Stationary performance evaluation measures in multi-dimensional Markov chains and applications in Queueing Theory

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University of Athens

Department of Mathematics

Section of Statistics and Operations Research

Stationary performance evaluation

measures in multi-dimensional Markov

chains and applications in Queueing Theory

Stella Kapodistria

0 1 2 · · · n − 2 n − 1 n !! "" !! "" (n j)pn−jqj×ν "" Athens 2009

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Queueing Theory

Stella Kapodistria

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g

To Antonis, Vaso, Espi and Apostolos

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encouragement and support. His ideas, his intuition and his unique way of treating mathematical problems has given me inspiration and guidance throughout my stud-ies.

To continue, I also take great pleasure in thanking professor Ivo Adan of Eind-hoven University of Technology for his support and contribution in our joint work covered in [3], which led to chapters 4 and 5 of the thesis. He worked with us with eagerness opening new horizons to the problem we were studying.

I would also like to thank the juries of the committee who have honored me with their participation and remarks. Moreover, I would like to thank the department of Mathematics, and particularly all professors of the section of Statistics and O.R. for the education they have provided me during my studies.

I wish to thank the State Scholarships Foundation (I.K.Y.) for the financial sup-port during my PhD studies.

I thank my family for their love and constant support. Finally, I owe a great debt of thanks to Alex, Panos, Stav and George for their friendship and for their companionship during my endless hours of studying in office 118.

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Introduction

1.1 Background and motivation

This dissertation deals with the modeling and analysis of certain queueing systems with a kind of synchronization and their applications. Queueing theory provides an efficient mathematical framework for the study of several congestion phenomena arising in diverse application areas such as telecommunications, production lines, etc. For the accurate description of a queueing system, we need to provide its following basic elements:

The input process. It refers to the arrivals to the system. It describes the distri-bution and dependencies of the interarrival times. The most common input process is the Poisson process.

The service mechanism. The basic characteristics of the service mechanism include the number of parallel servers, their identity (homogeneous or heterogeneous, their service speed etc.) and the distribution and dependencies of the service times.

The system capacity. It concerns the number of customers that can wait at any given time in a queueing system.

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cus-tomers for service. The most common queue disciplines are the “first-come, first-served” (FCFS), the “last-come, first-served” (LCFS), and the “service in random order” (SIRO). There are many other queueing disciplines which have been introduced for the efficient operation of computers and communication systems.

Also, there are other factors of customer behavior such as balking, reneging and jockeying, that should be exactly specified for the accurate mathematical description of a model. The shorthand notation of these elements facilitates the classification and reference to queueing systems with a variety of system characteristics. The basic classification-notation that is currently used in queueing theory was introduced by Kendall. According to Kendall’s notation, a queue is described by a sequence of five letter combinations - numbers A/B/s/c ( ): input process/service times/number of servers/capacity (discipline). For instance, M is used for exponential (memoryless-Markovian), D for constant (deterministic), Ekfor Erlang-k and G or GI for general

(independent) interarrival-service times in the positions A and B of Kendall’s nota-tion. For example,

M/G/1: Poisson arrival process, general service times, single server

Ek/M/1: Renewal arrival process with Erlang-k interarrival times, exponential

service times, single server

M/D/s: Poisson arrival process, constant service times, s servers. These symbolic representations are modified when other factors are involved.

The ultimate objective of the analysis of queueing systems is the understanding and quantification of their underlying processes. In the context of a queueing sys-tem, the most important process concerns the number of customers in system. If Q(t) denotes the number of customers in system at time t, t > 0, then the process {Q(t)} is a continuous-time stochastic process with discrete state-space {0, 1, 2, . . .}. This process is referred also as the queue length process.

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1.1 Background and motivation 3

Two other important processes from the viewpoint of customers are the sojourn time and waiting time processes. For a given customer we define his sojourn time to be the time from his arrival till his departure, while the waiting time is the time from his arrival till the beginning of service. The sojourn time and waiting time processes which record the corresponding times for the sequence of customers are discrete-time stochastic processes with continuous state-space [0, ∞]. Other important processes are related with the busy period of the system which is defined as the time from the arrival at an empty system till the next time that the system is empty again. Since time is an important factor, the analysis has to make a distinction between the time dependent, also known as transient, and the limiting behavior of a process of interest. Under certain conditions a stochastic process may settle down to what is commonly called steady state or state of equilibrium, in which its distribution properties are independent of time.

One of the first models that have been studied is the M/M/1 queue (Poisson arrival process, exponential service times, single server, infinite waiting room). It has been shown that under statistical equilibrium, the state balance equations are very simple and the limiting distribution of the queue length is obtained by recur-sive arguments. Introducing probability generating function techniques in several variants of the M/M/1 queue has been shown to be a very powerful method for studying the limiting behavior of the models. In general, the probability generating functions of the number of customers in system is such models satisfy certain linear algebraic equations that can be efficiently solved. An integrated theory has been developed for this class of models.

On the other hand, the application of generating function methods to Markovian models with infinite servers - variants of the M/M/∞ queue (Poisson arrival process, exponential service times, infinite servers) yields linear differential equations for the corresponding probability generating functions. This occurs because of the transition rates out of states n to the states n − 1, n ≥ 1, which are proportional to n. Models with transition rates dependent on the state of the system are referred to as state-inhomogeneous.

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In the queueing literature, there exists a significant number of papers dealing with state-inhomogeneous Markovian models, i.e. when there are transitions out of a state n to a state n" with rates proportional to n. This phenomenon usually appears due to features such as retrials, reneging or infinite number of servers. This is also the rule in stochastic models in Mathematical Biology, since every individual is associated with births and deaths.

For Markovian models, this type of state-inhomogeneous transition rates com-plicates the computation of the performance measures. For non-Markovian models, the basic idea is to use the methodology from the study of the M/G/∞ queue. In both cases, however, it seems fair to say that most of the models are analytically in-tractable. To overcome this difficulty a variety of methods has been developed. Gen-erating function techniques (see e.g. Gail et al. (2000), Grassmann(2002) and Mi-trani and Chakka (1995)) that have been proved very efficient for state-homogeneous models have been extended to deal with state-inhomogeneous models. Indeed, such systems can be solved in certain cases by applying results from the theory of hyper-geometric series (see e.g. Altman and Yechiali (2006), Artalejo and Gomez-Corral (1997), Baykal-Gursoy and Xiao (2004), Krishnamoorthy et al. (2005), Keilson and Servi (1993)), Perel and Yechiali (2009) and Yechiali (2007)).

In the present dissertation we aim to study certain state-inhomogeneous models that result from synchronized service or reneging. To clarify the basic idea, we will consider the following simple example. Consider a system with Poisson arrivals where each arriving customer sets his own clock and as soon as the clock expires the customer leaves the system. We assume that customers arrive according to a Poisson process at rate λ and their sojourn times are identical and independent random variables exponentially distributed with rate ν. This system is the standard M/M/∞ queue and its transition rate diagram can be seen in figure1.1.

Now, instead of the standard assumption that the customers have their indepen-dent clocks, let us consider the case where the customers are served simultaneously, i.e. there exist one clock for all of them. This is, for example, the case in remote

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1.1 Background and motivation 5 0 λ ##1 ν $$ λ ##2 2ν $$ λ ##3 3ν $$ λ ##· · · 4ν %%

Fig. 1.1: Transition rate diagram for the M/M/∞ queue

systems where customers decide to leave the system or not after the end of a service cycle, when a transport facility becomes available. We consider, again the model in which customers arrive according to a Poisson process at rate λ, but now we suppose that the departures can only occur at the epochs of a Poisson process at rate ν. At these departure opportunities, every present customer remains in the system with probability q, q ∈ (0, 1) or leaves the system with probability p = 1 − q, independently of the others. To distinguish this type of service we coin the term synchronized service. The similar idea can apply to systems with reneging. Instead of the standard assumption of abandonments due to independent reneging, we will study system with synchronized reneging. In general, this type of synchronization leads to rates from a state n to a state n" proportional to a binomial probability, as can be seen in figure 1.2.

0 λ ##1 (1 1)pν $$ λ ##2 (2 1)pqν $$ (2 2)p2ν && λ ##3 (3 1)pq2ν $$ (3 2)p2qν && (3 3)p3ν '' λ ##· · ·

Fig. 1.2: Transition rate diagram for a model with synchronized departures Therefore, our primary aim is to introduce and study synchronized actions in several queueing systems that imply binomial transition rates, from states n to all states n"_{, for 0 ≤ n}" _{≤ n. Similar Markov chains occur in Mathematical Biology in} the study of population processes subject to binomial catastrophes (see e.g. Artalejo et al. (2006), Brockwell et al. (1982), Economou (2004) and Economou and Fakinos (2008)). Moreover, Neuts (1994) studied a 1-dimensional discrete-time model with

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similar dynamics.

