Approximate Analysis of Tandem Blocking QueueingNetworks with Correlated Arrivals and Services

Guidance

Professor Masao FUKUSHIMAProfessor Tetsuya TAKINE

Kentaro OKAZAKI

2003 Graduate Course

in

Department of Applied Mathematics and Physics

Graduate School of Informatics

Kyoto University

KY

OTO

UNIVERSITY

FO

UN DED 1

897KYOTO JAPAN

February 2005

Abstract

Many real systems such as production lines are modeled as queueing networks with finitebuffers. In such a queueing model, blocking occurs if a customer attempts to enter the nextqueue whose buffer is full and not available. In this case, the customer is forced to waituntil the next queue can be entered. General queueing networks with blocking do not havea product-form solution. Therefore, it is very difficult to obtain the exact solution becauseof the explosion of state space. Accordingly, many approximation methods have beenproposed so far. Most of them are based on the node decomposition method, where thequeueing network is decomposed into several subsystems, and each subsystem is analyzedseparately. Also, most of approximation methods proposed in the past consider renewalarrivals and services in each node. However, they can be correlated in real systems.It is known that the correlations in arrivals and services have a great impact on theperformance of queueing networks.

In this thesis, we develop an approximate method for tandem queueing networks withfinite buffers and blocking, taking account of correlations in arrivals and services. Weassume that the arrival process is a two-state Markov Arrival Process (MAP), which canrepresent correlation in interarrival times. In addition, we apply MAP to the serviceprocess in each node. In our method, a tandem queueing network with n nodes is decom-posed into n−2 subsystems, each of which consists of three nodes in tandem. We developan efficient algorithmic procedure for analyzing each subsystem and propose an iterativeprocedure for computing the steady-state probabilities of the number of customers in eachnode. Through numerical experiments, we examine the accuracy of the approximationand confirm that our method well approximates the performance of tandem queueingnetworks with correlated arrivals and services.

Contents

1 Introduction 1

2 Mathematical model 22.1 Blocking mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Arrival and service processes . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Tandem configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Decomposition 5

4 Analysis of subsystems 64.1 The generator of the underlying Markov chain . . . . . . . . . . . . . . . . 64.2 Stationary distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.3 The statistics of interarrival and service times in the second node . . . . . 8

4.3.1 Service times in the second node . . . . . . . . . . . . . . . . . . . 84.3.2 Interarrival times in the second node . . . . . . . . . . . . . . . . . 10

4.4 Two moments and lag-1 autocorrelation approximation method . . . . . . 11

5 Algorithm 135.1 The first node has an infinite capacity . . . . . . . . . . . . . . . . . . . . 135.2 The first node has a finite capacity . . . . . . . . . . . . . . . . . . . . . . 14

6 Numerical experiments 15

7 Conclusion 20

A The generator of the underlying Markov chain of a subsystem 21

1 Introduction

Many real systems such as production lines are modeled as queueing networks with finitebuffers and blocking. A tandem queueing network with blocking is shown in Figure 1.External arrivals join the first node. Customers receive services successively from the firstnode to the last node. In such a queueing model, blocking occurs if a customer attemptsto enter the next queue whose buffer is full and not available. In this case, the customeris forced to wait until the next queue can be entered.

Figure 1: Tandem queueing network.

To design production lines, it is important to reduce the lead-time and in-process in-ventory. Small buffer reduces the in-process inventory, while it increases the lead-time.Hence, to design production lines, it is necessary to estimate these measures in advance.However, it is very difficult to obtain the exact solution of the queueing network becauseof the explosion of state space. Accordingly, many approximation methods have beenproposed [3, 5, 8]. Most of them are based on the node decomposition method, where thequeueing network is decomposed into several subsystems, and each subsystem is analyzedseparately. Upstream subsystems influence downstream subsystems, while downstreamsubsystems also influence upstream subsystems by blocking. Therefore, iterative algo-rithms are employed.

Most of approximation methods proposed in the past consider renewal arrivals and ser-vices in each node, and approximate them by the method of moment matching. It is knownthat the correlations in arrivals and services have a great impact on the performance ofqueueing networks. Recently some works propose approximation methods considering thecorrelation in the arrival process for queueing networks with infinite buffers or customerloss [6, 7, 11]. They report that the correlation in the arrival process makes considerabledifference in the average number of customers. It is conceivable that the same is truein the case there is blocking. In [1], Altiok and Melamed analyze the influence of thecorrelations on performance of the queueing networks. In addition, they give examplesof real manufacturing systems with correlated arrivals or services. They alert users topotential deleterious implications stemming from unfounded independence assumptions.

In this thesis, we develop an approximate method for tandem queueing networks withfinite buffers and blocking taking account of the correlations. The method estimates thedistributions of the queue length of the nodes in tandem queueing networks. The methodis based on the node decomposition. In our method, a tandem queueing network withN nodes is decomposed into N − 2 subsystems, each of which consists of three nodesin tandem. We focus our attention on the autocorrelation of interarrival times in eachnodes, as well as the mean and variance. This makes it possible to take account of theeffect caused by the correlations in arrivals and services. In our method, we assume thatthe arrival process is a two-state Markov Arrival Process (MAP), which can represent

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correlation in interarrival times. In addition, we apply MAP to the service process ineach node.

The rest of this thesis is organized as follows. In section 2, we explain the mathematicalmodel. In section 3, we describe the decomposition of the tandem queueing network. Insection 4, we give the method for analyzing each subsystem. We describe the iterativealgorithms in section 5. The result of numerical experiments is given in section 6. Finally,the conclusion is given in section 7.

2 Mathematical model

In this section, we introduce blocking mechanisms. After that, we present the arrival andservice processes used in this thesis. Finally, we describe the tandem queueing networkwith blocking.

2.1 Blocking mechanism

In a queueing network with blocking, a customer who attempts to go into the next queuewhose buffer is full has to wait. This phenomenon is referred to as blocking. Variousblocking mechanisms are propounded to model real systems [4]. We introduce three ofthe most commonly used blocking mechanisms.

