-lular manufacturing.

M ost production and communication systems can be mod-eled as queueing networks. An analysis of queueing net-works provides a basis for the optimal design of severalreal-life systems such as assembly production lines, net-work-structured cellular manufacturing systems, and pack-et-switching telecommunication networks. Inherent in suchreal-life systems are several complexities that make an exactanalysis of the underlying queueing models almost impos-sible. Some of the major complexities include the blockingphenomenon resulting from finite waiting spaces in therespective queues in a network, the network topology(open/closed), general service-time distributions, and thesize of the network. The importance of these issues in prac-tical systems design has led to considerable attention in theliterature on approximation techniques for queueing-systemanalysis.

In this paper, we develop a unified framework for ap-proximately solving a queueing system under blocking. Toillustrate our methodology better, we focus on tandem net-works and cyclic networks with limited waiting space ateach service point. We develop approximation strategies foropen and closed queueing systems under four types ofblocking: manufacturing, communication, minimal, and general.

The proposed algorithms in these cases are based on aunified model of serial queueing systems. We assume gen-eral service-time distributions at the server nodes and bothsaturated input supply and output demand. The saturationassumption is intended to simplify our development, andthe proposed framework can easily be modified to handleunsaturated supply and demand (Di Mascolo et al. 1996).The framework is unified for a general class of blockingprotocols for the following reasons: (i) a common set ofmodeling tools such as the kanban characterization ofqueueing systems, synchronization stations that link theflows from a production cell to the next, closed queueinganalysis of circulating kanbans within a cell, and open sys-tem analysis of the production stage queues in the cells areused in each blocking protocol; (ii) a common strategy forcell-level analysis and a common iterative algorithm forapproximating over the cells in a serial line are used; and (iii)the same approach is used for approximating both open andclosed queueing systems. However, the individual protocolsdiffer in how these models are formulated; accordingly, thespecific models of analysis and approximation within eachcell are unique to each protocol, although developed fromthe same set of modeling tools.

Existing work can be classified as: Blocked Open Queue-ing Networks (BOQN) and Blocked Closed Queueing Net-works (BCQN). The existing literature addresses primarilyBOQN systems, and considers mainly manufacturing andcommunication blocking (Perros 1994). Studies on BCQNsystems are much fewer in number because of the additionalcomplexity arising out of a fixed number of pallets circulat-ing in such systems (Onvural 1990). The work of Di Mascoloet al. (1996) is perhaps a first approximation scheme forBOQN systems under minimal blocking. General blocking,which encompasses all types of blocking was introduced byCheng and Yao (1994). To our knowledge, there exists noapproximation algorithm for analyzing BOQN or BCQNsystems under general blocking. However, general blockingforms the basis of a large set of queueing-system designconfigurations that could significantly outperform designsunder the other blocking schemes (Ramesh et al. 1997a,1997b). Based on these considerations, there is a need for a

Subject classifications: Queues-approximations, queues-tandem, queues-algorithmsOther key words: Inventory-production: approximations-heuristics

299INFORMS Journal on Computing 0899-1499� 100 �1204-0299 $05.00Vol. 12, No. 4, Fall 2000 © 2000 INFORMS

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unified and efficient approximation framework for bothopen and closed queueing networks under all types of block-ing. This need motivated our research.

Before we present our approach, we review relevant lit-erature. An exact analysis of BOQN systems with more thantwo servers is difficult, so approximations and simulationsare widely used. Several approximation schemes are basedon decompositions of queueing networks into subsystemsthat are analyzed separately. Iterative procedures that com-bine the subsystem analyses into overall network analysesconstitute the core of such approximation strategies. Someimportant works of this type include Dallery (1990), Dalleryand Frein (1993), Di Mascolo et al. (1996), and Perros et al.(1988). Perros (1994) presents a detailed classification ofapproximation schemes for BOQN systems. Onvural (1990)presents a survey of exact and approximate methods forBCQN systems. While two server BCQN systems have beenwell analyzed (Akyildiz 1987, Gordon and Newell 1967a,1967b, Van Dijk and Tijms 1986), studies on systems withmore than two servers are rather few (see e.g. Onvural 1990).In the following discussion, we address the past researchthat constitutes the building blocks of the proposed frame-work and develop its underlying rationale.

Di Mascolo et al. (1996) developed an approximationscheme for a kanban-controlled open production line underminimal blocking by decomposing the line into productionstages, analyzing the stages independently, and then linkingthe analyses into an overall iterative approximation proce-dure. Each stage is modeled as a Nonblocking ClosedQueueing Network (NBCQN) with a fixed number of kan-bans circulating within a production stage constituting theloop. The authors introduce the notion of a synchronizationstation between adjoining stages, and analyze each NBCQNas a subsystem constituting a production stage and twoadjoining synchronization stations, one on either side of thestage. The synchronization stations yield closed-formMarkov solutions under minimal blocking, and the produc-tion stage is analyzed as a �(n)/C2/1/N queue using themethod of Marie (1980). In the analysis of Marie (1980), thedeparture rates from a server are state-dependent, althoughthe service time follows a fixed phase-type (coxian) distri-bution. This is due to the fact that the arrival rates to thequeue in front of the server are state-dependent. Further-more, as long as the distributions of the service times arephase-type (combination of exponential distributions), thesystem can be modeled as a continuous-time Markov chainwith a product form solution (Gordon and Newell 1967b, DiMascolo et al. 1996). Therefore, by integrating the threeanalyses together with a product-form network solution forNBCQN systems, an iterative procedure for the overall anal-ysis of the subsystem is developed. Details and references onthese product-form methods can be found in Baynat andDallery (1993), Bruell and Balbo (1980), and Baskett et al.(1975). Finally, linking the subsystem analyses in sequentialdownstream and upstream orders, a successive-iterationscheme for the approximation of the entire line is obtained.

The proposed unified framework for BOQN and BCQNsystems builds on and generalizes the above decompositionstrategy. Queues in tandem under each blocking protocol

have an equivalent kanban system configuration (Mitra andMitrani 1990, 1991, Cheng and Yao 1994). Consequently, weadopt this equivalence and model BOQN and BCQN sys-tems under various blocking protocols using their kanbansystem counterparts. First, we generalize the decompositionstrategy of Di Mascolo et al. (1996) to approximate kanbansystems under various blocking protocols, yielding a frame-work for the analysis of BOQN systems in general. Next, weextend this framework to BCQN systems by embedding ananalysis of the corresponding BOQN system within an NB-CQN analysis of the pallets in circulation. The strategy inthis case is to realize BOQN flow characteristics equivalentto those of the BCQN. As a result, we obtain a unifiedframework for analysis of BOQN as well as BCQN systemsunder any blocking mechanism.

Extensive empirical evaluations of the proposed frame-work to assess the computational effort and the quality ofthe approximations have been carried out. The quality of theapproximations has been measured in terms of two param-eters: system throughput (ST) and the total work in process(WIP) inventory. The approximations have been evaluatedin comparison with measures obtained through simulation.The results show that the proposed framework yields reli-able approximations with minimal computational effort fora significant class of queueing networks. Furthermore, theresults are especially significant as the approximations workwith any general service-time distribution, whereas someknowledge of the service distribution is required for simu-lation.

The organization of the paper is as follows. Section 1presents the approximation framework for BOQN systems.Section 2 develops the framework to approximate BCQNsystems and concludes with the unified model. Section 3presents the computational results, and Section 4 presentsthe conclusions and directions for future research.