We will concentrate on 2-dimensional queueing systems with synchronized ser-vices and synchronized abandonments. We first study systems with synchronized services as a variation/extension of the model with infinite servers. However, the synchronization increases the complexity for obtaining the stationary distribution of the system as well as the other performance measures, such as the distribution of the sojourn time and the busy period analysis.

We then proceed to the modeling and analysis of queueing systems with syn-chronized reneging. The idea of customers independent reneging goes back to the pioneering work of Palm (1953, 1957) who was the first to study the M/M/c queue with exponential patience times. Another important researcher was Daley (1965), who did some important work on the GI/G/1 queue with independent reneging. Takacs (1974) considered the M/G/1 queue with a threshold waiting time, where the customers leave the system as soon as this threshold is exceeded. Later on, Boxma and de Waal (1994) studied the M/M/c queue with generally distributed patience times.

Recently, several authors studied the case of queueing systems where customers reneging is associated with the temporal absence of the server. We refer to mod-els in which the server becomes temporarily unavailable as server vacation modmod-els. Typically a vacation of the server starts as soon as the system remains empty after the end of a busy period. We refer to the time interval during which the server is unavailable as a vacation period. A vacation period is usually terminated by a condition that depends on the arrival process during that vacation period. For ex-ample, in some situations it is reasonable to assume that the server takes multiple vacations as long as the system remain empty, while in other situations it seems better to assume that the server takes only a single vacation and then stays in the system ready to serve, even when there are no waiting customers. In other sys-tems, the vacations happen because of random failures of the server (servers with on-off periods, unreliable servers, servers with failures and repairs). Models based

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1.1 Background and motivation 7

on the M/G/1 queue with vacations were studied by Altiok (1987), Cooper (1970), Fuhrmann (1984), Harris and Marshal (1988), Levy and Yechiali (1975), Shaked and Shanthikumar (1986), Yadin and Naor (1963), and many others. If we add the feature of independent reneging to a vacation model, we have a model that is referred to as a server vacation model with independent reneging. These models reflect human behavior in certain real life situations and were initially introduced by Altman and Yechiali (2006). In this thesis, we aim to extend their framework by introducing and analyzing server vacation models with synchronized abandonments. Another source for customer impatience is the possible failures of the system. Cases in which the service of the customers is interrupted by some random mech-anism were initially studied for single server systems, (see e.g. Gaver (1962) and Keilson (1962)) and then extended to the multiple servers case (see e.g. Mitrany and Avi-Itzhab (1968)).

Introducing the effect of synchronization and using generating function tech-niques to obtain the probability generating function of the number of customers in the system usually leads to systems that can be solved by applying the theory of q–hypergeometric series, known also as basic hypergeometric series (see Gasper and Rahman (2004)). However, in the queueing theory literature, there exists only few papers where this theory has been applied (see e.g. Ismail (1985), Kemp (1990, 1992, 1998, 2005)).

We will show that models with synchronized actions represent more realistically some service systems and that the framework of q–hypergeometric series enables us to express in closed form the main performance measures of such a system. In gen-eral, the theory of q–hypergeometric series can facilitate the computations regarding systems with this kind of binomial transitions arising from synchronization.

There exists a rich theory for the class of q–hypergeometric series and their q– calculus which enables fast calculations and simplifications. For this reason and for the sake of self-completeness we will briefly summarize the basic definitions and

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results of this theory in the next section. The interested reader can find more details on the definitions and the results below (with proofs and extensions) in Gasper and Rahman (2004), Chapters 1-3 and Appendices I-III.

1.2 Basic Hypergeometric series

Basic hypergeometric series are series of the form !∞

n=0un where u0 = 1 and u_un+1_n

a rational function of qn for a deformation parameter q, which is usually taken to satisfy |q| < 1. They were initially introduced by Heine who developed their basic theory, following Gauss’ fundamental paper on hypergeometric series.

Observing that the ratio un+1/un, being rational in qn, can be written in the form

un+1 un = (1 − a1q n_{)(1 − a} 2qn) · · · (1 − arqn) (1 − qn+1_{)(1 − b} 1qn) · · · (1 − bsqn)(−q n₎1+s−r_z, _(1.1)

we have that every such series assumes the form

rφs " a1, a2, . . . , ar b1, . . . , bs ; q, z # ≡ rφs(a1, a2, . . . , ar; b1, b2, . . . , bs; q; z) = ∞ $ n=0 (a1; q)n(a2; q)n· · · (ar; q)n (q; q)n(b1; q)n· · · (bs; q)n % (−1)nq(n2) &1+s−r zn, (1.2) where (a; q)n is referred as the q–shifted factorial and is given as

(a; q)n=    1, n = 0 (1 − a)(1 − aq)(1 − aq2_{) · · · (1 − aq}n−1_), _{n = 1, 2, . . . .}

It is assumed in the definition of the rφs series, equation (1.2), that bi (= q−m for

m = 0, 1, . . . and for every i = 1, 2, . . . , s. Using the ratio test, for 0 < |q| < 1, we can easily see that the rφs series converges for all z if r ≤ s and for |z| < 1 if

r = s + 1. In what follows it is assumed that 0 < q < 1. We also define (a; q)∞= ∞ * i=0 (1 − aqi).

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1.2 Basic Hypergeometric series 9

The following easily derived identities will be frequently used in this manuscript:

(a; q)n = (a; q)∞ (aqn_{; q)} ∞ (1.3) (a−1q1−n; q)n = (a; q)n(−a−1)nq−( n 2). (1.4)

Since products of q–shifted factorials occur so often, to simplify them we shall fre-quently use the more compact notations

(a1, a2, . . . , am; q)n = (a1; q)n(a2; q)n· · · (am; q)n

(a1, a2, . . . , am; q)∞ = (a1; q)∞(a2; q)∞· · · (am; q)∞.

1.2.1 The q–binomial theorem

A q–calculus has been developed that parallels the theory of hypergeometric func-tions. The most important summation formula for the q–hypergeometric series is given by the q–binomial theorem

1φ0(a; −; q; z) = ∞ $ n=0 (a; q)n (q; q)n zn= (az; q)∞ (z; q)∞ , _{|z| < 1.} (1.5)

Of particular importance are the two q–analogues of the exponential function ez_.

They are the functions eq(z) = _(z;q)1_∞ and Eq(z) = (−z; q)∞. The q–binomial

theorem enables to express the q–exponential functions in the form of q–series. If we set a = 0 in (1.5) we get eq(z) = 1φ0(0; −; q; z) = ∞ $ n=0 zn (q; q)n = 1 (z; q)∞ , _{|z| < 1.} (1.6)

If we replace z by −az in (1.5) and then let a → ∞ to get

Eq(z) = 0φ0(−; −; q; z) = ∞ $ n=0 q(n2) (q; q)n zn_{= (−z; q)}∞. (1.7)

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We observe that, as q → 1− _{the q–analogues reduce to their standard counterparts.}

In particular we have the relationships lim q→1−eq(z(1 − q)) = e z _(1.8) lim q→1−Eq(z(1 − q)) = e z _(1.9) lim q→1− (qaz; q)∞ (z; q)∞ = (1 − z) −a_. _(1.10)

1.2.2 Transformation formulas for 2φ1 series

Heine (1847,1878) showed that

2φ1(a, b; c; q, z) = (b, az; q)∞ (c, z; q)∞ 2 φ1(c/b, z; az; q, b) (1.11) = (c/b, bz; q)∞ (c, z; q)∞ 2 φ1(abz/c, b; bz; q, c/b) (1.12) = (abz/c; q)∞ (z; q)∞ 2 φ1(c/a, c/b; c; q, abz/c). (1.13)

Jackson (1910) proved that

2φ1(a, b; c; q, z) =

(az; q)∞

(z; q)∞ 2

φ2(a, c/b; c, az; q, bz). (1.14)

1.2.3 The q–integral

Finally Thomae (1869,1870) and Jackson (1910,1951) introduced the q–integral + 1 0 f (t)dqt = (1 − q) ∞ $ n=0 f (qn)qn (1.15)

and Jackson gave the most general definition + b a f (t)dqt = + b o f (t)dqt − + a 0 f (t)dqt, (1.16) where + a 0 f (t)dqt = a(1 − q) ∞ $ n=0 f (aqn)qn. (1.17)

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1.3 Overview of the thesis 11

If f is continuous on [0, a], then it is easily seen that lim q→1− + a 0 f (t)dqt = + a 0 f (t)dt. (1.18)

We will use the q–integral convergence to the ordinary integral to prove limiting results in the case when q → 1−, since a r+1φr series can be expressed as a q–

integral through the relation

r+1φr " a1, . . . , ar+1 b1, . . . , br ; q, qz # = (a1, . . . , ar+1; q)∞ (1 − q)(q, b1, . . . , br; q)∞ × + 1 0 tz−1(qt, b1t, . . . , brt; q)∞ (a1t, . . . , ar+1t; q)∞ dqt, (1.19)

when Re z > 0, and the series on the left side does not terminate.