Blocking after service (BAS): when a customer attempts to go into a downstreamnode, if the downstream node is full, the customer has to wait in the former node. Whilethe customer is waiting, the upstream node is blocked, so that other customers can notget service. As soon as the number of customers in the downstream node decreases, thewaiting customer moves, and the upstream node is unblocked.

Blocking before service (BBS): before a customer receives service in a node, the nextnode is checked. If the next node is full, the customer can not start receiving service andthe former node is blocked. As soon as a departure occurs in the next node, the formernode is unblocked. BBS is divided into two classes according to the behavior of waitingcustomers: waiting customers can be in the server (BBS-SO (server occupied)) and not(BBS-SNO (server not occupied)).

Repetitive service (RS): when a customer finishes the service, if the downstream nodeis full, the customer can not proceed and it receives service again.

In this thesis, we assume BAS. BAS is used to model production systems etc. Thetandem queueing network with BBS-SNO can be converted to the tandem queuing net-work with BAS. For example, we consider the tandem queueing network with BBS-SNOincluding three nodes whose capacities are m1, m2, and m3. The system is equivalent tothe tandem queueing network with BAS whose capacities are m1, m2 − 1, and m3 − 1.

2.2 Arrival and service processes

In this thesis, we assume that the arrival process is Markovian Arrival Process (MAP).MAP is an arrival process modulated by an underlying Markov chain with a finite statespace Ω = {1, . . . , M} [10]. We assume the underlying Markov chain is irreducible.

2

MAP is characterized by two M × M matrices C, and D. Diagonal elements of Care negative, and other elements are nonnegative. Elements of D are nonnegative. Thesojourn time in state i follows an exponential distribution with mean (−C ii)−1. Theunderlying Markov chain changes its state from i to j with rate Ci,j(i �= j) and nocustomer arrives. With rate Di,j, the Markov chain changes its state from i to j and acustomer arrives. Therefore, the elements of C and D satisfy the following equations.

M∑j=1j �=i

Ci,j +M∑

j=1

Di,j = −C i,i, for all i ∈ Ω.

Let MAP(C, D) denote MAP with representation (C, D).We explain about the statistical property of MAP(C , D) below. (C+D) is the genera-

tor of the underlying Markov chain. By the assumption that the underlying Markov chainis irreducible, the steady-state probability vector π is the unique nonnegative solution of

π (C + D) = 0, πe = 1,

where e denotes a column vector with an appropriate dimension, whose elements are allequal to one. Let λ denote the arrival rate of MAP(C, D):

λ = πDe.

We define πe as the steady-state probability vector of the underlying Markov chain im-mediately after arrivals. Thus we have

πe =πD

πDe.

Let X be the interarrival time in steady-state of MAP(C, D). The probability densityfunction of X is shown as

f(x) = πeexp(Cx)De.

The Laplace-Stieltjes transform is represented as

f(s) = E[e−sX ] = πe∫ ∞0

e−sxexp(Cx)dxDe = πe (sI − C)−1 De, Re(s) > 0.

Let (X1, X2, . . .) denote a sequence of interarrival times in steady state of MAP(C, D).The first and second moments of Xl are expressed as follows.

E[Xl] =πD

πDe(−C)−1 e = 1

πDe,

E[X2l ] = 2πD

πDe

((−C)−1

)2e =

2

πDeπ (−C)−1 e.

The state of the underlying Markov chain immediately before an arrival correlates withthat of immediately after the arrival. Therefore, there are correlations among interarrivaltimes. The joint density function of (X1, X2, . . . , Xn) is given by

g(x1, x2, . . . , xn) = πeexp(Cx1)Dexp(Cx2)D . . . exp(Cxn)De.

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Hence, the lag-k autocorrelation of Xl is given by

E[XlXl+k] =πD

πDe(−C)−1

((−C)−1 D

)k(−C)−1 e. (1)

It is known that MAP can approximate any stationary point process with arbitraryaccuracy [2]. The more states of the MAP, the more accurately the approximation is.However, the computational time in analysis grows with the number of states. Therefore,we assume that MAP has two states in this thesis.

In what follows, we describe the service process employed in this thesis. We applyMAP to the service process. Let Ωs = {1, . . . , Ms} be the state space of the serviceprocess. The service process is expressed by two Ms × Ms matrices Cs and Ds. Amongelements of Cs and Ds, the same relations are applied as in C and D of MAP(C, D).One customer receives the service at once. While there is no customer in the server, thestate of the service process never changes. If we observe the process only for periodsduring which there are some customers in the server, the service process is identical withMAP(Cs, Ds). We assume that the underlying Markov chain is irreducible. The servicetime of the lth customer is denoted by Yl. The sequence of service times (Y1, Y2, . . . , Yn)has the joint probability density function

g(y1, y2, . . . , yn) = πsexp(Csy1)Dsexp(Csy2)Ds . . . exp(Csxn)Dse,

where πs denotes the steady-state probability vector of the underlying Markov chainimmediately after services. This is similar to the joint probability density function of(X1, X2, . . . , Xn). Hence, the statistics of service times are expressed in the same way asthose of interarrival times of MAP(Cs, Ds).

We name this process Markovian Service Process (MSP). We denote by MSP(Cs, Ds)MSP with representation (Cs, Ds).

2.3 Tandem configuration

We consider open queueing networks consisting of N queues in series. We number thenodes from 1 to N . The first node has a finite or infinite capacity buffer, and othernodes have finite capacity buffers. Let mk denote the capacity of node-k. We assumethe blocking mechanism is BAS. The arrival process to the first node is MAP(Ca, Da)and the service process in node-k is MSP(Ck, Dk). The arrival rate of MAP(Ca, Da) isdenoted by λa. We define nk(t), and pk(t) as the number of customers and the state ofthe service process, respectively, in node-k at time t. In addition, nk(t) = mk + 1 meansthat there are mk customers in node-k and node-k is blocking the upstream node at timet. Then the process {(n1(t), . . . , nN(t), p1(t), . . . , pN(t)), t ≥ 0} is the continuous timeMarkov chain.