1. Blocked Open Queueing NetworksWe first introduce some basic notation in the followinganalysis of BOQN systems. Let i � 1, . . . , N denote asequence of servers operating in series with finite bufferspace at each service cell. The input to the system (rawmaterial) enters cell 1 and the output (finished products)leaves cell N. We assume saturated input and output con-ditions in the queueing system and general service-timedistributions at the cells. Let ki denote the buffer spaceavailable at service cell i, for i � 1, . . . , N. We develop thedifferent blocking conditions in this queueing system asfollows.

The parts arriving at a cell form a waiting line in thebuffer area. Assume that each cell has a roving server, whomoves from buffer to buffer to complete the processingrequired. A completed part is immediately transferred to thenext cell if a buffer is available and the blocking protocoladmits the transfer. Otherwise, the finished part is retainedin its existing buffer and the roving server is free to move onto the next buffer in that cell. In this operational scheme, wedefine two more control parameters for each cell as follows.Let ai and bi denote the maximum number of parts at cell i

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that are awaiting service and awaiting outward transferafter service completion, respectively. It follows that, ai � ki,bi � ki and ai � bi � ki, i � 2, . . . , N � 1, a1 � k1 � b1 (dueto supply saturation), aN � kN, bN � 0 (due to demandsaturation), and ki � 0 @ i. These conditions define thegeneral blocking control structure in tandem queues (Chengand Yao 1994). The other well-known blocking structuresare obtained from the general blocking defining conditionsas follows: {ai � ki, bi � 1 for i � 1, . . . , N � 1, aN � kN, bN �0} defines manufacturing blocking, {ai � bi � ki, for i � 1, . . . ,N � 1, aN � kN, bN � 0} defines minimal blocking, and thesame conditions of manufacturing blocking modified withbi � 0 for i � 1, . . . , N � 1 define communication blocking(Ramesh et al. 1997a, Buzacott 1988, Zipkin 1989).

In general, let (a, b, k) � (ai, bi, ki)i�1i�N define the control

parameters of the general blocking tandem queue system.We denote the sets of control parameters for manufacturing,minimal, and communication blocking protocols as (k, 1, k),(k, k, k), and (k, 0, k), respectively. Furthermore, it has beenshown that any general blocking system (a, b, k) has equiv-alent systems (a, k, k) and (k, b, k) that yield identical timeepochs in the flow of parts throughout the line (Ramesh etal. 1997a, 1997b). These equivalent configurations reducegeneral blocking systems to just two sets of parameters, andare termed selective input and selective output control systems,respectively. We address general blocking systems usingtheir selective output control equivalents. The above controlparameters also have specific functionalities in the kanbanequivalents of queues in tandem as follows. The parametersai, bi, and ki denote the number of input buffers, outputbuffers and the circulating kanban cards at each cell i (Mitraand Mitrani 1990). The proposed approximation frameworkemploys the kanban equivalents, and includes algorithmsfor (k, b, k), (k, 1, k) and (k, 0, k) blocking configurations.

In the kanban version of queues in tandem analysis, DiMascolo et al. (1996) introduce the concept of synchronizationstations. A synchronization station is a virtual station posi-tioned between adjoining service points, and is intended tocapture the coordination between the two cells in the trans-fer of parts from one to the other. A synchronization stationbetween cells i and (i � 1) consists of two buffer areas:upstream and downstream. The upstream area correspondsto the output buffers of cell i and the downstream areacorresponds to the bulletin board to which the releasedkanbans return in cell (i � 1). A finished part in the up-stream area is matched with a kanban in the downstreamarea, and together are taken to the input queue awaitingservice at cell (i � 1). In the decomposition strategy, theentire line is broken into subsystems, with each subsystemconsisting of the synchronization station preceding a servicepoint, the input queue at the service point, and the followingsynchronization station. Analyzing each subsystem inde-pendently, the entire line is approximated using a forward-backward iterative passing scheme among the subsystems(Di Mascolo 1996).

We employ the above scheme in the proposed generalizedframework. We develop the structure of the synchronizationstations, their methods of analysis, and the iterative schemesof line approximation under different blocking protocols.

We first present the framework for approximation, and sub-sequently specialize it for different protocols.

1.1 BOQN Approximation FrameworkThe BOQN approximation framework is based on the de-composition strategy of DiMascolo et al. (1996). Consider theopen tandem queueing system shown in Figure 1(a). Usingthe kanban version of this line, the system is decomposedinto a sequence of subsystems as shown in Figure 1(b). Asynchronization station SSi(i�1) participates in the individualanalyses of subsystems i and (i � 1). The queueing system ateach service cell in Figure 1(a) is indicated as the subnetworkin Figure 1(b). The upstream arrivals correspond to the flowof parts through the line, and the downstream arrivals per-tain to the returning kanbans at each cell. The arrival ratesare state-dependent, and are denoted as follows. The up-stream arrival rates at subsystem i are �i

u(niu), 0 � ni

u � ki�1,where ni

u is the number of parts awaiting removal at theoutput buffer area of subsystem i. The downstream arrivalrates at subsystem i are �i

d(nid), 0 � ni

d � ki�1, where nid is

the number of kanbans already at the bulletin board ofsubsystem (i � 1). The output buffer area of cell i and thebulletin board of cell (i � 1) together constitute the synchro-nization station SSi(i�1). Let �i(n), 1 � n � ki denote the statedependent service rates at cell i. Let �j(n), 0 � n � ki denotethe state-dependent arrival rates of the combination of partsand kanbans at subnetwork j. For the sake of simplicity, weassume saturated input supply and output demand in thetandem queueing system in this presentation. This is not alimitation of the proposed framework, and unsaturated sup-ply and demand can easily be incorporated by adding asynchronization station at the beginning of the line to feedinputs according to a saturated supply and another synchro-nization station at the very end to pull the flow with asaturated demand (see Di Mascolo et al. 1996 for details).The structures of these synchronization stations are inde-pendent of the blocking protocols that may be used withinthe queueing system. We now present the overall approxi-mation framework as follows. Subsequently, we specializethis framework to specific blocking protocols.

Framework: BOQN

Step 0: {Initialization}Fix all downstream arrival rate parameters �d’sto some initial values. The suggested initialvalues could be the mean service rates at thecorresponding subsystems.

Step 1: {Forward Pass}Starting from subsystem 1, successively solvefor each subsystem till subsystem (N � 1); inthis analysis, solve for the parameters �u

(i�1)

given the parameters �ui and �d

i while analyzingsubsystem i.

Step 2: {Backward Pass}Starting from subsystem N, successively solvefor each subsystem till subsystem 2; in thisanalysis, solve for the parameters �d

(i�1) giventhe parameters �u

(i�1) and �di while analyzing

subsystem i.

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Step 3: {Convergence Test}Check for convergence of the �d and �u

parameters at all the subsystems. Ifconvergence is reached according to the criteriaused, then stop; else return to Step 1.

The above framework involves the independent analyses ofthe subsystems in steps 1 and 2. Modeling each subsystem asan NBCQN with the circulating kanbans, the overall strat-egy of these analyses is presented as follows.

Algorithm: Subsystem_Analysis

Step 0: {Initialization}Consider subsystem i. Assume the unknownstate-dependent service rates (�) at each stationof subsystem i (SS(i�1)i, subnetwork i, SSi(i�1)).Initially, all values of � at each station can beset at the mean service rate of the subsystem(see Di Mascolo et al. 1996 for a discussion onsetting initial values). Note that these valueswill be recomputed until stability is reached inthe following steps.

Step 1: {NBCQN Modeling}Modeling the circulation of kanbans withinsubsystem i as an NBCQN, determine the state-dependent arrival rates (�) at each station usingthe algorithm of Buzen (1973) (also see Bruelland Balbo 1980).

Step 2: {Throughput Determination}For each station in subsystem i, determine thestate-dependent throughput rates (v), given thearrival rates (�). In this case, employ Markovanalyses for synchronization stations and thealgorithm of Marie (1980) for the subnetwork.The individual stationwise analysis requires theunderlying subsystem be modeled as anNBCQN.