1.2.4 _{Limiting regimes as q → 0}+

As q → 0+ we can easily see that lim q→0+eq(z) = 1 1 − z (1.20) lim q→0+Eq(z) = 1 + z. (1.21)

We can also prove the limiting relation lim q→0+ r+1φr " a1, · · · , ar+1 b1, · · · , br ; q, q # = 1. (1.22)

1.3 Overview of the thesis

This thesis is organized according to the specific models that we study. More con-cretely, in every chapter we first motivate the model under consideration and then describe the previous contributions reported in the literature. Subsequently, we provide a mathematical description of the model and begin to study several perfor-mance measures such as the stationary distribution of the system state, the sojourn time and the busy period distributions. Several mean performance measures are

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also considered and studied. We also report results concerning the behavior of the model under several limiting regimes.

In chapter 2, we consider a single server Markovian queue with synchronized services and setup times. Customers arrive according to a Poisson process and are served simultaneously. At a service completion epoch, every customer remains sat-isfied with probability p (independently of the others) and departs from the system; otherwise he stays for a new service. Moreover, the server takes multiple vacations whenever the system is empty. To study the model we introduce a 2-dimensional Markov chain and observe that the transition rates of the underlying Markov chain are state-inhomogeneous, because the number of customers n is reduced according to a binomial (n, p) distribution at each service completion epoch. We show that the model can be efficiently studied using the framework of q-hypergeometric series and we carry out an extensive analysis including the stationary, the busy period and the sojourn time distributions. Exact formulas and numerical results show the effect of the level of synchronization to the performance of the system. More concretely, chapter 2 is organized as follows: In section 2.2 we describe the model and introduce the appropriate notation. In section 2.3 we carry out the equilibrium analysis of the system state with the use of partial probability generating functions and we derive several exact formulas and iterative algorithmic schemes. Some limiting regimes are discussed in more detail. The busy period and sojourn time distributions are studied in section 2.4. Finally, in section 2.5, we provide several numerical studies and we discuss the effect of the level of synchronization on the system performance.

In chapter 3 we consider a single server unreliable queue represented by a 2-dimensional continuous time Markov chain. Customers arrive according to a Poisson process and find the server in one of two modes: on or off. The server is subject to failures that remove all present customers from the system. Moreover, as long as the server is down, we assume that the arrivals continue to come, but the cus-tomers become impatient and perform synchronized abandonments. This model was motivated by remote systems where customers have to wait for a certain transport facility to abandon the system. Then, whenever the facility visits the system, the

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1.3 Overview of the thesis 13

present customers can decide whether to leave the system or not. More specifically, we assume that the abandonment opportunities occur according to a Poisson pro-cess, whenever the server is down (so we can think that the transportation facility’s arrivals occur according to this process). Then, at an abandonment opportunity epoch, every customer decides to abandon the system with probability p or remains in the system waiting for service with probability q = 1 − p, independently of the others. In section 3.2 we describe the model and introduce the appropriate notation. In section 3.3 we carry out the equilibrium analysis of the system state and derive several exact formulas and iterative algorithmic schemes. Some limiting regimes with a particular interest are discussed in more detail. In section 3.4 we study the conditional mean sojourn times of a customer, while in section 3.5 we treat the sys-tem busy period distribution of the model.

In chapters 4 and 5 we present a detailed analysis of two fundamental queueing models with vacations and impatient customers, where the source of the impatience is the absence of the server. Instead of the standard assumption that customers perform independent abandonments, we consider situations where customers aban-don the system simultaneously. In chapter 4 we study the Markovian case, while in chapter 5 we study the non-Markovian counterparts. In both chapters we study two models. The first model is the single-server queue with multiple vacations, where customers decide whether to abandon the system or not when the vacation periods finish. In the second model, we suppose that the abandonments opportunities oc-cur according to a Poisson process during vacation periods. At the abandonment opportunities, every present customer remains in the system with probability q or abandons the system with probability p = 1 − q, independently of the others. Chap-ter 4 is organized as follows: In section 4.2, we describe the dynamics of the models. In section 4.3 we carry out a mean value analysis of the two Markovian models. In section 4.4 we present, separately, the stationary analysis of the two models. We also obtain more explicit results under various limiting regimes concerning the pa-rameters of the models. Along the same lines, chapter 5 is organized as follows: In section 5.2, we describe the dynamics of the models. In section 5.3 we carry out a mean value analysis of the two models, while in section 5.4 we study their

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station-ary distributions by using a generating function approach. In section 5.5 we present several numerical results that illustrate the effect of the various parameters on the performance measures of the model.

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Chapter

2

Synchronized services in a

single server vacation queue

We consider a single server Markovian queue with synchronized services and setup times. Customers arrive according to a Poisson process and are served simultane-ously. At a service completion epoch, every customer remains satisfied with prob-ability p (independently of the others) and departs from the system; otherwise he stays for a new service. Moreover, the server takes multiple vacations whenever the system is empty.

The transition rates of the underlying 2-dimensional Markov chain are state - in-homogeneous, because the number of customers n is reduced according to a binomial (n, p) distribution at each service completion epoch. We show that the model can be efficiently studied using the framework of q-hypergeometric series and we carry out an extensive analysis including the stationary, the busy period and the sojourn time distributions. Exact formulas and numerical results show the effect of the level of synchronization to the performance of such systems.

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2.1 Introduction

In the queueing literature, there exists a significant number of papers dealing with 2-dimensional Markovian models. In this kind of queueing models, the state of the system is represented by a vector (n, i), where n records the number of customers, while i gives information about some special feature of the system (e.g. state of the server(s), number of priority customers, phase of the arrival or the service process etc.). These models have received considerable attention, because on the one hand they can easily represent various queueing characteristics and on the other hand many of them are analytically tractable, thus providing effective computations of performance measures.

However, most of the analytically tractable models have two limitations: First, only one of the two variables is unlimited (usually the number of customers), while the other is assumed to take only a finite number of values. Second, the underlying Markov chain exhibits -at least to a certain degree- spatial homogeneity. A variety of methods has been developed for such models. Matrix analytic methods (see e.g. Bini et al. (2005), Latouche and Ramaswami (1999) and Neuts (1981, 1989)) and generating function techniques (see e.g. Gail et al. (2000), Grassmann(2002) and Mitrani and Chakka (1995)) have been proved very efficient.

In systems that do not satisfy the above constraints the analysis is much more complicate. When only the first constraint is violated, that is both variables are unlimited but the Markov chain is space-homogeneous, the matrix analytic methods still apply, although there exist serious difficulties (see e.g. Kroese et al. (2004)). An alternative is to use analytic tools. One widely used approach is the reduction to a Riemann-Hilbert boundary value problem (see e.g. Cohen and Boxma (1983)). The compensation method (Adan(1991)) has also been used for the efficient solution of a number of models.

The second constraint, that is the spatial homogeneity, is usually violated due to features such as retrials, reneging or infinite number of servers. Indeed, in these

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2.1 Introduction 17

situations, there are transitions out of a state (n, i) to a state (n − 1, i"_{) with rates}

proportional to n. This is also the rule in stochastic models in Mathematical Bi-ology, since every individual is associated with births and deaths. There are only few works trying to extend matrix analytic methods within this framework. In most cases the authors use truncation or generalized truncation ideas to study the sys-tems (see e.g. Artalejo and Pozo (2002) and the references therein) or they apply generating function methods. However, we have to note that in this case the partial generating functions of the number of individuals in system satisfy a system of linear differential equations in contrast with the state homogeneous case where they satisfy linear algebraic equations. Such systems are usually intractable or can be solved in terms of hypergeometric series (see e.g. Altman and Yechiali (2006), Artalejo and Gomez-Corral (1997), Baykal-Gursoy and Xiao (2004), Keilson and Servi (1993) and Krishnamoorthy et al. (2005) ).