General tandem configurations do not have a closed-form solution. One way of an-alyzing such queueing networks is to solve the Markov chain and gain the stationaryprobability vector. However, it is not practical since the number of states in the Markovchain explosively increases with the number of nodes.

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3 Decomposition

In this section, we present our approximate approach to tandem queueing networks withblocking. The algorithm approximates the stationary distribution of the queue length ineach node.

We assume that the first node has an infinite capacity. Therefore, if the system isstable, all arrival customers can go into the system, and the throughput of the queueingnetwork is equal to λa.

２１ ４３

２１ ３

subsystem-1

subsystem-2

３２ ４

Figure 2: Decomposition to subsystems.

First, the tandem queueing network is decomposed into N −2 subsystems as in Figure2. Each subsystem consists of three nodes. Subsystem-1 includes node-1, node-2, andnode-3. Subsystem-2 includes node-2, node-3, and node-4, etc. Let mk,i(k = 1, 2, 3) bethe maximum capacities of the ith node in subsystem-k, and the capacities of these nodesare identical with those of the corresponding nodes in the original queueing network, thatis, mk,i = mk+i−1.

We define a two-state MAP(Ck,a, Dk,a) as the arrival process to subsystem-k andMSP(Ck,d, Dk,d) as the service process in the third node of subsystem-k. The serviceprocesses in the first and the second nodes are identical with those of the correspondingnodes in the original tandem queueing network. We define ak(t) as the state of the arrivalprocess at time t. Let pk,i(t) be the state of service process in the ith node of subsystem-kat time t. Also let nk,i(t) denote the number of customers in the ith node in subsystem-kat time t. In addition, let nk,i(t) = mk,i + 1(i = 2, 3) represent that the ith node insubsystem-k is full and the i − 1st node is blocked.

Our intent is that the behavior of nk,2(t) closely matches that of nk+1(t). For thatpurpose, MAP(Ck,a, Dk,a) emulates the arrival process to node-k in the original network,and MSP(Ck,d, Dk,d) emulates the service process of node-k + 2 including the effect ofblocking. These are unknown processes at first, except the arrival process to the firstnode and service process in the last node. Therefore, we take an iterative algorithm.

The first node of each subsystem has a finite buffer with the exception of subsystem-1. Therefore, there are customer losses in subsystems. However, in the actual queueing

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network, all customers go through the network without being lost. When the downstreamnode is full, if upstream server completes the service, blocking occurs. As soon as thedownstream server completes the service, the blocked customer enters the downstreamnode. To simulate this behavior we assume that when the first node is full, if the servicein the first node is finished, new arrival occurs with the probability q. The throughput ofsubsystem-k is a function of q. Therefore, let λk(q) denote the throughput of subsystem-k. We choose a suitable value of q so that λk(q) = λa. λk(q) is monotonically increasingwith q. If q = 0, λk(q) ≤ λa because of customer losses. q = 1 means that there arealways customers in the first node of subsystem. Hence, λk(q) ≥ λa unless the system isunstable. Therefore, if the original queueing network is stable, λk(q) = λa, 0 ≤ q ≤ 1 hasa solution.

Let S(t) = {nk,1(t), nk,3(t), ak(t), pk,1(t), pk,2(t), pk,3(t)}. The state of subsystem-k attime t is expressed as (nk,2(t), S(t)). The process {(nk,2(t), S(t)), t ≥ 0} is a continuoustime Markov chain. Let Qk denote the infinitesimal generator of the Markov chain. Thenwe can solve the balance equations,

πkQk = 0, πke = 0,

where πk denotes the steady-state probability vector of the Markov chain. In the nextsection, we describe the analytical method in details. The steady-state queue lengthdistribution of the second node in subsystem-k approximately represents that of node-k + 1 in the original queueing network. In addition, the arrival process and the serviceprocess in the second node correspond to the arrival process and the service process,respectively, in node-k + 1 in the original network.

Once the stationary state distribution of the Markov chain is given, we can evaluate thefirst two moments and the lag-1 autocorrelation of interarrival times and service timesin the second node of subsystem-k. Thus we can construct MAP(Ck+1,a, Dk+1,a) andMAP(Ck−1,d, Dk−1,d) to emulate these statistics. In the next section, we describe how toevaluate these statistics and to fit statistics of a MAP to those.

4 Analysis of subsystems

In this section, we obtain the stationary state probability vector of subsystem-k. Next, weobtain the statistics of the arrival and service processes in the second node of subsystem-k.Finally, we describe the method for constructing the arrival and service processes of theadjacent subsystems.

4.1 The generator of the underlying Markov chain

In subsystem-k, the underlying Markov chain can move from (i, S) to (i + 1, S ′) for0 ≤ i ≤ mk,2 and from (i, S) to (i − 1, S ′) for 1 ≤ i ≤ mk,2 + 1. Hence, the infinitesimal

6

generator of subsystem-k has the (m2+2)×(m2+2) block tridiagonal structure as follows.

Qk =

B1 B2B0 A1 A2 O

A0 A1. . .

. . .. . . A2

O A0 A1 H2H0 P 2

.

Each element of the (i, j)th block corresponds to the rate with which the state of theMarkov chain moves from (i, S) to (j, S ′), except the diagonal elements of Qk. Thestructure of each block is given in Appendix.

4.2 Stationary distribution

We analyze the underlying Markov chain to obtain the steady-state probability vector πk.For that purpose, we use the uniformization technique. We define θk and Qku as

θk = maxi

|(Qk)ii|,

and

Qku = I +1

θQk,

respectively. Note that Qku is a nonnegative matrix. In addition,

Qkue =(I +

1

θQk

)e = 1.

Therefore, Qku is considered as the transition matrix of a discrete time Markov chainY (t). Since πkQk = 0,

πkQku = πk

(I +

1

θQk

)= πk.

Therefore, πk is identical with the steady-state probability vector of the discrete timeMarkov chain Y (t). We then solve the following balance equations to obtain πk.

πkQku = πk, πke = 1. (2)

Qku has the same structure as Qk, that is, if (Qk)i,j = 0, then (Qku)i,j = 0. Qk is asparse matrix. There are at most nine nonzero elements in each low. Therefore, we canutilize the sparse structure to compute πk.