Step 3: {Service Rate Determination}Set the service rates (�) of each station to theircorresponding throughput rates (v).

Step 4: {Convergence Test}Check for convergence of the � parameters ateach station of the subsystem. If convergence isreached according to the criteria used, thenstop; else return to Step 1.

The key to the above analysis lies in the ability to model eachsubsystem as an NBCQN under a variety of blocking pro-tocols. The requirements of NBCQN modeling can be sum-marized as follows:

(a) When a part and a kanban are found at a synchroniza-tion station, the protocol should permit an immediatetransfer of them together to the following subnetwork.This implies no blocking at the synchronization stations.

(b) When a service is completed at a subnetwork, the partshould be transferred immediately to the next waiting

Figure 1. The decomposition framework.

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station (which could be either within the subnetwork orat the following synchronization station). This impliesno blocking at the subnetwork.

In the following sections, we develop architectures of sub-systems that satisfy these requirements under all types ofblocking protocols.

1.2 General Blocking SystemsWithout loss of generality, the decomposition strategy undergeneral blocking is illustrated in Figure 2 using three stages.Extensions to more than three stages is straightforward. Inthis Figure, the line is decomposed into three subsystems,corresponding to the three service cells. Subsystems 1 and 3are special in the sense that they mark the beginning andend of the line, respectively. The decomposition structure ofsubsystem 2 will be repeated at other intermediate sub-systems if the line has more than three stages. Each sub-system is modeled as a NBCQN as follows.

First consider subsystem 2, which represents any interme-diate stage in the line. The subnetwork component of thissubsystem is buffered between the synchronization stations

SS12 and SS23. This subsystem is a closed queueing networkwith the kanbans of cell 2 circulating within the subsystem.This is shown in the circulation returning from SS23 to SS12

in Figure 2. Note that the bulletin board (downstream com-ponent) of a synchronization station SS23 has a capacity toaccommodate up to k3 kanbans, and its output buffer area(upstream component) has b2 spaces. Now consider SS12.When a part and a kanban are matched at SS12, they togethermove into the input buffer area of cell 2. Since we employthe selective output control equivalent of general blocking,we have ai � ki @i, and hence, there will always be a spacein the input buffer area when this match occurs. However, itis necessary to ensure that the roving server does not beginprocessing on an input part if the output buffers in the cellare already full. Note that this situation can arise since b2 �k2 in the selective output control protocol. Consequently,blocking could arise within the subsystem. Therefore, inorder to model the subsystem as an NBCQN, we develop thefollowing strategy.

First, we classify the parts entering the input area withkanbans into two sets: those for which a space in the output

Figure 2. General blocking decomposition.

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area is available for transfer after service completion, andthose for which no such space is available. These sets aredenoted as A and U, respectively. Let x2 � b2 denote thenumber of finished parts awaiting removal in the outputbuffer area of SS23. Therefore, if �U� � 0, then �A� � x2 � b2,and if �A� � x2 � b2, then �U� � 0. Using these conditions, westructurally segregate the sets A and U as follows. We in-troduce another type of kanban, termed the virtual kanbanswithin each subsystem. The cell 2 is assigned b2 virtualkanbans, and are posted at a virtual bulletin board, which ispositioned between the main bulletin board and the servicestation. When a part and a kanban leave synchronizationstation S12, they first enter a queue called the virtual queue.The virtual queue and the virtual bulletin board togetherconstitute a virtual synchronization station, and operate thesame way as the other synchronization stations. A part witha kanban at the virtual queue is matched with a virtualkanban at the virtual bulletin board, and together are takento the real input queue. The server is allowed to process onlythose parts from the real input queue. Upon service comple-tion, the part and the two kanbans are moved to the outputbuffer area in SS23. When a kanban from cell 3 arrives at thebulletin board of SS23, the part is detached from the twocurrent kanbans, matched with a kanban from cell 3, andmoved to the virtual queue of subsystem 3. Finally, thedetached kanbans are returned to their respective bulletinboards in subsystem 2.

The above design, illustrated using subsystem 2, is for-malized for any intermediate subsystem 1 � i � N in Figure3. Each cell i has ki kanbans, ai � ki input buffers and bi � ki

output buffers. The subsystem modeling cell i is composedof the following sequence of stations: synchronization sta-tion SS(i�1)i, a virtual synchronization station denoted asVSSi, a real input queue denoted as RIQi, and a synchroni-zation station SSi(i�1). The output buffer area and the bulle-tin board of a synchronization station SSi(i�1) have bi andki�1 waiting spaces, respectively. The virtual queue and thevirtual bulletin board at VSSi have (ki � bi) and bi waiting

spaces, respectively. The real input queue RIQi has bi wait-ing spaces.

As can be seen from Figures 2 and 3, a subsystem 1 � i �N involves two nested CQNs. The kanbans among the sta-tions SS(i�1)i, VSSi, RIQi, and SSi(i�1) constitute the real CQN,while the virtual kanbans circulating among VSSi, RIQi, andSSi(i�1) constitute a virtual CQN. The following theoremestablishes the correctness of modeling cell i under generalblocking using the nested CQN structure.

Theorem 1. The nested CQN structure in subsystem 1 � i � Nyields the same time epochs and flow control as in general block-ing. Furthermore, each CQN in the nested scheme is nonblocking.

Proof: Consider SS(i�1)i. The capacities of the output areaand the bulletin board in SS(i�1)i are as specified in thegeneral blocking protocol. When a part and a kanban arematched at SS(i�1)i, they are immediately transferred to thevirtual queue in VSSi without blocking. This is because: (i)the virtual queue has a capacity of ki � bi, (ii) when thevirtual queue is non empty, the virtual bulletin board shouldbe empty (otherwise a match would occur at VSSi), andhence, (iii) when the virtual queue is full, there can be nokanbans at the bulletin board of SS(i�1)i. This can be ob-served from the circulations in the two CQNs. Next, therecan be no blocking in the virtual CQN as there are exactly bi

spaces in each station of this network. Finally, since theflows in the two CQNs are identical between VSSi, RIQi, andSSi(i�1), both the CQNs are nonblocked.

The nonblocked transfers out of SS(i�1)i are the same timeepochs as in the underlying general blocking protocol. Theadmitted parts are simply partitioned into two sets U and A,with U waiting at the virtual queue and A at the real queue.Since the CQNs are nonblocking and the virtual CQN en-sures that there can be at most bi parts between the realqueue and the output buffer area of cell i, the time epochs inthe virtual CQN correspond to those in the underlying pro-tocol. �

Figure 3. Analysis of subsystem 1 � i � N under general blocking.

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The above virtual system modeling is not needed for cells1 and N, as they are basically nonblocking networks them-selves (see Figure 2). Consequently, subsystems 1 and N canbe analyzed using the procedure given in Di Mascolo et al.(1996). We develop an algorithm to analyze a subsystem 1 �i � N using the framework presented earlier, as follows.

First, consider SS(i�1)i. The state-dependent arrival ratesof the returning kanbans at SS(i�1)i are denoted as �I

i(nIi), nI

i �0, . . . , ki, where nI

i is the number of kanbans already presentat SS(i�1)i. Similarly, let the state-dependent arrival rates ofparts into SS(i�1)i be denoted as �u

i (nui ), nu

i � 0, . . . , bi�1,where nu

i is the number of parts already at SS(i�1)i. Thestation SS(i�1)i can be independently modeled as a Markovchain fed by two Markovian processes, and yields a closedform solution (Di Mascolo et al. 1996). The Markov analysisof SS(i�1)i yields the state-dependent throughput rates fromthis station denoted as vI

i(nIi), nI

i � 0, . . . , ki. The combinationof a part and kanban from SS(i�1)i enter the virtual queue,whose capacity is (ki � bi). Let �VQ

i (nVQi ), nVQ

i � 0, . . . , (ki �bi) denote the state-dependent arrival rates at the virtualqueue where nVQ

i is the number of parts already in thisqueue. The following Lemma establishes a mapping from nI

i

to nVQi .