The primary aim of this chapter is the study of a synchronization characteristic that also leads to spatially inhomogeneous Markov chains. Indeed, suppose that all the n present customers of a given system are served concurrently, according to an exponential distribution with parameter µ. If at the service completion epoch every customer is satisfied and departs with probability p or repeats his service with probability q, independently of the others, then there are binomial transition rates of the form µ,_n

n#-pn−n #

qn#, from a state (n, i) to states (n"_{, i), for 0 ≤ n}" _{≤ n. Similar} Markov chains occur in Mathematical Biology in the study of population processes subject to binomial catastrophes (see e.g. Artalejo et al. (2006), Brockwell et al. (1982), Economou (2004) and Economou and Fakinos (2009) ). Moreover, Neuts (1994) studied a 1-dimensional discrete-time model with similar dynamics.

The model of this chapter is a queueing system with synchronized services and setup times (also known as multiple server’s vacations). The literature on queueing systems with vacations is vast (see e.g. Takagi (1991), Tian and Zhang (2006) and the references therein). However, to the best of our knowledge, there are no papers dealing with vacation queueing systems with some kind of synchronization. A po-tential application of this type of models is the representation of distance-learning

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service systems (e.g. webseminars), where all customers are served concurrently and then decide independently whether to repeat their service or not.

The chapter is organized as follows. In section 2.2 we describe the model and introduce the appropriate notation. In section 2.3 we carry out the equilibrium analysis of the system state and we derive several exact formulas and iterative al-gorithmic schemes. Some limiting regimes are discussed in more detail. The busy period and sojourn time distributions are studied in section 2.4. Finally, in section 2.5, we provide several numerical studies and we discuss the effect of the level of synchronization to the system performance.

2.2 Model description and notation

We consider a queueing system in which customers arrive according to a Poisson process at rate λ. The service is provided by a single server, who serves simultane-ously all present customers. The successive service times of the server are assumed to be exponential random variables with rate µ. At a service completion epoch, each customer is satisfied and departs with probability p or repeats his service with probability q = 1 − p, independently of the others. Whenever the system becomes empty, the server is deactivated immediately. As soon as an arrival enters to an empty system, the server begins a setup time to reactivate. The setup times are exponentially distributed at rate ϑ. An alternative interpretation is that the server takes exponentially distributed multiple vacations with parameter ϑ as long as the system remains empty.

The system is represented by a continuous-time Markov chain {(N(t), I(t)) : t ≥ 0}, where N(t) is the number of customers in the system at time t and I(t) denotes the state of the server at time t (0=off and 1=on), t ≥ 0. The corresponding transition diagram is given in Figure 2.1.

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2.3 The equilibrium state distribution 19 1, 1 (1 1)pµ (( λ ##2, 1 (2 1)pqµ )) (2 2)p2µ ** λ ##3, 1 (3 1)pq2µ )) (3 2)p2qµ ++ (3 3)p3µ ,, λ ##· · · 0, 0 λ ##1, 0 ϑ --λ ##2, 0 ϑ --λ ##3, 0 ϑ --λ ##· · ·

Fig. 2.1: Transition rate diagram of {(N(t), I(t))}.

We also define the partial probability generating functions Π0(z) and Π1(z) by

Π0(z) = ∞ $ n=0 π(n, 0)zn and Π1(z) = ∞ $ n=1 π(n, 1)zn .

In section 2.3 we will determine Π0(z) and Π1(z) in terms of q-hypergeometric

series. Moreover, in section 2.4 we will see that the Laplace-Stieltjes transform of the busy period of this system is also expressed in terms of q-hypergeometric series.

2.3 The equilibrium state distribution

The balance equations of the model are given as follows: λπ(0, 0) = µ ∞ $ n=1 pnπ(n, 1) = µΠ1(p) (λ + ϑ)π(n, 0) = λπ(n − 1, 0), n ≥ 1 (2.1) (λ + µ)π(1, 1) = ϑπ(1, 0) + µ ∞ $ j=1 .j 1 / pj−1qπ(j, 1) (2.2) (λ + µ)π(n, 1) = ϑπ(n, 0) + λπ(n − 1, 1) + +µ ∞ $ j=n . j n / pj−nqn_{π(j, 1), n ≥ 2.} (2.3)

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These equations can be solved efficiently by employing generating function methods and we obtain the following.

Theorem 2.1. The equilibrium state probability of an empty system π(0, 0) is given by π(0, 0) = 0 λ + ϑ ϑ + λ λ + µEq . λ µ / 3φ2 . −λ ϑ, q, 0; − λ ϑq, − λ µq; q, q /1−1 . (2.4) The partial probability generating functions Π0(z) and Π1(z) are given by

Π0(z) = λ + ϑ λ + ϑ − λzπ(0, 0), |z| < 1 + ϑ λ (2.5) Π1(z) = 0 1 −λ + ϑ_ϑ π(0, 0) 1 eq . −λ_µ(1 − z) / − λ(λ + ϑ)(1 − z) (λ + ϑ − λz)(λ + µ − λz) × π(0, 0)3φ2 " −λϑ(1 − z), q, 0 −λϑq(1 − z), −λµq(1 − z) ; q, q # , |z| < min{1 +ϑ_λ, 1 +µ λ}. (2.6)

The convergence of the series is absolute in the corresponding open disks and uniform in every compact subset of them.

Proof. Using (2.1) we have immediately that π(n, 0) = . λ λ + ϑ /n π(0, 0), n ≥ 0. (2.7)

We have now that equation (2.5) follows easily from (2.7).

Multiplying both sides of equations (2.2) and (2.3) by z and zn respectively and summing for all n = 1, 2, . . . taking into account (2.7), we obtain

∞ $ n=1 (λ + µ)π(n, 1)zn = ϑ ∞ $ n=1 . λ λ + ϑ /n π(0, 0)zn+ ∞ $ n=2 λπ(n − 1, 1)zn + ∞ $ n=1 µ ∞ $ j=n . j n / pj−nqnπ(j, 1)zn. (2.8) After some algebraic manipulation in (2.8) we obtain

(λ + µ − λz)Π1(z) = µΠ1(1 − q + qz) + λ(λ + ϑ)(z − 1)

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2.3 The equilibrium state distribution 21 Defining A(z) = λ(λ + ϑ)(z − 1) µ(λ + ϑ − λz) , we arrive at Π1(z) = µ λ + µ − λzΠ1(1 − q + qz) + µ λ + µ − λzA(z)π(0, 0). (2.10) From the normalization equation Π0(1) + Π1(1) = 1 and (2.5) we obtain that

Π1(1) = 1 −

λ + ϑ

ϑ π(0, 0). (2.11)

By iterating (2.10) we can prove inductively that

Π1(z) = Π1(1 − qn+1+ qn+1z) n * k=0 µ µ + λ(1 − z)qk + n $ j=0 A(1 − qj+ qjz) j * k=0 µ µ + λ(1 − z)qkπ(0, 0), n ≥ 0. (2.12)

Taking limit as n → ∞ in (2.12) and using (2.11) results in

Π1(z) = . 1 − λ + ϑ ϑ π(0, 0) / ∞ * k=0 µ µ + λ(1 − z)qk + ∞ $ j=0 A(1 − qj + qjz) j * k=0 µ µ + λ(1 − z)qkπ(0, 0), (2.13)

provided that the series and the infinite product converge. To prove the conver-gence of the series and of the infinite product we will express (2.13) in terms of q–hypergeometric series. Let aj(z) = A(1 − qj+ qjz) j * k=0 µ µ + λ(1 − z)qk, j ≥ 0.

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that the ratio of two successive terms assumes the form aj+1(z) aj(z) = ϑ + λq j_{(1 − z)} ϑ + λqj+1_{(1 − z)} µ µ + λqj+1_{(1 − z)}q = 1 + λ ϑqj(1 − z) 1 +λ_ϑqj+1_{(1 − z)} 1 1 +λ_µqj+1_{(1 − z)}q = (1 − qq j_{)(1 − (−}λ ϑ(1 − z))qj)(1 − 0qj) (1 − qj+1_{)(1 − (−}λ ϑq(1 − z))qj)(1 − (−µλq(1 − z))qj) q, (2.14)

and hence is a rational function of qj_{, of the form stated in equation (}_1.1_{) for}

r = s + 1 = 3.

We define bk(z) = λ_µ(1 − z)qk and it is clear that in {z ∈ C : |z| < 1 +µ_λ} we have

1 + bk(z) (= 0 for k = 0, 1, 2, . . .. So we obtain that ∞ * k=0 1 (1 + bk(z)) = ∞ * k=0 µ µ + λ(1 − z)qk = 1 2 −λµ(1 − z); q 3 ∞ = eq(− λ µ(1 − z)) (2.15)

is a non-vanishing analytic function. We conclude that (2.13) assumes the form (2.6). It remains to prove (2.4). To this end we set z = 0 in (2.6). Then we obtain

0 = Π1(0) = 0 1 − λ + ϑ_ϑ π(0, 0) 1 eq . −λ_µ / − λ λ + µπ(0, 0)3φ2 . −λ ϑ, q, 0; − λ ϑq, − λ µq; q, q / . (2.16)

We multiply (2.16) by Eq(λ_µ), take into account that Eq(λ_µ)eq(−λ_µ) = 1, solve for

π(0, 0) and we obtain (2.4).