We use the Gauss-Seidel Iteration to solve (2). The method converges for ordinaryMarkov chains. In case that the method does not converge, we employ the power method.The power method converges for any Markov cahins that have steady-state probabilityvector. However, the convergence of the power method is slower than that of the Gauss-Seidel Iteration. In the algorithm, we assume that mk,2 ≥ 2.

7

Step 1. Initialization:

• Define π(0)k arbitrary, set n = 0.Step 2. n := n + 1.

• Solve

π(n)k,0B1u + π

(n−1)k,1 B0u = π

(n)k,0,

π(n)k,0B2u + π

(n)k,1A1u + π

(n−1)k,2 A0u = π

(n)k,1,

π(n)k,l−1A2u + π

(n)k,l A1u + π

(n−1)k,l+1 A0u = π

(n)k,l , l = 2, . . . , mk,2 − 1,

π(n)k,mk,2−1A2u + π

(n)k,mk,2

A1u + π(n−1)k,mk,2+1

H0u = π(n)k,mk,2

,

π(n)k,mk,2

H2u + π(n)k,mk,2+1

P 2u = π(n)k,mk,2+1

.

• Normalizeπ

(n)k =

π(n)k

π(n)k e

.

If ‖π(n)k − π(n)k Qku‖ ≤ �, then stop. Otherwise, goto step2.

4.3 The statistics of interarrival and service times in the secondnode

4.3.1 Service times in the second node

We consider the departure process from the second node under the condition that there aresome customers in the second node. Therefore, we construct the censored Markov chainunder the condition n2(t) ≥ 1 so that we can obtain MSP(C2s, D2s), which representsthe service process in the second node.

The generator of the censored Markov chain Qcens is given by

Qcens =

A1 + B0(−B1)−1B2 A2A0 A1 A2

A0. . .

. . .. . . A1 A2

A0 A1 H2H0 P 2

.

We extract the elements of Qcens corresponding to the rates for departure from the second

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node, and construct D2s. C2s is composed of other elements.

C2s =

A1 A2

A1. . .. . . A2

A1 H2P 2

, D2s =

B0(−B1)−1B2A0 O

A0. . .. . . O

A0 OH0 O

.

Let T+ denote the subset of the all states corresponding to the positive numbers of cus-tomers in the second node, and πk+ denote the sub-vector of πk over T+. The stationaryprobability vector πcens of the censored process is proportional to πk+ [9], i.e.,

πcens = (πk+e)−1 πk+.

Let (V1, V2, . . .) be a sequence of service times of MSP(C2s, D2s). The statistics of(V1, V2, . . .) are

E[Vl] =1

πcensD2se,

E[V 2l ] =2

πcensD2seπcens (−C2s)−1 e,

E[VlVl+1] =1

E[Vl]πcens(−C2s)−1D2s(−C2s)−1e.

The inversion of C2s is the most expensive operation. Since C2s is a block upper triangularmatrix, we can calculate it efficiently as follows.

We define x = (x1, . . . , xmk,2+1) so that

(x1, . . . , xmk,2+1) = πcens(−C2s)−1.Using the block matrices, (x1, . . . , xmk,2+1) can be written as

x1 = π0(−A1)−1, (3)xl = (xl−1A2 + πl)(−A1)−1, l = 2, . . . , mk,2, (4)

xmk,2+1 = (xmk,2H2 + πmk,2+1)(−P 2)−1. (5)We described the system of equations from (3) to (5) using the block elements of the

generator. However, we can write the equations by the uniformized generator.The block elements of the uniformized generator are derived as follows. For a diagonal

block A, uniformized block Au is given by

Au = I +1

θA.

Therefore,

(−A)−1 = 1θ

(I − Au)−1 .

9

For a non-diagonal block B, we define Bu as

Bu =1

θB.

We define u = (u1, . . . , umk,2+1) as θ−1x. Using the uniformized generator, we can

rewrite the system of equations from (3) to (5) as below,

u1 = πcens,0(I − A1u)−1,ul = (ul−1A2u + πcens,l)(I − A1u)−1, l = 2, . . . , mk,2,

umk,2+1 = (xmk,2H2u + πcens,mk,2+1)(I − P 2u)−1.

The inverse matrix in each equation has already been obtained in analysis of the stationarystate distribution.

We define y = (y1, . . . , ym2+1) as

y = πcens(−C2s)−1D2s = xD2s.

Consequently,

y1 = u1B0u(I − B1u)−1B2u + u2A0u,yl = ul+1A0u, l = 2, . . . , mk,2 − 1,

ymk,2 = umk,2+1H0,

ymk,2+1 = O.

Now we can compute πcens(−C2s)−1D2s(−C2s)−1 in a similar way to compute πcens(−C2s)−1replacing πcens by y.

4.3.2 Interarrival times in the second node

The statistics of interarrival times in the second node can be obtained as the precedingsubsection. The arrival process is MAP (C2a, D2a) described as follows.

C2a =

B1 O

B0 A1. . .

A0. . .

. . .. . .

. . . OA0 A1 O

H0 P 2

, D2a =

O B2O O A2

. . .. . .

. . .. . . O A2

O O H2O O

.

The elements of D2a correspond to the rates for service completions in the first node.Note that the elements of H2 do not correspond to the arrival to the second node in fact.

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We denote by (W1, W2, . . .) a sequence of interarrival times of MAP (C2a, D2a). Wethen have

E[Wl] =1

πkD2ae,

E[W 2l ] =2

πkD2aeπk (−C2a)−1 e,

E[WlWl+1] =1

E[Wl]πk(−C2a)−1D2a(−C2a)−1e.

We define v = (v0, . . . , vmk,2+1) as θ−1(−C2a)−1e. Since C2a is a block lower triangular

matrix, we can compute v as follows.

v0 = (I − B1u)−1e,v1 = (I − A1u)−1(B0uv0 + e),vl = (I − A1u)−1(A0uvl−1 + e), l = 2, . . . , mk,2,

vmk,2+1 = (I − P 2u)−1(H0uvmk,2 + e).