Lemma 1. nVQi � max{0, ki � bi � nI

i}, 1 � i � N

Proof. Note that nIi can be any number in the discrete

interval between 0 and ki. First, consider the case where ki �bi � nI

i � 0. If nIi kanbans are at the bulletin board of SS(i�1)i,

then ki � nIi kanbans, along with their accompanying parts,

can be found anywhere in the virtual CQN. However, sincebi virtual kanbans are available and ki � nI

i � bi, clearly theki � nI

i kanbans should exist between RIQi and the outputarea. Hence, nVQ

i � 0 in this case. Now, consider the casewhen ki � bi � nI

i � 0. Since ki � nIi � bi, all virtual kanbans

are in use between RIQi and the output area. Consequently,nVQ

i � ki � bi � n1i . �

The above Lemma is used to match the throughput ratesfrom SS(i�1)i with the arrival rates at the virtual queue in aniterative approximation scheme for the nested CQNs. SincenVQ

i is uniquely mapped onto nIi when nI

i � ki � bi, we have

nVQi � ki � bi � n I

i (1)

�VQi �nVQ

i � v Ii�n I

i @n Ii � ki � bi. (2)

However, nVQi � 0 @nI

i � ki � bi. Consequently, we approx-imate �VQ

i (nVQi ) in this case as follows.

�VQi �0 � � �

��ki�bi

ki

p Ii��v I

i���� ���ki�bi

ki

p Ii�� (3)

where pIi(�) is the probability of � kanbans at the bulletin

board of SS(i�1)i. This approximation is a weighted averageof the throughput rates vI

i(�), � � ki � bi, weighted by thesteady state probabilities pI

i(�) derived from the Markovanalysis of SS(i�1)i.

Now, consider VSSi. The state-dependent arrival ratesof the returning virtual kanbans at VSSi are denoted as

�VBi (nVB

i ), nVBi � 0, . . . , bi, where ni

VB is the number of virtualkanbans already present at the virtual bulletin board. Thefollowing Lemma establishes a mapping from nI

i to nVBi .

Lemma 2. nVBi � max{0, nI

i � ki � bi}, 1 � i � N.

Proof: We analyze again in terms of two cases: (i) nIi � ki �

bi and (ii) nIi � ki � bi. First, consider case (i). As in Lemma

1, since nVQi � 0 and ki � nI

i kanbans can be found betweenRIQi and the output buffer area, we should have bi � ki � nI

i

virtual kanbans at VSSi. Now, consider case (ii). Since ki �nI

i � bi, all virtual kanbans are in use between RIQi and theoutput area. Hence, nVB

i � 0. �

The above Lemma provides the basis for a reverse mappingfrom the arrival rates at the virtual bulletin board to that atthe real bulletin board in an analysis of the nested CQNs.Since nVB

i is uniquely mapped onto nIi when nI

i � ki � bi, wehave

nVBi � n I

i � ki � b I (4)

� Ii�n I

i � �VBi �nVB

i @n Ii � ki � b I. (5)

However, nVBi � 0 @nI

i � ki � bi. Nevertheless, both the realand virtual kanbans always travel together within the vir-tual CQN. Consequently, the arrival rate at the virtual bul-letin board when nVB

i � 0 is the same as the arrival rates atthe real bulletin board for any nI

i � ki � bi. Applying this, wedetermine

� Ii�n I

i � �VBi �0 @n I

i ki � bi. (6)

The mapping from vIi(nI

i) to �VQi (nVQ

i ) is termed the forwardmapping of rates (equations 1–3), while the mapping from�VB

i (nVBi ) to �I

i(nIi) is termed the reverse mapping of rates

(equations 4–6). The forward and reverse mappings areused in an alternating manner within an iterative frameworkfor the analysis of subsystem 1 � i � N. Finally, putting allthe above pieces together, we present the iterative algorithmfor any intermediate subsystem under general blocking asfollows.

Algorithm: General Blocking Subsystem Analysis(GBSA)

Step 0: {Initialization}Consider subsystem i. Fix parameters �u

i (nui ),

nui � 0, . . . , bi�1 and �d

i (ndi ), nd

i � 0, . . . , ki�1.Initialize �I

i(nIi), nI

i � 0, . . . , ki to some initialvalues.

Step 1: {Solve SS(i�1)i}Solve the Markov model of SS(i�1)i (Di Mascoloet al. 1996). Determine the throughput ratesvI

i(nIi), nI

i � 0, . . . , ki.Step 2: {Determine arrival rates at virtual queue:

Forward Mapping}Determine �VQ

i (nVQi ), nVQ

i � 0, . . . , ki � bi fromvI

i(nIi), nI

i � 0, . . . , ki from the forward mappingequations (1)–(3).

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Step 3: {Solve the virtual CQN}Solve the virtual CQN using the iterativeapproximation scheme given in Di Mascolo etal. (1996), by fixing �VQ

i (nVQi ), nVQ

i � 0, . . . ,ki � bi at values determined from step 2 and�d

i (ndi ), nd

i � 0, . . . , ki�1 at values already fixedin Step 0. Determine �VB

i (nVBi ), nVB

i � 0, . . . , bi

from this analysis.Step 4: {Determine arrival rates of real kanbans at

SS(i�1)i: Reverse Mapping}Determine �I

i(nIi), nI

i � 0, . . . , ki from �VBi (nVB

i ),nVB

i � 0, . . . , bi from the reverse mappingequations (4)–(6).

Step 5: {Convergence Test}Determine whether the recomputed �I

i valuesare within of their old values. If they arewithin the chosen interval, stop; else return toStep 1.

Algorithm GBSA solves the problem of analyzing a sub-system by partitioning it into two components: SS(i�1)i andthe virtual CQN. Note that these two components basicallyconstitute the real CQN underlying the subsystem. Algo-rithm GBSA completes the analysis of the nested CQN sys-tem by embedding the virtual CQN analysis within the realCQN analysis, using the mappings between SS(i�1)i and thevirtual CQN. Finally, the overall analysis of the entireBOQN system under general blocking is completed usingthe framework presented earlier, with algorithm GBSA forthe analysis of all subsystems 1 � i � N, and the method ofDi Mascolo et al. (1996) for subsystems 1 and N.

1.3 Manufacturing and Minimal Blocking SystemsManufacturing blocking systems are easily analyzed by

using the algorithm for general blocking developed above,by simply setting bi � 1 for any cell i where manufacturingblocking is involved. The minimal blocking systems havebeen addressed in Di Mascolo et al. (1996).

1.4 Communication Blocking SystemsIn this protocol, service on an input part in a cell can begin

if and only if there is a space in the input queue at thefollowing cell. In the kanban version, this translates to bi �0 at cell i where communication blocking is used. Conse-quently, service on an awaiting part can begin if and only if:(i) a space is available in the input area, and (ii) a kanban isavailable at the bulletin board of the following cell. Clearly,setting bi � 0 will destroy the nested CQN structure ofgeneral blocking systems developed earlier. Therefore, aspecial nested CQN structure is developed for communica-tion blocking.