The absolute convergence of the series (2.6_{) is guaranteed for z ∈ {z ∈ C : |z| <} min{1 + ϑλ, 1 +

µ

λ}}. Indeed, we first observe that the 3φ2 . −λ ϑ(1 − z), q, 0; − λ ϑq(1 − z), − λ µq(1 − z); q, q /

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2.3 The equilibrium state distribution 23

convergences absolutely (see the comments after the definition (1.2)) for all z such that −λq(1−z)ϑ (= q−m and −

λq(1−z)

µ (= q−m, for m = 0, 1, . . ., consequently it

con-verges for z with |z| < min{1 +λqϑ, 1 + µ

λq}. Moreover the denominator of (2.6) does

not vanish for z (= 1 + ϑλ, 1 + µ

λ. Hence the partial probability generating function

Π1(z) as given in equation (2.6) converges for |z| < min{1 + ϑ_λ, 1 + µ_λ}.

The uniform and absolute convergence of the series in (2.6) can be also directly proved using the Weierstrass M–test (see e.g. Ahlfors (1979), Chapter 2, §2.2.3). !

We can now proceed to the calculation of the moments for the equilibrium dis-tribution of the number of customers in the system. Note that the moments of all orders exist since the partial probability generating functions Π0(z) and Π1(z)

converge inside an open disk with radius of convergence strictly greater than 1. We have the following.

Theorem 2.2. The factorial moments m_(n)_{= E[N (N − 1)(N − 2) · · · (N − n + 1)]} of the equilibrium number of customers in the system are given by

m_(n) = (λ + ϑ)λ n_n! ϑn+1 π(0, 0) + n $ k=1 (λ + ϑ)λnn! ϑk_µn−k+1 (q; q)k−1 (q; q)n π(0, 0) + λ n_n! µn_{(q; q)} n 0 1 − λ + ϑ ϑ π(0, 0) 1 , n ≥ 1. (2.17) In particular E[N ] = (λ + ϑ)λ ϑ2 π(0, 0) + λ µ(1 − q), (2.18) V ar[N ] = 2(λ + ϑ)λ 2 ϑ3 π(0, 0) + 2(λ + ϑ)λ2 ϑ2_{µ(1 − q}2₎π(0, 0) + 2λ2 µ2_{(1 − q)(1 − q}2₎ +(λ + ϑ)λ ϑ2 π(0, 0) + λ µ(1 − q)− . (λ + ϑ)λ ϑ2 π(0, 0) + λ µ(1 − q) /2 . Proof. The factorial moment generating function P (z) =!∞

n=0m(n)z

n

n! is given by

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We have already shown in theorem 2.1 that Π0(z) and Π1(z) converge in a

neigh-borhood of 1, hence P (z) is well defined in a neighneigh-borhood of 0. Using the following relation 3φ2 " a, q, 0 aq, bq ; q, q # = (1 − a)(1 − b) ∞ $ n=1 n−1 $ k=0 akbn−k−1(q; q)k (q; q)n , (2.20)

the proof of which can be found in the appendix, we obtain that

3φ2 " _λ ϑz, q, 0 λ ϑqz,µλqz ; q, q # = (1 −λ_ϑz)(1 −_µλz) ∞ $ n=1 n−1 $ k=0 (λz)n−1(q; q)k ϑk_µn−k−1_{(q; q)} n = (ϑ − λz)(µ − λz) ϑµ ∞ $ n=1 n $ k=1 (λz)n−1(q; q)k−1 ϑk−1_µn−k_{(q; q)} n . (2.21) Moreover eq. λ µz / = ∞ $ n=0 2 λ µz 3n (q; q)n . (2.22)

Plugging (2.21) and (2.22) into (2.6) and using (2.5), (2.19) yields, after some

sim-plifications, (2.17). _!

Remark 2.1. Differentiating (2.9) n times and setting z = 1 yields Π(n)₁ (1) = (λ + ϑ)λ n_{n!π(0, 0)} µϑn_{(1 − q}n₎ + nλ µ(1 − qn₎Π (n−1) 1 (1), n ≥ 1. (2.23)

Equation (2.23) forms an iterative scheme with initial conditions for Π(0)₁ (1) = Π1(1)

given by (2.11). The first order scheme given by (2.23) can be used to obtain (2.17). The exact inversion of Π1(z) given by (2.6), although possible using (2.20), is

practically useless, since the corresponding formulas for the stationary probabilities involve infinite sums. Therefore, we cannot obtain the equilibrium state distribution of the system in closed form, but we can exploit (2.6) using some numerical inversion algorithm (see e.g. Abate et al. (2000)) to obtain the equilibrium distribution for given values of the parameters up to any desired degree of accuracy. Nevertheless,

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we can obtain closed form expressions for some limiting regimes. We present them in theorems2.3,2.4 and 2.5below.

To emphasize the dependence on the parameters of the model in the rest of this section, we will denote π(n, i), Π0(z) and Π1(z) by π(n, i; λ, µ, p, ϑ), Π0(z; λ, µ, p, ϑ)

and Π1(z; λ, µ, p, ϑ) respectively. Note that µp can be thought of as the effective

ser-vice rate per customer. Indeed the overall serser-vice time of a customer is a geometric sum of exponentially distributed random variables with rate µ and so we can easily see that it is also exponentially distributed with parameter µp. Under this perspec-tive, if we have two models with the same parameters λ and ϑ that differ only in µ and p, but with µp = µ∗ _{fixed, we can think that the models have identical arrival}

rates, effective service rates per customer and setup rates λ, µ∗ and ϑ and differ only in the ‘level of synchronization’ p. Indeed, the case p → 0+ corresponds to no synchronization since the customers depart almost singly at the service completion epochs. On the contrary, the case p → 1− corresponds to full synchronization since almost all present customers depart simultaneously.

We are interested in studying the equilibrium behavior of the system for the case where λ, µ∗ _{and ϑ are kept fixed in the two limiting cases p → 0}+_{(q → 1}−) and p → 1−_{(q → 0}+_{). The corresponding results are presented in theorems}_2.3_and _2.4_.

Theorem 2.3. For a system with arrival rate λ, effective service rate per customer µ∗ and setup rate ϑ, the equilibrium state distribution π(1)(n, i) = lim_q→1−π(n, i; λ,

µ∗

1−q, 1 − q, ϑ) in the limiting case of no synchronization is given by

π(1)(0, 0) = 4 λ + ϑ ϑ + (λ + ϑ)λ µ∗ + 1 0 exp(_µλ∗s) ϑ + λs ds 5−1 (2.24) π(1)(n, 0) = . λ λ + ϑ /n π(1)_{(0, 0), n ≥ 1} (2.25) π(1)(n, 1) = . λ λ + ϑ /n 1 n! n $ k=1 . λ + ϑ µ∗ /k (n − k)! π(1)(0, 0), n ≥ 1. (2.26)

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Proof. We use (1.19) and we obtain that (1 − q)3φ2 . −λ_ϑ(1 − z), q, 0; −λ_ϑq(1 − z), −λ_µq(1 − z); q, q / = (− λ ϑ(1 − z), q, 0; q)∞ (q, −λϑq(1 − z), −λµq(1 − z); q)∞ + 1 0 (qt, −λϑq(1 − z)t, −λµq(1 − z)t; q)∞ (−λϑ(1 − z)t, qt, 0; q)∞ dqt. (2.27) By simplifying several terms, taking into account the relations (0; q)∞ = 1 and

(a; q)∞= (1 − a)(aq; q)∞ we have that (2.27) assumes the form

(1 − q)3φ2 . −λ_ϑ(1 − z), q, 0; −λ_ϑq(1 − z), −λ_µq(1 − z); q, q / = 1 + λ ϑ(1 − z) (−λµq(1 − z); q)∞ + 1 0 (−λµq(1 − z)t; q)∞ 1 + λ_ϑ_{(1 − z)t} dqt = (ϑ + λ(1 − z))eq . −_µλq(1 − z) / + 1 0 Eq(λ_µq(1 − z)t) ϑ + λ(1 − z)t dqt. (2.28) Replacing µ by _1−qµ∗ and using (1.8) we have

lim q→1−eq . −λ µq(1 − z) / = lim_q→1−e_q 2 −µλ∗q(1 − z)(1 − q) 3 = e−µ∗λ(1−z)_{, (2.29)} lim q→1−Eq . λ µq(1 − z)t / = lim_q→1−E_q 2 λ µ∗q(1 − z)t(1 − q) 3 = eµ∗λ(1−z)t_{. (2.30)}