We define w as

w =

w0...

wmk,2+1

= D2a(−C2a)−1e.

Therefore, we have

w0 = B2uv1,

wl = A2uvl+1, l = 1, . . . , mk,2 − 1,wmk,2 = H2uvmk,2+1,

wmk,2+1 = 0.

Then, we can compute (−C2a)−1D2a(−C2a)−1e in a similar way to compute (−C2a)−1ereplacing e by w.

4.4 Two moments and lag-1 autocorrelation approximation method

It may be difficult to analyze a queueing model involving a MAP with a large number ofphases. Therefore, such a MAP is approximated by a MAP with two phases.

Given the first two moments e1, e2 and the lag-1 autocorrelation l1 of interarrival timesof the MAP with a large number of phases, we construct MAP(Cf , Df) with two phasesby an approximation method given below. We assume that

Cf =

[ −α δα0 −β

], Df =

[α − γ γ − δα

γ β − γ],

α, β > 0, min (α, β) > γ > δα, 0 ≤ δ ≤ 1.

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The steady-state probability vector of the underlying Markov chain is πf = [0.5, 0.5]. Let(Z1, Z2, . . .) denote a sequence of interarrival times of this MAP. The first two momentsand the lag-1 autocorrelation of (Z1, Z2, . . .) are obtained as follows.

E[Zl] =1

πfDfe=

2

(1 − δ)α + β ,

E[Z2l ] = 2πfDfπfDfe

((−Cf)−1

)2e =

2

αβ

(1 + δ)α + β

(1 − δ)α + β =(1 + δ)α + β

αβE[Zl],

E[ZlZl+1] =πfDfπfDfe

(−Cf )−1((−Cf)−1 Df

)(−Cf )−1 e

=(1 + δ) α + β

((1 − δ) α + β)αβ −γ

((α − β)2 − α2δ2

)((1 − δ) α + β)α2β2

=1

2E[Z2l ] − γE[Zl]

((α − β)2 − α2δ2

)α2β2

.

We assign values to parameters α, β, δ, and γ so that these statistics approximate e1, e2,and l1. First, we determine values of α, β, and δ satisfying the following equations.

2

(1 − δ)α + β = e1, (6)2

αβ

(1 + δ)α + β

(1 − δ)α + β = e2. (7)

After that, we obtain γ that minimizes

τ =(1 + δ) α + β

((1 − δ) α + β)αβ −γ

((α − β)2 − α2δ2

)((1 − δ) α + β) α2β2 − l1.

Ife2−e21

e21≥ 1, we assign zero to δ, and solve (6) and (7) for α and β. After that, we

determine γ that minimizes τ .However, if δ = 0, then we have

cv2 =E[Z2l ] − E[Zl]2

E[Zl]2≥ 1.

Therefore, δ must be greater than zero whene2−e21

e21< 1. In what follows, we consider the

case in which 0.5 <e2−e21

e21< 1.

We solve (6) for β, and we have

β =2

e1− (1 − δ) α. (8)

By replacing β by the right side of (8) in equation (7), this equation becomes

2δα + 2e1

2e1

α − α2 + δα2 e1 = e2.

12

We solve this equation for delta. We then have

δ =α2 − 2

e1α + 2

e2

α2 − 2e1e2

α=

(α − 1

e1

)2+

(2e2− 1

e21

)α

(α − 2e1

e2

) . (9)

This equation means that ife2−e21

e21< 1, then δ > 0. We obtain the minimum value of δ

such that equations (6) and (7) are satisfied. Differentiate (9) with respect to α, we have

dδ

dα=

2((e21 − e2)α2 + 2e1α − 2e

21

e2

)e1e2

(α2 − 2e1

e2α

)2 .

Therefore, the value of α that minimizes δ is obtained as follows.

α =e1 +

√2

e41e2

− e21e2 − e21

=1

cv2

1

e1+

√1 − cv2

e2

. (10)

We replace α by the right side of (10) in equation (6) and (7), and solve those equationsfor β and δ. After that, we obtain γ that minimizes τ .

5 Algorithm

We present iterative algorithms for performance evaluation of tandem queueing networkswith blocking. Let p

(n)k,j denote the approximate steady-state probability that the number

of customers in node-k is equal to j in the nth iteration. Also let ea1(k), ea2(k), and l

a1(k)

denote the first two moments and the lag-1 autocorrelation of the sequence of interarrivaltimes, respectively, in the second node of subsystem-k, and ed1(k), e

d2(k), and l

d1(k) denote

those of the sequence of service times, respectively, in the second node of subsystem-k.Our approximate method is described below.

5.1 The first node has an infinite capacity

Step 1. Initialization:

• (C1,a, D1,a) := (Ca, Da).• (Ck,s, Dk,s) := (Ck+2, Dk+2), k = 1, . . . , N − 2.• n := 1.

Step 2. For k = 1, . . . , N − 1:• Analyze subsystem-k.

13

• Derive ea1(k), ea2(k), and la1(k).• Find MAP(Ck+1,a, Dk+1,a) using the fitting procedure.

Step 3. For k = N, . . . , 2:

• Analyze subsystem-k.• Derive ed1(k), ed2(k), and ld1(k).• Find MAP(Ck−1,d, Dk−1,d) using the fitting procedure.

Step 4. Convergence test:

If

maxi,j

|p(n)i,j − p(n−1)i,j |p

(n)i,j

< �,

then stop, otherwise n := n + 1, and go to Step 2.

The stationary queue length distribution of node-k is approximated by that of the sec-ond node in subsystem-k−1 except node-1 and node-N . The stationary queue length dis-tributions of node-1 and node-N are approximated by that of the first node in subsystem-1and that of the third node in subsystem-N − 2, respectively.

We do not have a proof of convergence. In our experiences, the algorithm alwaysconverged within the reasonable number of iterations unless the original queueing networkis unstable. The criterion for determining convergence of the Gauss-Seidel Iteration hasto be two orders magnitude less than �. Let pk,j(x) denote the approximate steady-stateprobability that the number of customers in node-k is equal to j when � = x. In exampleswe examined, we have

maxi,j

|pi,j(10−5) − pi,j(10−6)|pi,j(10−5)

< 0.01.