The decomposition structure of communication blockingsystems is shown in Figure 4. To begin with, consider anysubsystem 1 � i � N � 1. This subsystem is also modeledalong the lines of intermediate subsystems in the generalblocking case, using the concept of nested CQNs. We intro-duce the virtual synchronization station and the virtual kan-bans as before. In addition, we also introduce another set ofkanbans termed Communication Blocking (CB)-kanbans with aCommunication Blocking Bulletin Board (CBB) at subsystem i.

The bulletin board is appended to SS(i�1)i, and a part canleave SS(i�1)i if and only if it is matched with a real kanbanand a CB-kanban. The triplet joins the virtual queue at VSSi,and is matched with a virtual kanban to enter RIQi. At thisstage, the input part carries three kanbans with it. Afterservice completion, the CB-kanban is detached and returnedimmediately to CBBi. The completed part proceeds to theoutput area of SSi(i�1) with the other two kanbans. The partflow from SSi(i�1) to the next cell and the return of the realand virtual kanbans to their respective bulletin boards arethe same as in the general blocking model. In this model,subsystem i is assigned (ki � 1) CB-kanbans, and bi is setequal to 1. While the above decomposition structure is uni-formly used for all subsystems i � 2, . . . , N � 2, subsystem1, (N � 1) and N are modeled as shown in Figure 4.

The CB-kanbans are a kind of virtual kanbans themselves.They are specifically intended to model the flow controlprocess in communication blocking. In fact, communicationblocking is closely related to manufacturing blocking, and itis illustrative to derive the modeling logic from a compari-son of the two protocol models shown in Figure 5. Wedevelop this analysis as follows.

Consider a cell i with ki waiting spaces. The communica-tion blocking protocol specifies that: (i) cell i is blocked ifthere are exactly ki�1 parts in the next cell i � 1, and (ii) anypart on which service is completed at cell i should immedi-ately be transferred to cell i � 1. In comparison, the manu-facturing blocking protocol specifies that cell i is blocked if:(i) there are exactly ki�1 parts in the next cell i � 1, and (ii)there is one part at cell i, on which service is completed andretained in its current buffer. Based on the above, it caneasily be seen that: (i) the input waiting area at cell i undercommunication blocking consists of the output buffer atSS(i�1)i, the virtual queue at cell i, and RIQi, and (ii) the inputwaiting area under manufacturing blocking consists of onlythe virtual queue at i and RIQi in the subsystem configura-tions. This is shown in Figure 5. While the general blockingmodel ensures that there can be at most ki parts in the inputarea of cell i under manufacturing blocking, the CB-kanbanCQN and the output buffer of SS(i�1)i together ensure thisunder communication blocking. In the communicationblocking model, cell i � 1 is blocked when (i) a part exists inthe output buffer of SS(i�1)i and (ii) the (ki � 1) CB-kanbansexist between the virtual queue at i and RIQi. Finally, it isalso easy to observe that both the CB-kanban and virtualkanban CQNs are nonblocked. Based on the above analysisand Theorem 1, the following theorem is obtained.

Theorem 2. The nested CQN structure involving the real kan-bans, virtual kanbans, and CB-kanbans yields the same timeepochs and flow control as in communication blocking. Further-more, each CQN is nonblocked.

We now develop an analysis of the communication block-ing subsystems using a set of mappings as before. Let�CB

i (nCBi ), nCB

i � 0, . . . , ki � 1 denote the state-dependentarrival rates at CBBi. Further, we now have two downstreamarrival processes at SSi(i�1). Let �d

i (ndi ), nd

i � 0, . . . , (ki�1 � 1)denote the state-dependent downstream arrival rates atCBBi�1 of SSi(i�1). The other arrival and throughput rate

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parameters are the same as before, and these parameters areindicated in Figure 4. The subsystem analysis begins withSS(i�1)i, which is modeled as a Markov chain fed by threearrival processes as follows. Let (nu

i , nIi, ni

CB) define a state ofthe SS(i�1)i Markov chain. Note that nu

i is equal to either 0 or1. When nu

i � 0, nCBi � nI

i � 1 when nIi � ki, and (nCB

i � nIi)

or (nCBi � nI

i � 1) when nIi � ki. Similarly, when nu

i � 1, theonly feasible configurations are (nI

i � nCBi � 0) and (nI

i � 1,nCB

i � 0). With these conditions the state-transition diagramof the Markov chain is shown in Figure 6. This Markov chainyields a closed form solution as follows. Let Pijk denote thesteady-state probability of state (i, j, k). These probabilitiesare obtained using the following recursive path:

P0,ki,ki�1 � C (7)

P0,ki�2,ki�2 �P0,ki�1,ki�2�u

i �0 � �CBi �0,ki � 1,ki � 2�

� Ii�0, ki � 2, ki � 2

(8)

P0,ki�1,ki�1 ��u

i �0C� I

i�0, ki � 1, ki � 1(9)

P0,ki�1,ki�2 �P0,ki�1,ki�1 � �u

i �0

�CBi �0, ki � 1, ki � 2

(10)

P100 �P110�CB

i �110

� Ii�100

(11)

Figure 4. Communication blocking decomposition.

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The equilibrium probabilities determined as above are usedto map the throughput rates of SS(i�1)i to the input arrivalrates at the virtual queue as follows. First, note that Lemma1 is still valid in this case. Now consider that case when nu

i �0, 1 � nI

i � ki and 1 � nCBi � ki�1. The values of nVQ

i in thiscase are in the discrete interval between 0 and ki � 3.Further, in this case, there is always a kanban and a CB-kanban available when a part arrives at the output buffer inSS(i�1)i. Consequently, the arriving part would immediatelyacquire a kanban from each set and move on to the virtualqueue. Hence, we have from the above,

�VQi �nVQ

i � � iu�0 , 0 � nVQ

i � ki � 3. (12)

Next, we consider the case nIi � nCB

i � 0, nui � 0 or 1. In this

case, nVQi � ki � 1. Clearly, there can be no arrival into the

virtual queue unless a part moves out of it. Hence,

�VQi �ki � 1 � 0. (13)

Finally, consider the case when nIi � 1. This case represents

three possible states in the Markov model of SS(i�1)i: (0, 1, 0),(1, 1, 0), and (0, 1, 1). Further, nVQ

i � ki � 2 in this case. Usingthe equilibrium probabilities, we estimate the arrival ratehere as follows:

�VQi �ki � 2 �

P110�CBi �110 � P011�u

i �0

P110 � P011 � P010(14)

Note that the state (0, 1, 0) does not contribute to the arrivalrate in the above equation since it requires a concurrentarrival of a part and a CB-kanban for an output to occur.

Now consider the reverse mapping case as in generalblocking. Again, Lemma 2 holds, and equations (4)–(6) holdin the reverse determination of �I

i(nIi), 0 � nI

i � ki. The arrivalof CB-kanbans to CBBi follows the service pattern at serveri. Consequently, if vRIQ

i (nRIQi ), nRIQ

i � 0, 1 are the throughputrates at server i,

�CBi �nCB

i � vi�1 , 0 � nCBi � ki � 1 (15)

�CBi �ki � 1 � 0 (16)

Putting all these together, the iterative algorithm for theanalysis of an intermediate subsystem under communica-tion blocking is as follows.

Algorithm: Communication Blocking SubsystemAnalysis (CBSA)

Step 0: {Initialization}Consider subsystem i. Fix parameters �u

i (nui ),

nui � 0, 1, �d

i (ndi ), nd

i � 0, . . . , ki�1 and �di (nd

i ),nd

i � 0, . . . , ki�1 � 1. Initialize �Ii(nI

i), nIi �

0, . . . , ki and �CBi (nCB

i ), nCBi � 0, . . . , ki � 1 to

some initial values.Step 1: {Solve SS(i�1)i}

Solve the Markov model of SS(i�1)i.Step 2: {Determine arrival rates at virtual queue:

forward mapping}Determine �VQ

i (nVQi ), nVQ

i � 0, . . . , ki � 1 usingthe forward mapping equations (12)–(14).