We can then take the limit as q → 1− _{in (}_2.28_{), using (}_2.29_{), (}_2.30_{) and (}_1.18_{). We}

obtain lim q→1−(1 − q)3φ2 . −λ_ϑ(1 − z), q, 0; −λ_ϑq(1 − z), −_µλq(1 − z); q, q / = (ϑ + λ(1 − z))e−µ∗λ(1−z) + 1 0 eµ∗λ(1−z)t ϑ + λ(1 − z)tdt = (ϑ + λ(1 − z))e−µ∗λ(1−z) + 1−z 0 eµ∗λs ϑ + λs 1 1 − zds. (2.31)

For z = 0 we have in particular lim q→1−(1 − q)3φ2 . −λ_ϑ, q, 0; −λ_ϑq, −_µλq; q, q / = (ϑ + λ)e−µ∗λ + 1 0 eµ∗λs ϑ + λsds. (2.32)

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Taking the limit as q → 1− _{in (}_2.4_{) and taking into account (}_2.32_{) yields easily}

(2.24). Equation (2.25) is obvious in light of (2.7). To obtain (2.26) we begin by taking limit as q → 1− in (2.6). Using (2.29) and (2.31) we have

Π(1)₁ (z) = lim q→1−Π1(z) = 0 1 −λ + ϑ_ϑ π(1)(0, 0) 1 e−µ∗λ(1−z) −λ(λ + ϑ)_µ_∗ e−µ∗λ(1−z) + 1−z 0 eµ∗λs ϑ + λsds π (1)_{(0, 0).} _(2.33)

We differentiate (2.33) with respect to z and we obtain d dzΠ (1) 1 (z) = λ µ∗Π (1) 1 (z) + λ(λ + ϑ) µ∗_{(ϑ + λ(1 − z))}π (1)_{(0, 0).}

We expand _dzdΠ(1)₁ (z), Π(1)₁ (z) and _µ∗_{(ϑ+λ(1−z))}λ(λ+ϑ) in power series and we equate the

coefficients of zn. We obtain the stable recursive scheme π(1)(1, 1) = λ µ∗π (1)_{(0, 0)} _(2.34) π(1)(n, 1) = λ nµ∗π (1) (n − 1, 1) +_nµλ∗ . λ λ + ϑ /n−1 π(1)_{(0, 0), n ≥ 2. (2.35)} Iterating (2.35) and using (2.34) yields (2.26). _! Theorem 2.4. For a system with arrival rate λ, effective service rate per customer µ∗ _{and setup rate ϑ, the equilibrium state distribution π}(2)_{(n, i) = lim}

q→0+π(n, i; λ,

µ∗

1−q, 1 − q, ϑ) in the limiting case of full synchronization is given by

π(2)(0, 0) = ϑµ ∗ ϑµ∗_{+ λµ}∗_{+ λϑ} (2.36) π(2)(n, 0) = . λ λ + ϑ /n π(2)_{(0, 0), n ≥ 1} (2.37) π(2)(n, 1) =    µ∗ λ+µ∗n 2 λ λ+µ∗ 3n π(2)(0, 0), if µ∗ = ϑ ϑ µ∗_−ϑ 22 λ λ+ϑ 3n −2λ+µλ ∗ 3n3 π(2)(0, 0), if µ∗ _{(= ϑ, n ≥ 1.}(2.38). Proof. We take the limit as q → 0+ in (2.4), using (1.21) and (1.22). This yields

π(2)(0, 0) =0 λ + ϑ ϑ + λ λ + µ∗ . 1 + λ µ∗ /1−1 , (2.39)

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which is easily reduced to (2.36). Equation (2.37) is immediate from (2.7). Taking q → 0+ in (2.6), taking into account (1.20), (1.22) and (2.36) implies, after some simplifications, that Π(2)₁ (z) = lim q→0+Π1(z) = 0 1 − λ + ϑ_ϑ π(2)(0, 0) 1 1 1 +_µλ∗(1 − z) − λ(λ + ϑ)(1 − z) (λ + ϑ − λz)(λ + µ∗_{− λz)}π (2)_{(0, 0)} = ϑλz (λ + µ∗_{− λz)(λ + ϑ − λz)}π (2)_{(0, 0).} _(2.40)

Expanding Π(2)₁ (z), (λ + µ∗_{− λz)}−1 _{and (λ + ϑ − λz)}−1 in power series and equating the coefficients of zn _yields

π(2)(n, 1) = ϑλn n $ k=1 . 1 λ + µ∗ /k. 1 λ + ϑ /n−k+1 π(2)_{(0, 0), n ≥ 1.} (2.41)

By computing the geometric sum in (2.41) for the two cases µ∗= ϑ and µ∗ _{(= ϑ we}

obtain the two branches of (2.38). _!

The last limiting regime that we consider is for fixed λ, µ and q, when ϑ → ∞. In that case the server is not deactivated. This system can be seen as the Poisson arrival process subject to binomial catastrophes and it reduces to a special case of Economou (2004). We then have the following.

Theorem 2.5. For a system with arrival rate λ, service rate µ and service repeat probability q, the equilibrium state distribution π(3)(n, i) = limϑ→∞π(n, i; λ, µ, 1 −

q, ϑ) in the limiting case of no deactivation of the server is given by π(3)(0, 0) = eq . −λ µ / (2.42) π(3)_{(n, 0) = 0, n ≥ 1} (2.43) π(3)(n, 1) = ∞ $ k=n (−1)n2−λµ 3k ,k n -(q; q)k , n ≥ 1. (2.44)

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Proof. Taking limit as ϑ → ∞ in (2.4) implies that π(3)(0, 0) = 0 1 + λ λ + µEq . λ µ / 3φ2 . 0, q, 0; 0, −λ_µq; q, q /1−1 . (2.45) However,3φ2 2 0, q, 0; 0, −λµq; q, q 3 is simplified to2φ1 2 q, 0; −λµq; q, q 3 . Setting a = 0 in (2.20), we obtain 2φ1(q, 0; bq; q, q) = 1 − b b (eq(b) − 1). (2.46)

Then using (2.46) we have that (2.45) yields

π(3)(0, 0) = 4 1 + λ λ + µEq . λ µ /_{1 +}λ µ −λµ . eq . −λ_µ / − 1 /5−1 , (2.47)

which gives (2.42) after some simplifications. Taking ϑ → ∞ in (2.5) and using (2.47) yields

Π(3)₀ (z) = π(3)(0, 0) = eq

. −λ_µ

/

, (2.48)

so we have immediately (2.43_{). We now take ϑ → ∞ in (}2.6) and we obtain Π(3)₁ (z) = %_{1 − π}(3)(0, 0)&eq . −λ_µ(1 − z) / − λ(1 − z) λ + µ − λzπ (3)_{(0, 0)} 3φ2 . 0, q, 0; 0, −_µλq(1 − z); q, q / . (2.49) But by (2.46), we have 3φ2 . 0, q, 0; 0, −λ_µq(1 − z); q, q / = 2φ1 . q, 0; −λ_µq(1 − z); q, q / = 1 + λ(1−z) µ −λ(1−z)µ . eq . −λ_µ / − 1 / . (2.50)

Plugging (2.50) in (2.49) results after some simplification to Π(3)₁ (z) = eq . −λ µ(1 − z) / − eq . −λ µ / . (2.51)

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Expanding (2.51_{) in power series of (1 − z) using (}1.6) yields Π(3)₁ (z) = ∞ $ k=0 2 −λµ 3k (q; q)k k $ n=0 .k n / (−z)n− eq . −λ µ / = ∞ $ n=1 ∞ $ k=n 2 −µλ 3k (q; q)k .k n / (−z)n, which proves (2.44). _!