Therefore, it seems reasonable to assume � = 10−5.

5.2 The first node has a finite capacity

We assumed that the first node of the tandem queueing network has an infinite buffer sofar. However, we can approximate the performance of tandem queueing networks whosefirst node has a finite buffer in the similar way. We denote by m1 the capacity of node-1,and other assumptions do not change. The approximate algorithm for such a queueingnetwork is as follows.

Step 1. Initialization:

• (C1,a, D1,a) := (Ca, Da).• (Ck,s, Dk,s) := (Ck+2, Dk+2), k = 1, . . . , N − 2.• n := 1.

14

Step 2. Derive the throughput of the network λ(n)a :

• Analyze subsystem-1.• Derive λ(n)a .• Derive ea1(1), ea2(1), and la1(1).• Find MAP(C2,a, D2,a) using the fitting procedure.

Step 3. For k = 2, . . . , N − 1:• Analyze subsystem-k.• Derive ea1(k), ea2(k), and la1(k).• Find MAP(Ck+1,a, Dk+1,a) using the fitting procedure.

Step 4. For k = N, . . . , 2:

• Analyze subsystem-k.• Derive ed1(k), ed2(k), and ld1(k).• Find MAP(Ck−1,d, Dk−1,d) using the fitting procedure.

Step 5. Convergence test:

If

maxi,j

|p(n)i,j − p(n−1)i,j |p

(n)i,j

< �,

then stop, otherwise n := n + 1, and go to Step 2.

6 Numerical experiments

In the following experiments, we use two-state MAP as the arrival process. The serviceprocess in each node is two-state MSP. The average queue lengths in nodes obtained by ourmethod and simulations are represented in the following figures. The simulation time wasgreater than 107 times the largest mean service time of the system. We simulated 30 times,and displayed 95% confidence interval in the figures. Also for comparison, we plotted theresults obtained by Algorithm-4 in [12], which does not consider the correlations in thearrival and service processes.

Example 1We start with an example with no correlated arrivals and services, where N = 10, and

mk = 4 for k = 2, . . . , N . The first node has an infinite capacity. The arrival process isMAP(Ca, Da):

Ca =

[ −0.02 0.000.00 −0.13

], Da =

[0.0026 0.01730.0173 0.1126

],

15

and the service process of node-k is MSP(Ck, Dk):

Ck =

[ −0.15 0.000.00 −0.45

], Dk =

[0.0375 0.11250.1125 0.3375

], k = 1, . . . , N.

We denote by (X1, X2, . . .) a sequence of interarrival or service times of the process. Thestatistics of the sequence of interarrival times and that of service times are shown in Table1.

Table 1: The statistics of the sequence of times.E[Xl] Var[Xl] Cov[Xl, Xl+1]

MAP(Ca, Da) 13.33 591.5 0.000MSP(Ck, Dk) 3.333 18.52 0.000

The results for Example 1 are shown in Figure 3.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1 2 3 4 5 6 7 8 9 10

aver

age

queu

e le

ngth

node

simulationproposed method

algorithm4

Figure 3: With no correlated arrivals and services.

Figure 3 shows that both the proposed method and Algorithm-4 work well.

Example 2We now consider a situation where the arrival process is correlated. Example 2 has

the following parameters: N = 10, and mk = 4 for k = 2, . . . , N . The first node has aninfinite capacity. In addition, the service processes in each node is identical with that ofExample 1. Example 2 differs from Example 1 in the arrival process. The arrival processis MAP(Ca, Da):

Ca =

[ −0.02 0.000.00 −0.13

], Da =

[0.01827 0.0017330.001733 0.1283

].

16

The statistics of the sequence of interarrival times are shown in Table 2.

Table 2: The statistics of the sequence of interarrival times.E[Xl] Var[Xl] Cov[Xl, Xl+1]

MAP(Ca, Da) 13.33 591.5 186.2

Note that the mean and variance of interarrival times of Example 2 are identical withthose of Example 1, while the covariance differs. Figure 4 shows the results obtained forExample 2.

0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4 5 6 7 8 9 10

aver

age

queu

e le

ngth

node

simulationproposed method

algorithm4

Figure 4: With correlated arrivals.

The data obtained by simulations in Figure 3 and Figure 4 demonstrates the effectof correlation in arrivals. The correlated arrival process raises the average queue lengthsof nodes. The proposed method reflects this effect, while Algorithm-4 obtains the sameresults as Example 1. However, it should be noted that the approximate average queuelength obtained by our method are lower than the results of simulations. We believe thatit is because of the correlation between arrivals and services in each node. In a node ofthe tandem queueing network with blocking, if many arrivals occur in a short time, theservice times tend to be long because of the blocking. In our method, we assumed thatthe service process of the third node in each subsystem is independent of arrivals.

Example 3Here, we examine the effect of the correlations in the service processes of nodes. In

Example 3, there are N = 10 servers. The first node has an infinite capacity, and thecapacity of node-k is mk = 4 for k = 2, . . . , N . The arrival process is identical with that

17

of Example 1, and the service process in node-k is MSP(Ck, Dk):

Ck =

[ −0.15 0.000.00 −0.45

], Dk =

[0.1388 0.011250.01125 0.4388

], k = 1, . . . , N.

The mean, variance and covariance of the sequence of service times in node-k are shownin Table 3.

Table 3: The statistics of the sequence of service times.E[Xl] Var[Xl] Cov[Xl, Xl+1]

MAP(Ck, Dk)(k = 1, . . . , N) 3.333 18.52 3.333

Note that the mean and variance of the sequence of service times in each node are identicalwith those of Example 1.

0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4 5 6 7 8 9 10

aver

age

queu

e le

ngth

node

simulationproposed method

algorithm4

Figure 5: With correlations in services of all nodes.

The results of simulations in Example 1 and Example 3 show that the correlationsin service processes have the effect on the expected queue lengths in nodes. The resultsobtained by the proposed method are very close to those obtained by simulations.