Step 3: {Solve the virtual CQN}Solve the virtual CQN problem as in thegeneral blocking case. Use the Markov modelfor the communication blockingsynchronization stations in the analysis ofSSi(i�1). Determine �VQ

i (nVQi ), nVQ

i � 0, 1, andvi(1).

Step 4: {Determine arrival rates at SS(i�1)i: reversemapping}Determine �I

i(nIi), nI

i � 0, . . . , ki from equations(4)–(6). Determine �CB

i (nCBi ), nCB

i � 0, . . . , ki � 1from equations (15)–(16).

Step 5: {Convergence test}Determine whether the recomputed �I

i and �CBi

values are within the chosen interval. If theyare, then stop; else, return to Step 1.

While the above algorithm is applied in the analysis ofsubsystems 1 � i � N � 1, the remaining subsystems areanalyzed as follows. Subsystem N � 1 is also analyzed usingalgorithm CBSA, except that the Markov model for synchro-

Figure 5. Modeling logic in communication and manufacturing blocking.

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nization stations given in Di Mascolo et al. (1996) is used inthe analysis of SS(N�1)N. This is because there are no CB-kanbans in subsystem N (see Figure 4). The subsystem 1 and

N are analyzed as in the previous models, since they requireno special designs for communication blocking.

2. Blocked Closed Queueing NetworksThe BCQN systems are analyzed using an encapsulation anddecomposition framework as presented in Figure 7. Considera BCQN system composed of N cells in tandem as shown inFigure 7(a). Let K denote the number of pallets in circulation.To begin, we introduce a synchronization station SS01 beforecell 1. The station SS01 consists of a pallet buffer area to holdthe returning pallets from the system, and a bulletin board forthe kanbans in cell 1. Since the supply of raw material issaturated, the returning kanbans at cell 1 can be assumed topick up the necessary raw material and wait at the bulletinboard. When a kanban at the bulletin board is matched witha pallet, they are immediately transferred to the input queueat call 1. As before, we let a1 � b1 � k1 and aN � kN, and anytype of blocking can be used in the line. The service cellsbetween the endpoints of this configuration are encapsu-lated and treated as a BOQN system as shown in Figure 7(b).Now, let nI

1 � 0, . . . , k1, np � 0, . . . , K and nN � 0, . . . , kN

denote the number of kanbans at SS01, number of pallets atSS01, and number of input parts awaiting service at cell N,respectively. Correspondingly, let �I

1(nI1), �p(np), and �N(nN)

denote their state-dependent arrival rates. Let vN(nN) denotethe throughput rate of cell N. We develop an analysis of theencapsulated and decomposed BCQN system as follows.

First, note that the system comprising of SS01, the encap-sulated BOQN within, and cell N is a nonblocking CQNwith the circulating pallets defining the loop. This followsfrom the nonblocking transfers out of SS01, the nonblockingdecompositions of the encapsulated system under any typeof blocking protocol, and the nonblocked transfers in andout of cell N. Consequently, the resulting system configura-tion can be analyzed as a Markov chain with the followingstate space: {�np, nI

1, nN � np � 0, . . . , K, nI1 � 0, . . . , k1, nN �

0, . . . , kN}. Further, note that not all possible combinations ofthe three variables yield feasible states. The constraints onthe possible combinations are as follows.

0 � np � K � k1 (17)

npn1I � 0 (18)

K � np� � k1 � nI1� � nN � �

i�2

N�1

ki (19)

Constraint (17) is straightforward, constraint (18) specifiesthat when np � 0, nI

1 � 0, and when nI1 � 0, np � 0. This

follows from the nonblocked transfer of a pallet and a kan-ban out of SS01 as soon as they are matched. Constraint (19)stipulates that the maximum number of pallets that could bein circulation among cells 2 through N � 1 at any timecannot exceed their cumulative maximum holding capacity.Figure 8 presents the maximum possible state space underthese constraints for the underlying Markov chain. Figure 9illustrates the state space for a specific example where N �3, k1 � k2 � k3 � 3, and K � 6.

Figure 6. Markov chain of SS(i�1)i under communicationblocking.

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The above Markov chain does not yield a convenientclosed form solution as in the BOQN systems, due to itsstructure. However, the transition-probability matrix isfairly sparse, and its solution in practical instances is quiteinexpensive. This is discussed in the following section. Afterobtaining the equilibrium probabilities from the Markovsolution, the arrival rates �I

1, �N, and the throughput rate vN,the arrival rates �p(np) are computed as follows.

�p�np ��nN P�np, nI

1, nN � vN�nN

�nN P�np, nI1, nN

, @np � 0 (20)

�p�0 ��nI

1 �nN P�0, nI1, nNvN�nN

�nI1 �nN P�0, nI

1, nN(21)

where P(np, nI1, nN) is the equilibrium probability of sate �np,

nI1, nN . Linking all these together, we develop an approxi-

mation framework for BCQN systems as follows.

Step 0: {Initialization}Initialize �p(np), np � 0, . . . , K � k1, to someinitial values.

Step 1: {Solve BOQN}

Incorporate SS01 within subsystem 1. Hence,�p(np) values provide the upstream rateparameters. Solve the BOQN problem for theline from cell 1 to cell N. Depending on thetype of blocking involved, use the appropriatealgorithm from the BOQN framework.Determine �I

1(nI1), nI

1 � 0, . . . , k1, and �N(nN),vN(nN), nN � 0, . . . , kN from this analysis.

Step 2: {Solve Markov Model}Solve the Markov model �np, nI

1, nN using the�I

1(nI1), �N(nN) parameters determined from

above. Determine �p(np), np � 0, . . . , K � k1

from equations (20), (21).Step 3: {Convergence test}

Determine whether the recomputed �p(np)values are within a of their previous values. Ifso, stop; else, return to Step 1.

3. Computational ResultsDetailed computational investigations on the proposed al-gorithms under the unified framework have been carriedout. The focus of these studies has been on the accuracy ofthe approximations and the computational effort involved.The algorithms have been evaluated using benchmark com-

Figure 7. Encapsulation and decomposition framework for BCQN systems.

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parisons with simulation results. The algorithms have beenprogrammed in Fortran and implemented on a SUN/UNIXserver with a 248 MHz SUNW, UltraSPARC-II CPU. Thesimulation models have been developed in the SIMAN sim-ulation language (Pegden et al. 1995) and implemented on a300 MHz IBM/PC. The algorithms on the UNIX server arefully portable to the PC environment. We present our resultsby organizing them into BOQN and BCQN studies as fol-lows.

3.1 BOQN StudiesThe experimental studies on BOQN systems are based on

a fully crossed statistical design involving the followingparameters and their associated levels:

(1) Blocking Protocol, BP: {Communication, Manufacturing,General}

(2) Number of cells, N: {5, 6, 7, 8, 9, 10}(3) Number of kanbans in each cell, k: {5, 8, 10, 12}(4) Number of output buffers in each cell 1 � i � N, under

general blocking, bi: {2, . . . , k � 1 � k}(5) Coefficient of variation in service time distribution,

CV2 � {0.5, 1, 2}

In total, the above design yielded 72 trials in communicationand manufacturing blocking each, and 504 trials in generalblocking. Simulations of the line under all these configura-tions have been carried out for the production of 50,000 partsafter a warmup of 5,000 initial parts. Further, the simulations

Figure 8. Maximum state space Markov model.