2.4 Busy period and sojourn time distributions

We now study the busy period of the model, i.e. the time from the arrival of a customer at an empty system till the next epoch that the system is empty again. The busy period L is a first passage time starting from the state (1, 0) to reach (0, 0). Let L(n,i), i = 0, 1 and n ≥ i be a generic random variable representing a first

passage time to state (0, 0) starting from (n, i) and denote by ϕ_(n,i)(s) = E[e−sL(n,i)_]

its Laplace-Stieltjes transform. By conditioning on the time of the next event (first-step analysis) we obtain that ϕ(n,i)(s) satisfy the system

ϕ_(0,0)(s) = 1 (2.52) ϕ_(n,0)(s) = λ λ + ϑ + sϕ(n+1,0)(s) + ϑ λ + ϑ + sϕ(n,1)(s), n ≥ 1 (2.53) ϕ_(n,1)(s) = λ λ + µ + sϕ(n+1,1)(s) + µpn λ + µ + s + µ λ + µ + s n $ j=1 .n j / pn−jqjϕ(j,1)(s), n ≥ 1. (2.54)

We define the mixed transforms Φi(s, z), i = 0, 1, by

Φi(s, z) = ∞

$

n=i

ϕ(n,i)(s)zn, i = 0, 1. (2.55)

These mixed transforms carry information for all first passage times distributions from an arbitrary state to (0, 0) and in particular for the busy period of the system. They can be expressed in terms of q–hypergeometric series as follows.

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2.4 Busy period and sojourn time distributions 31

Theorem 2.6. The Laplace-Stieltjes transforms ϕ_(1,1)(s), ϕ_(1,0)(s) and the mixed transforms Φ1(s, z), Φ0(s, z) are given by the equations

ϕ_(1,1)(s) = µ(1 − q)1φ1(− λ µ+s; − λq2 µ+s; q, µq2 µ+s) (λq + µ + s)1φ1(−_µ+sλ ; −_µ+sλq ; q,_µ+sµq ) (2.56) Φ1(s, z) = µ(1 − q)z22φ2 2 q, −1−zz ; (µ+s)zq λ(1−z), − zq2 1−z; q, µzq2 λ(1−z) 3 ((λ + µ + s)z − λ)(1 − (1 − q)z) − λzϕ_(1,1)(s)2φ2 2 q, −1−zz ; (µ+s)zq λ(1−z), − zq 1−z; q, µzq λ(1−z) 3 (λ + µ + s)z − λ (2.57) ϕ_(1,0)(s) = ϑ λΦ1 . s, λ λ + ϑ + s / (2.58) Φ0(s, z) = 1 + ϑz (λ + ϑ + s)z − λ . Φ1(s, z) − Φ1 . s, λ λ + ϑ + s // . (2.59) The involved q–hypergeometric series converge absolutely in {(s, z) ∈ C2 : |z| < 1, Re(s) ≥ 0}.

Proof. For the sake of notational convenience we introduce here an operator no-tation. Let T (z) = _1−(1−q)zqz This is a linear fractional transformation and we define its k-th compositions by T0(z) = z and Tk(z) = T (Tk−1(z)) for k ≥ 1. Then it can

be easily proved inductively that Tk(z) =

qkz

1 − (1 − qk_)z, k ≥ 0. (2.60)

We also define the quantities

a0(z) = 1 (2.61) ak(z) = (−1)k . µz λ(1 − z) /k q(k2) 6k i=1(1 −(µ+s)zλ(1−z)qi) . (2.62)

Multiplying (2.54) with (λ + µ + s)zn _{and adding for all n ≥ 1 results after some}

manipulations to [(λ + µ + s)z − λ] Φ1(s, z) = µpz2 1 − pz − λzϕ(1,1)(s) + µz 1 − pzΦ1(s, T (z)). (2.63)

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By iterating (2.63) we obtain [(λ + µ + s)z − λ]Φ1(s, z) = µpz2 1 − pz − λzϕ(1,1)(s) + n $ k=1 4 _k * i=1 µTi(z) q[(λ + µ + s)Ti(z) − λ] 5 0 µpT2 k(z) 1 − pTk(z) 1 − n $ k=1 4 _k * i=1 µTi(z) q[(λ + µ + s)Ti(z) − λ] 5 λTk(z)ϕ(1,1)(s) + n+1 * i=1 µTi(z) q[(λ + µ + s)Ti(z) − λ][(λ + µ + s)z − λ]Φ1 (s, Tn+1(z)) (2.64)

and taking limits as the number of iterations n goes to infinity results similarly with the derivation of (2.13) from (2.12) to

[(λ + µ + s)z − λ]Φ1(s, z) = µpz2 1 − pz − λzϕ(1,1)(s) + ∞ $ k=1 4 _k * i=1 µTi(z) q[(λ + µ + s)Ti(z) − λ] 5 0 µpT_k2(z) 1 − pTk(z) 1 − ∞ $ k=1 4 _k * i=1 µTi(z) q[(λ + µ + s)Ti(z) − λ] 5 λTk(z)ϕ(1,1)(s). (2.65)

The absolute convergence of the series can be proved straightforward by applying the ratio test. We can now easily verify that

Ti(z) (λ + µ + s)Ti(s) − λ = (−1)q i_z λ(1 − z)(1 −(µ+s)zλ(1−z)qi) , i ≥ 1 so k * i=1 µTi(z) q[(λ + µ + s)Ti(s) − λ] = ak(z). (2.66)

We plug (2.66) into (2.65) and we obtain [(λ + µ + s)z − λ]Φ1(s, z) = ∞ $ k=0 ak(z) 0 µpT_k2(z) 1 − pTk(z) 1 − ∞ $ k=0 ak(z)λTk(z)ϕ(1,1)(s). (2.67)

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2.4 Busy period and sojourn time distributions 33

By setting z = _λ+µ+sλ in equation (2.67) we obtain

ϕ_(1,1)(s) = !∞ k=0ak 2 λ λ+µ+s 3 4 µpT2 k “ λ λ+µ+s ” 1−pTk “ λ λ+µ+s ” 5 !∞ k=0ak 2 λ λ+µ+s 3 λTk 2 λ λ+µ+s 3 . (2.68)

By writing the ratio of two consecutive terms of the sums in (2.68) as in (1.1), we can express the sums in φ-notation. Then (2.68) yields (2.56).

We then solve (2.67) for Φ1(s, z) and express the sums involved in the standard

manner to put them in φ-notation. This yields (2.57).

We now multiply the equations (2.52) and (2.53) with z0 _{and z}n _{respectively and}

add for all n ≥ 0. After some algebraic manipulations we obtain that

[(λ + ϑ + s)z − λ] Φ0(s, z) = (λ + ϑ + s)z − λ − zλϕ(1,0)(s) + ϑzΦ1(s, z). (2.69)

We set z = _λ+ϑ+sλ in (2.69) and solve for ϕ_(1,0)(s) so we obtain (2.58). Solving (2.69) for Φ0(s, z), taking into account (2.58) results in (2.59). !

We now consider a tagged customer and let S denote its sojourn time in the system. The following theorem shows that the distribution of S is either a mixture of exponential distributions with parameters µ(1 − q) and ϑ (in the case where µ(1−q) (= ϑ) or a mixture of an exponential distribution and an Erlang-2 distribution with parameter ϑ (in the case where µ(1 − q) = ϑ). More specifically we have the following.

Theorem 2.7. The distribution of the sojourn time of a customer in the system is given by FS(x) = 7 (1 − p1)(1 − e−µ(1−q)x) + p1(1 − e−ϑx), if ϑ (= µ(1 − q) (1 − p2)(1 − e−ϑx) + p2(1 − e−ϑx− xe−ϑx), if ϑ = µ(1 − q), x ≥ 0, (2.70) where p1= (λ + ϑ)µ(1 − q) ϑ(µ(1 − q) − ϑ)π(0, 0), p2 = λ + ϑ ϑ π(0, 0). (2.71)

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Proof. The sojourn time of a customer in the system depends on whether the server is active or not at the time of his arrival. Because of the PASTA property we have that the probability that the server is active at an arrival instant is Π1(1)

given by (2.11) while the probability that the server is deactivated is Π0(1). Then

the sojourn time S has the representation S = 7 Y +!J j=1Xj, with probability λ+ϑ_ϑ π(0, 0) !J j=1Xj, with probability 1 − λ+ϑ_ϑ π(0, 0), (2.72) where Y and X1, X2, . . . are independent exponentially distributed random variables

with rates ϑ and µ respectively and J is independent of them, representing the number of services that will be required until the customer leaves the system. We have that J is geometrically distributed with P (J = j) = (1 − q)qj−1, j = 1, 2, . . .. Because of (2.72) the Laplace-Stieltjes transform of S assumes the form

˜ FS(s) = λ + ϑ ϑ π(0, 0) ϑ ϑ + s µ(1 − q) µ(1 − q) + s+ . 1 −λ + ϑ_ϑ π(0, 0) / µ(1 − q) µ(1 − q) + s. (2.73) The partial fraction expansion of (2.73) yields

˜ FS(s) =    (1 − p1)_µ(1−q)+sµ(1−q) + p1_ϑ+sϑ , if ϑ (= µ(1 − q) (1 − p2)_ϑ+sϑ + p2 2 ϑ ϑ+s 32 , if ϑ = µ(1 − q)

where p1, p2 are given by (2.71) and we invert to obtain (2.70). !