Example 4There are N = 10 nodes in the network. The first node has an infinite capacity, and

mk = 4 for k = 2, . . . , N . The moments of service times are identical among all nodes.However, only the service process in node-5 is correlated. The service processes are MSPsrepresented as follows.

C5 =

[ −0.15 0.000.00 −0.45

], D5 =

[0.1388 0.011250.01125 0.4388

],

18

Ck =

[ −0.15 0.000.00 −0.45

], Dk =

[0.0375 0.011250.01125 0.3375

], k = 1, . . . , 4, 6, . . . , N.

The arrival process is identical with that of Example 1. The statistics of service times areshown in Table 4. The average queue length in each node is shown in Figure 6.

Table 4: The statistics of the sequence of service times.E[Xl] Var[Xl] Cov[Xl, Xl+1]

MSP(C5, D5) 3.333 18.52 3.333MSP(Ck, Dk)(k = 1, . . . , 4, 6, . . . , N) 3.333 18.52 0.000

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1 2 3 4 5 6 7 8 9 10

aver

age

queu

e le

ngth

node

simulationproposed method

algorithm4

Figure 6: With a correlation in services of node-5.

Figure 6 shows that the correlation in the service process of node-5 raises the averagequeue lengths of node-5 and its upstream nodes. The proposed method reflects the effectof the correlation.

Example 5We have presented examples in which the capacity of the first node is infinite. Recall

that, we also developed the algorithm for tandem queueing networks whose first nodehas a finite capacity. In Example 5, the first node has a capacity of m1 = 4, and otherconditions are identical with those of Example 2. The results for Example 5 are shown inFigure 7.

19

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1 2 3 4 5 6 7 8 9 10

aver

age

queu

e le

ngth

node

simulationproposed method

algorithm4

Figure 7: The first node has a finite capacity.

Figure 7 suggests that our method for tandem queueing networks whose first node hasa finite capacity performs accurately.

7 Conclusion

We proposed an approximation method for performance evaluation of tandem blockingqueueing networks with correlated arrivals and services. The method is based on themethod of decomposition. The algorithm decomposes the tandem queueing network intoseveral subsystems and separately analyzes each subsystem. Also, we developed an effi-cient algorithmic procedure for analyzing each subsystem.

The results of simulation suggested that correlations in arrivals and services had agreat impact on the performance of tandem queueing networks with blocking. Throughnumerical experiments, we examined the accuracy of the approximation and confirmedthat our method well approximated the performance of tandem queueing networks withcorrelated arrivals and services.

Acknowledgement

First of all, I wish to thank Professor Tetsuya Takine for his invaluable guidance, sup-port and encouragement throughout my studies. I would like to thank Professor MasaoFukushima who has given me careful advice and guidance. I am also very grateful tomembers of Professor Fukushima’s laboratory for their camaraderie.

References

[1] Tayfur Altiok and Benfamin Melamed (2001) “The case for modeling correlation inmanufacturing systems,” IIE Transactions, 33, 779–791.

20

[2] Søren Asmussen and Ger Koole (1993) “Marked point processes as limits of Markovianarrival streams,” Journal of Applied Probability, 30, 365–372.

[3] Simonetta Balsamo (2000) “Closed queueing networks with finite capacity queues:approximate analysis,” Proceedings of the 14th European Simulation Multiconferenceon Simulation and Modelling: Enablers for a Better Quality of Life, 593–600.

[4] Simonetta Balsamo (2003) “A review on queueing network models with finite capacityqueues for software architectures performance prediction,” Performance Evaluation,51, 269–288.

[5] Alexandre Brandwajn and Yung-Li Lily Jow (1988) “An approximation method fortandem queues with blocking,” Operations Research, 36, 73–83.

[6] Muckai K. Girish and Jian-Qian Hu (2000) “Higher order approximation for the singleserver queue with splitting, merging and feedback,” European Journal of OperationalResearch, 124, 447–467.

[7] Armin Heindl (2003) “Decomposition of general tandem queueing networks withMMPP inputs and customer losses,” Performance Evaluation, 51, 117–136.

[8] Kyung P. Jun and Harry G. Perros (1990) “An approximate analysis of open tandemqueueing networks with blocking and general service times,” European Journal ofOperational Research, 46, 123–135.

[9] G. Latouche and V. Ramaswami (1999) “Introduction to Matrix Analytic Methods inStochastic Modeling,” American Statistical Association, Alexandria, Virginia.

[10] David M. Lucantoni, Kathleen S. Meier-Hellstern, and Marcel F. Neuts (1990) “Asingle-server queue with server vacations and a class of non-renewal arrival processes,”Advances in Aplied Probability, 22, 676–705.

[11] K. Mitchell and A. van de Liefvoort (2003) “Approximation models of feed-forwardG/G/1/N queueing networks with correlated arrivals,” Performance Evaluation, 51,137–152.

[12] Harry G. Perros (1994) “Queueing Networks With Blocking,” Oxford UniversityPress, New York, Oxford.

A The generator of the underlying Markov chain of

a subsystem

For the analysis of subsystem-k, we describe the generator of the underlying Markovchain. In what follows, we omit the subscript k. Let MAP(Ca, Da) denote the arrivalprocess to the subsystem, and MSP(C i, Di) denote the service process in the ith nodefor i = 1, 2, 3. We assume that when the first node is full, as soon as a departure occurs

21

in the first node, a customer arrives at the first node with probability q. The value of q ischosen such that the throughput of the subsystem is equal to that of the original network.Other parameters are as follows.

mi : The capacity of the ith node.ni : The number of customers in the ith node.a : The state of the arrival process.pi : The state of the service process in the ith node.

We denote by (n2, n1, n3, P ) = (n2, n1, n3, a, p1, p2, p3) the state of the subsystem. LetΞ denote the whole state set of (n2, n1, n3, P ), and let Φ denote the Cartesian product ofthe state sets of the arrival and service processes. Ξ is defined as follows.