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yielded steady performance characteristics under this runlength and warmup conditions. The chosen warmup is 10%of the simulation run length. We also experimented withlarger warmup conditions and it did not make any signifi-cant change to these characteristics. The simulations re-quired between 10 minutes to more than an hour of CPUtime on the PC, depending on the line parameters. Com-pared to this, the average CPU time required by the approx-imation algorithms on the UNIX server has been 0.01 CPseconds, with a maximum of 0.04 CP seconds. All algo-rithms converged in the trials conducted with � 0.001. Thevalue of is the upper limit of �(old value � new value)/oldvalue� of arrival and departure rates at each stage at conver-gence. The chosen value � 0.001 is therefore extremelysmall and signifies stable convergence conditions. Any fur-ther reduction in may not alter the convergence valuessignificantly. Further, increasing may also not help as itmay affect the quality of the estimates without yielding anycomputational advantages, since the computational load ofthe algorithms with � 0.001 is already minimal. The agree-ments between the approximation and simulation results

have been measured in terms of two performance measures:

%WIP �

Total WIP� Approximation� Total WIP�Simulation

Total WIP�Simulation� 100

%THP �

Throughput� Approximation� Throughput�Simulation

Throughput�Simulation� 100

where WIP stands for work-in-process inventory. The re-sults of these studies are summarized for representativeconfigurations in Table I. Similar results have been obtainedin the rest of the trials as well.

Table I also includes the confidence intervals for %WIPand %THP given as percentages of the simulation averageWIP and throughput, respectively. The confidence intervalsare used to determine the statistical precision of the simula-tion results. The confidence interval represents the range ofvalues in which the true average will be included with aprobability of 95%. For example, for N � 5, k � 5, and b �

Figure 9. An illustration of the Markov state space where N � 3, K � 6, and k1 � k2 � k3 � 3.

312Vroblefski, Ramesh, and Zionts

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1, we are 95% sure that the true average WIP is within 0.35%of the average WIP found through simulation.

These results indicate that the throughput approximationon the average is within 1.5%, 2.3%, and 5.6% of that ofsimulation for general, manufacturing, and communicationblocking systems, respectively. Similarly, the WIP approxi-mation is on the average within 2.2%, 2.5%, and 8.1% of thatof simulation for these protocols. We extensively tested theapproximation schemes for serial systems that commonlyoccur in practice. In general, the approximation algorithmsperformed very well. The results shown in Table I are rep-resentative of all values of N, k, and CV2 tested. The devia-tion ranges are comparable when these parameters are var-ied. In all our experiments, the deviations that we observedfrom simulation results in general blocking systems are be-tween 0.24% and 3.92% for throughput and between 0.43%and 7.8% for WIP. Therefore, from a qualitative point ofview, the approximations are close to actual simulation re-sults within acceptable limits (roughly, �4% deviation inthroughput and �8% deviation in WIP) for general blockingsystems under the range of configuration parameters tested.

The deviations we observed from simulation results in com-munication blocking systems are between 1.54% and 4.34%for throughput and between 0.67% and 18.23% for WIP.Similarly, the deviations of the manufacturing blocking ap-proximation are between 0.64% and 4.22% for the through-put and between 1.03% and 8.97% for the WIP. The onlysituation where the deviations from the simulation resultsare sizable is when the number of output buffers is very low,as in communication blocking. These results are consistentwith those observed by Di Mascolo et al. (1996) with mini-mal blocking systems. The deviations are basically due tothe assumption of state-dependent Markov arrival processesin the decomposition framework, which may not be accuratewhen the number of kanbans as well as the output buffersare very small. We also experimented with unbalanced se-rial systems by randomly generating the mean service timesin the configurations tested using the same convergencecriteria. We obtained similar performance results with thealgorithms in these cases as well. From these analyses, weconclude that: (i) the approximations work well for generalblocking systems of any configuration for serial systems that

Table I. Open Queueing Network Results for CV2 � 1

N k b % WIPConfidenceInterval (%) % THP

ConfidenceInterval (%)

COMMUNICATION �3.33 1.11 �4.36 2.815 5 1 �2.20 0.35 �0.61 1.755 5 2 �1.07 1.17 0.87 2.775 5 3 �0.79 1.04 �3.15 2.55MINIMAL �2.71 0.63 �1.87 2.93COMMUNICATION �0.88 1.19 �1.32 2.275 10 2 �2.20 0.31 �0.35 2.015 10 5 �4.00 0.65 �1.67 1.655 10 8 �4.93 1.18 �1.72 2.33MINIMAL 1.09 0.49 0.54 2.41COMMUNICATION �9.10 0.52 �7.21 2.347 5 1 �3.21 0.86 �1.04 1.977 5 2 �1.36 0.41 0.82 3.187 5 3 0.43 1.03 �3.47 1.55MINIMAL �4.59 0.44 �2.16 1.91COMMUNICATION �4.95 0.60 �1.80 2.917 10 2 �0.42 1.05 �0.27 3.077 10 5 �7.65 1.08 0.38 1.977 10 8 �0.55 1.10 �0.91 1.84MINIMAL 3.31 1.19 0.60 2.47COMMUNICATION �17.84 0.42 �11.25 2.3410 5 1 �1.61 0.36 0.92 1.8110 5 2 2.36 0.51 0.77 1.8210 5 3 �3.17 1.07 �3.42 2.64MINIMAL �6.31 0.68 �2.30 2.18COMMUNICATION �11.51 0.34 �2.87 3.4210 10 2 �0.56 1.07 �0.70 2.5710 10 5 �5.29 1.23 �0.65 2.1010 10 8 �2.05 1.04 �1.22 2.50MINIMAL 0.34 0.65 �0.75 2.19

313General Open and Closed Queueing Networks with Blocking: A Unified Framework for Approximation

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commonly occur in practice, and (ii) the approximations forcommunication blocking systems are not as good as those ofgeneral blocking, but nevertheless give fairly good assess-ments, given the computational effort involved. In this anal-ysis, we have chosen to compare the performance of theproposed algorithms with results of simulations incorporat-ing actual system parameters. Hence the deviations in theapproximations from the simulation results are the real de-viations from actual system performance. Given the aboveperformance results, we also conclude that the approxima-tion framework yields reliable and accurate results at almostnegligible computational effort for practical systems of rea-sonable size. Although it will be illustrative to compare theproposed approach with other approximations in the liter-ature, we are not aware of any approach for approximatinggeneral blocking systems. Most of the literature on queueingsystem approximation is concerned with minimal and man-ufacturing blocking systems (see Govil and Fu 1999 for asurvey of the state of the art in queueing systems in manu-facturing). These blocking protocols really do not yield acomparable basis, as they are special cases of the generalblocking systems considered in this research.

3.2 BCQN StudiesThe same experimental design as in the BOQN systems isused in this case as well, except with the addition of thefollowing parameter: number of pallets (K). This parameteris varied at four levels in each instance of the other para-metrical combinations: {3k, 3(k � 1), 3(k � 2), 3(k � 3)}. Notethat N � k is an upper bound on K. The above levels yielddifferent flow rates within a line as determined by the num-ber of pallets.