1 ≤ n1 ≤ m1, 1 ≤ n2 ≤ m2 + 1, 0 ≤ n3 ≤ m3 + 1, P ∈ Φ,1 ≤ n1 ≤ m1, n2 = 0, 0 ≤ n3 ≤ m3, P ∈ Φ,n1 = 0, 1 ≤ n2 ≤ m2, 0 ≤ n3 ≤ m3 + 1, P ∈ Φ,n1 = 0, n2 = 0, 0 ≤ n3 ≤ m3, P ∈ Φ.

Note that (n2, n1, n3, S) is the continuous time Markov chain on Ξ. We denote by Qthe generator of the underlying Markov chain. Q is an (m2 + 2)× (m2 + 2) block matrixthat takes a form:

Q =

B1 B2B0 A1 A2 O

A0 A1. . .

. . .. . . A2

O A0 A1 H2H0 P 2

.

We describe the structure of each block element matrix below.A2 includes the rates corresponding to the increase of the number of customers in the

second node, that is, completions of services in the first node. A2 is an (m1 +1)×(m1 +1)block matrix given below.

A2 =

O O

A(1)0

. . .

. . .. . .

A(1)0 O

O (1 − q)A(1)0 qA(1)0

,

with

A(1)0 = In3 ⊗ Ia ⊗ D1 ⊗ Ip2 ⊗ Ip3,

where I · denotes an identity matrix with appropriate dimension, and ⊗ denotes a Kro-necker product [9].

22

A0 includes the rates corresponding to the decrease of the number of customers in thesecond node, that is, completions of services in the second node. A0 is an (m1+1)×(m1+1)block matrix that is given by

A0 =

A′0 O. . .

O A′0

,

where A′0 is an (m3 + 2) × (m3 + 2) block matrix.

A′0 =

O A(3)2 O

. . .. . .. . . A

(3)2

O O

O A(3)0 O

,

where

A(3)2 = Ia ⊗ Ip1 ⊗ D2 ⊗ Ip3,

A(3)0 = Ia ⊗ Ip1 ⊗ Ip2 ⊗ D3.

A(3)0 represents unblocking of the second node, that is, departures from the third node.A1 is an (m1 + 1) × (m1 + 1) block matrix that takes a form:

A1 =

B(1)1 A

(1)2

A(1)1 A

(1)2 O

. . .. . .

O A(1)1 A

(1)2

P(1)1

.

A(1)2 includes the rates corresponding to arrivals to the first node. We then have

A(1)2 = In3 ⊗ Da ⊗ Ip1 ⊗ Ip2 ⊗ Ip3.

A(1)1 is an (m3 + 2) × (m3 + 2) block matrix.

A(1)1 =

B(3)1

A(3)0 A

(3)1 O

. . .. . .

A(3)0 A

(3)1 A

(3)2

O O P(3)3

.

23

Each diagonal block in A(1)1 corresponds to transitions in states of the arrival process and

service processes. We then have

A(3)1 = Ca ⊗ Ip1 ⊗ Ip2 ⊗ Ip3 + Ia ⊗ C1 ⊗ Ip2 ⊗ Ip3

+Ia ⊗ Ip1 ⊗ C2 ⊗ Id + Ia ⊗ Ip1 ⊗ Ip2 ⊗ C3,P

(3)3 = A

(3)1 − Ia ⊗ Ip1 ⊗ C2 ⊗ Ip3,

B(3)1 = A

(3)1 − Ia ⊗ Ip1 ⊗ Ip2 ⊗ C3.

If n1 = m1 + 1, then arriving customers are lost. If n1 = 0, then the state of the serviceprocess in the first node does not change. Therefore,

P(1)1 = A

(1)1 + A

(1)2 ,

B(1)1 = A

(1)1 − In3 ⊗ Ia ⊗ C1 ⊗ Ip2 ⊗ Ip3 .

Note that if n1 = 0, then n2 can not be m2 + 1 and that if n2 = m2 + 1, then the firstnode is blocked. Therefore, P 2 is an m1 × m1 block matrix given by

P 2 =

A(1)1

′′A

(1)2

. . .. . .

A(1)1

′′A

(1)2

P(1)1

′′

,

where

A(1)1

′′= A

(1)1 − In3 ⊗ Ia ⊗ C1 ⊗ Ip2 ⊗ Ip3,

P(1)1

′′= P

(1)1 − In3 ⊗ Ia ⊗ C1 ⊗ Ip2 ⊗ Ip3.

If n1 = 0, then n2 can not be m2 +1. Therefore, H2 is an (m1 +1)×m1 block matrixand H0 is an m1 × (m1 + 1) block matrix:

H2 =

O · · · OA

(1)0 O

. . .

O A(1)0

, H0 =

A′0 O. . .

. . .

A′0 OO (1 − q)A′0 qA′0

.

B1 is an (m1 + 1) × (m1 + 1) block matrix given by

B1 =

B(1)1 A

(1)2

′O

A(1)1

′A

(1)2

′

. . .. . .

A(1)1

′A

(1)2

′

O P(1)1

′

.

24

If n2 = 0, then n3 can not be m3 + 1 and the state of the service process in the secondnode does not change. Therefore, we have

A(1)1

′=

B(3)1 O

A(3)0 A

(3)1

. . .. . .

O A(3)0 A

(3)1

− In3−1 ⊗ Ia ⊗ Ip1 ⊗ Ip2 ⊗ Id,

where In3−1 denotes an identity matrix with m3 + 1 dimension. Furthermore,

B(1)1

′= A

(1)1

′ − In3−1 ⊗ Ia ⊗ C1 ⊗ Ip2 ⊗ Id,P

(1)1

′= A

(1)1

′+ In3−1 ⊗ Da ⊗ Ip1 ⊗ Ip2 ⊗ Id.

B2 and B0 are similar to A2 and A0, respectively, and described as follows.

B2 =

O O

A(1)0

′ . . .. . .

. . .

A(1)0

′O

O (1 − q)A(1)0′

qA(1)0

′

,

where

A(1)0

′=

1 O 0. . .

...O 1 0

⊗ Ia ⊗ D1 ⊗ Ip2 ⊗ Ip3,

and

B0 =

A′′0 O. . .

O A′′0

, A′′0 =

O A(3)2 O

. . .. . .. . . A

(3)2

O

O A(3)0

.

25