The results of the BCQN studies are summarized forrepresentative configurations in Tables II, III, and IV. Similarresults have been obtained with the rest of the trials as well.The BCQN results are comparable to those of BOQN sys-tems. The throughput approximations are on the averagewithin 3.3%, 3.8%, and 5.9%, and the WIP approximationswithin 2.8%, 3.3%, and 3.9% of those simulations for general,manufacturing, and communication blocking systems, re-spectively. The average CPU time for a BCQN approxima-tion is about 1 CPU second on the UNIX server. BCQNsystems require a greater computational load than BOQNsystems, and this is mostly due to the matrix inversionrequired in solving the BCQN Markov model. Nevertheless,

Table II. Minimal Blocking Closed Queueing Network Results

N k Mu CV2 K%

THPConfidenceInterval (%) % WIP

ConfidenceInterval (%)

5 5 5 1 12 �4.34 4.51 �2.29 7.605 5 5 1 15 �2.57 3.24 0.45 8.725 5 5 1 18 �1.91 4.96 �0.30 7.105 5 5 1 20 �3.06 2.45 �3.03 7.635 5 5 1 22 �0.96 3.98 �0.09 6.635 5 5 1 24 �1.16 2.44 �2.06 5.765 5 5 1 25 �1.58 4.29 �2.25 8.49

10 10 5 1 50 �2.11 2.96 �0.53 5.8110 10 5 1 60 �1.76 3.07 �3.21 8.9610 10 5 1 70 �1.35 2.60 �2.07 7.4210 10 5 1 80 �0.72 2.89 �0.76 8.6010 10 5 1 90 �0.62 4.75 �3.48 5.11

Table III. Communication Blocking Closed Queueing Network Results

N k Mu CV2 K%

THP

ConfidenceInterval

(%)%

WIPConfidenceInterval (%)

5 5 5 1 6 �11.32 2.45 �4.69 6.685 5 5 1 8 �10.37 3.55 �6.47 7.135 5 5 1 10 �13.40 2.45 �5.28 6.415 5 5 1 12 �9.59 2.68 �3.81 6.815 5 5 1 14 �3.27 3.02 3.13 5.70

10 10 5 1 40 �3.35 3.24 �4.13 5.6710 10 5 1 45 �2.35 2.55 �3.42 8.3310 10 5 1 50 �1.82 4.82 �2.49 5.1510 10 5 1 55 �1.81 4.47 �5.81 5.5610 10 5 1 60 �1.51 4.75 �6.14 7.76

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the total CPU time requirement is very small, compared tothe simulation alternative. Given the quality of the approx-imations as demonstrated from the above results and theminimal CPU loads, the proposed framework is viable, ac-curate, and efficient in approximating BOQN as well asBCQN systems under any general blocking protocol.

4. ConclusionIn this paper, we developed a unified framework for ap-proximating open as well as closed queueing serial systemsunder a variety of blocking protocols. Such network systemspredominate in several application areas such as cellularmanufacturing, telecommunications, and distributed controlsystems. The intrinsic complexities in the exact analysis ofsuch systems have focused considerable attention in theliterature on both approximation methods and simulationtechniques. While much of the approximation literature isconcentrated on open queueing networks, research onclosed queueing systems is relatively restricted.

Di Mascolo et al. (1996) present an approximation algo-rithm for open tandem queues under the minimal-blockingkanban configuration. We extend and generalize their workto open queueing systems under any general blocking pro-tocol, and synthesize the framework to approximate closedqueueing systems under any blocking protocol as well. Us-ing a unified framework for approximation, we develop

specific decomposition strategies and approximation algo-rithms for a variety of system configurations and controlstructures. Our approach is founded on two principles:equivalence between tandem queues and kanban systemsunder any general blocking protocol, and the decompositionof a line into isolated subsystems with an iterative analysislinking the subsystems. All the approximation algorithmsdeveloped in this work are derived from these principles,and can be cast from the unified framework for generalsystem approximation.

The proposed algorithms have been extensively tested.Our results indicate that the proposed framework yieldsrobust, reliable, and accurate estimates of system character-istics such as throughput and WIP in a wide range of systemconfigurations. Furthermore, the computational effort in-volved is minimal. Together, these two performance mea-sures indicate the attractiveness of the proposed frameworkin the analysis of complex queueing systems, especiallycompared with the popular and widely used simulationalternative.

The proposed unified framework presents a set of highlyuseful tools of analysis to queueing-systems designers,whether they are kanban-based or not. A major concern inthe design of queueing control systems is the tradeoff be-tween throughput and WIP, and a designer may be requiredto evaluate several alternate design scenarios in terms of

Table IV. General Blocking Closed Queueing Network Results

N k b Mu CV2 K%

THPConfidenceInterval (%)

%WIP

ConfidenceInterval

(%)

5 5 1 5 1 15 �4.75 4.15 4.11 7.655 5 1 5 1 18 �2.57 3.16 �1.64 8.915 5 1 5 1 21 �2.38 4.13 �2.10 7.225 5 1 5 1 24 �1.92 4.75 �1.35 6.395 5 2 5 1 15 �6.43 4.26 0.65 7.845 5 2 5 1 18 �2.76 3.48 �1.11 5.795 5 2 5 1 21 �2.04 4.50 �1.16 6.105 5 2 5 1 24 �1.58 4.99 �2.76 6.995 5 3 5 1 15 �6.74 4.15 1.98 6.295 5 3 5 1 18 �3.96 4.59 1.41 5.905 5 3 5 1 21 �1.76 2.27 0.56 8.305 5 3 5 1 24 �0.97 4.61 0.04 5.85

10 10 2 5 1 50 �0.94 2.30 3.82 7.5110 10 2 5 1 60 0.16 2.85 1.32 6.2210 10 2 5 1 70 0.87 3.50 �1.37 6.5510 10 2 5 1 80 �0.70 2.71 �0.45 6.4010 10 5 5 1 50 �0.70 3.26 �2.49 8.3810 10 5 5 1 60 �0.68 3.94 �1.60 5.4210 10 5 5 1 70 �0.56 3.92 �2.04 8.4310 10 5 5 1 80 �1.04 3.68 �2.69 7.0310 10 8 5 1 50 �3.29 3.24 1.63 8.1010 10 8 5 1 60 �2.33 2.54 2.93 5.3710 10 8 5 1 70 �1.23 4.93 �2.33 7.3110 10 8 5 1 80 �0.48 4.68 �3.09 8.06

315General Open and Closed Queueing Networks with Blocking: A Unified Framework for Approximation

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these tradeoffs before arriving at a good design. While sim-ulation is mostly used for this purpose, it is often expensive.It therefore restricts the number of alternatives a designermay consider. Furthermore, recent developments in the gen-eral blocking area suggest that general blocking systemscould provide superior performance characteristics in termsof throughput and WIP over the traditional manufacturingand minimal blocking systems (Ramesh et al. 1997a, b).While general blocking is promising, it adds another signif-icant dimension to the design space by enabling the selectivecontrol of either the input or output buffers at a cell. Con-sequently, a designer is faced with an exponentially increas-ing array of design options, indicating the need for quickand accurate assessments of queueing system configura-tions. In this respect, the proposed framework addresses thisneed.

We see several possible avenues of future research. First,the analyses of tandem BOQN and BCQN systems devel-oped here can be generalized to any network queueingstructure. In this case, the models developed above can begeneralized using visit ratios and other routing algorithmsthat may be used in the various service nodes of a network.This generalization particularly applies to the X.25 andISDN (Integrated Services Digital Networks) packet switch-ing wide area networks in telecommunications (Stallings1997). Second, the above analyses can be extended to mul-tiple product flows in group-technology-based cellular man-ufacturing systems and priority queueing systems in a va-riety of service areas. Finally, the equivalence of tandemqueues and kanban systems can be exploited to analyze avariety of control options such as common buffer areas andtransfer lines, and arrival behaviors such as balking, reneg-ing, group arrivals, etc. In some cases, the Markov analysesmay be generalized as renewal processes. While these re-search extensions are theoretically attractive, they are alsopractically relevant and important. We are currently inves-tigating some of these ideas.

Acknowledgements

We wish to thank Maria DiMoscolo for several useful discussionsduring the initial stages of this work. We are also indebted to SidJagabandhu for his help with modeling some of the blocking sys-tems. Finally, we thank the two anonymous reviewers for their thecritical comments, which have greatly contributed to this paper.

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