2.5 Numerical results

In this section we present some numerical results that shed further light in the be-havior of the model. We begin by illustrating the effect of each single parameter p, ϑ and λ in the mean number of customers in the system, E[N ], when the other parameters are kept fixed. In all numerical scenarios we assume that the time unit has been re-scaled to agree with the mean service time, i.e. we set µ = 1. Our results are based on the computation of the mean number of customers given from equation (2.18) of theorem2.2. In figure 2.2we present the graph of E[N ] with respect to p for λ = 3.5 and ϑ = 1.5. The function is decreasing convex as p varies from 0 to 1.

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2.5 Numerical results 35 0 0 .2 0 .4 0 .6 0 .8 1 0 5 10 15 20 25 30 35 40 45 50 p E [N ;p ] λ= 3.5 θ= 1.5 µ= 1

Fig. 2.2: E[N ; λ, µ, p, ϑ] versus p.

0 2 4 6 8 1 0 1 2 3.5 3 .5 5 3.6 3 .6 5 3.7 3 .7 5 3.8 θ E [N ;θ ] λ= 3 .5 µ= 1 p= 0 .5

Fig. 2.3: E[N ; λ, µ, p, ϑ] versus ϑ.

0 0.5 1 1 .5 2 0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4 1 .6 1 .8 2 λ E [N ;λ ] θ= 1.5 µ= 1 p= 0.5

Fig. 2.4: E[N ; λ, µ, p, ϑ] versus λ.

Figure 2.3shows the influence of the setup rate ϑ in the mean number of customers in the system, when λ = 3.5 and p = 0.5. The function is decreasing convex with a horizontal asymptote as ϑ → ∞. This agrees with our intuitive expectation and the result of theorem 2.5. Figure2.4shows the behavior of E[N ] as the arrival rate λ varies, when ϑ = 1.5 and p = 0.5. As expected, the mean number of customers in the system increases when λ increases.

The next figures2.5and2.6demonstrate the effect of the level of synchronization in the mean and the variance of the number of customers in the system. In these

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0 0 .2 0 .4 0 .6 0 .8 1 3 .6 3 .8 4 4 .2 4 .4 4 .6 4 .8 5 p E [N ;p ] λ= 3 .5 θ= 1.5 µp= 1

Fig. 2.5: E[N ; λ,µ_p$, p, ϑ] versus p.

0 0 .2 0 .4 0.6 0.8 1 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 p V a r[ N ;p ] λ= 3.5 θ= 1 .5 µp= 1

Fig. 2.6: V ar[N ; λ, µ_p$, p, ϑ] versus p.

two figures, the arrival rate λ, the effective service rate per customer µ$ = µp and the setup rate ϑ are kept fixed, λ = 3.5, ϑ = 1.5, µ$ = 1. We observe that both E[N ] and V ar[N ] are increasing convex functions of p, i.e. both the average number of customers and its variability increase as the level of synchronization increases.

0 1 2 3 4 5 6 7 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 x f( x ) λ= 3.5 θ= 1.5 µp= 1 p= 0 .9 p= 0 .6 5 p= 0.4

Fig. 2.7: Busy period densities.

Finally, we present the probability densities of the busy period for three numerical scenarios, by numerically invert the Laplace-Stieltjes transform of the busy period,

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2.5 Numerical results 37

ϕ_(1,0)(s), given in theorem2.6. Although there is no symbolic inversion of ϕ_(1,0)(s), the use of therφsnotation enables us to compute the numerical inversion very easily,

by exploiting the available procedures and algorithms that support the computation of q–hypergeometric series in all standard mathematical software packages. In these scenarios we assume that λ = 3.5, ϑ = 1.5, µp = 1 and we consider three cases for p: 0.4, 0.65 and 0.9. Again, the numerical findings in figure2.7 are in absolute accordance with the fact that the mean number of customers increases as the level of synchronization increases.

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Chapter

3

Synchronized abandonments in

a single server unreliable

queue

We consider a single server unreliable queue represented by a 2-dimensional contin-uous time Markov chain. At failure times, all present customers leave the system. Moreover, customers become impatient and perform synchronized abandonments, as long as the server is down. We analyze this model and derive the main performance measures using results from the basic q–hypergeometric series.

3.1 Introduction

In the queueing literature, there exists a significant number of papers dealing with queueing systems with abandonments. In the majority of the papers, the source of the impatience has been taken to be either a long wait already experienced at a queue, or a long wait anticipated by the customers. Recently, Altman and Yechiali (2006, 2008), Perel and Yechiali (2009) and Yechiali (2007) considered systems with server(s) alternating between on and off periods, where customers’ impatience is due to the absence of the server(s). Such systems can model satisfactorily the reneging

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behavior of waiting customers in real systems with servers that are temporarily un-available due to either scheduled vacations or failures.

Models with servers alternating between on and off periods and customers’ aban-donments are generally hard to analyze. In a Markovian framework, the state of such a system is typically represented by a vector (n, i), where n records the num-ber of customers and i the state of the server(s). However, the abandonments that the customers perform independently lead to transitions out of a state (n, 0) to a state (n − 1, 0) with rates proportional to n. In this sense, the independent aban-donments of the customers give rise to ‘spatially inhomogeneous’ continuous time Markov chains. This is also the distinctive feature in queueing models with an in-finite number of servers, retrial queueing models and population growth models in Mathematical Biology, since every individual is associated with births and deaths. In most cases these models are mathematically intractable and it is not possible to conclude with closed form results.

There are only a few research works trying to extend matrix analytic methods or other analytic tools for models with this type of spatial inhomogeneity. In general, the authors use truncation or generalized truncation ideas to study the systems (see e.g. Artalejo and Pozo (2002) and the references therein) or they apply generating function methods. However, we have to note that in this case the partial generating functions of the number of individuals in system satisfy a system of linear differential equations in contrast with the ‘spatially homogeneous’ case where they satisfy linear algebraic equations. Such systems are usually intractable or can be solved in terms of hypergeometric series (see e.g. Altman and Yechiali (2006, 2008), Artalejo and Gomez-Corral (1997), Baykal-Gursoy and Xiao (2004), Keilson and Servi (1993), Krishnamoorthy et al. (2005), Perel and Yechiali (2009) and Yechiali (2007)).

The aim of this chapter is to extend the study of queues with disasters and im-patient customers when the system is down that was introduced by Yechiali (2007). Yechiali (2007) considered Markovian queues subject to disasters that remove all the customers from the system and turn the server down. As long as the server is

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3.1 Introduction 41

down, he assumed that the arrivals continue to come, but the customers become impatient and perform independent abandonments, that is, every customer sets on his own exponential patience clock and leaves the system when his patience time expires. We assume that the customers are impatient but they perform synchronized abandonments. Such a model is motivated by remote systems where customers have to wait for a certain transport facility to abandon the system. Then, whenever the facility visits the system, the present customers can decide whether to leave the sys-tem or not. Therefore, we have synchronized departures for some of the customers. More specifically, we assume that the abandonment opportunities occur according to a Poisson process at rate ξ, whenever the server is down (so we can think that the transportation facility’s arrivals occur according to this process). Then, at an aban-donment opportunity epoch, every customer decides to abandon the system with probability p or remains in the system waiting for service with probability q = 1 − p, independently of the others. This implies that there exist binomial transition rates of the form ξ,_n

n#-pn−n #

qn#, from a state (n, 0) to states (n"_{, 0), for 0 ≤ n}" _{≤ n. Similar} Markov chains occur in Mathematical Biology in the study of population processes subject to binomial catastrophes (see e.g. Artalejo et al. (2007), Brockwell et al. (1982), Economou (2004) and Economou and Fakinos (2008)). Moreover, Neuts (1994) studied a 1-dimensional discrete-time model with similar dynamics.

We show that the framework of basic q–hypergeometric series enables us to ex-press in closed form the main performance measures of this system. In general, the theory of q–hypergeometric series can facilitate the computations regarding systems with this kind of binomial transitions arising from synchronization.

The chapter is organized as follows. In section 3.2 we describe the model and introduce the appropriate notation. In section 3.3 we carry out the equilibrium anal-ysis of the system state and derive several exact formulas and iterative algorithmic schemes. Some limiting regimes with a particular interest are discussed in more detail. In section 3.4 we study the sojourn time of a customer, while in section 3.5 we treat the system busy period distribution of the model.

Stationary performance evaluation measures in multi-dimensional Markov chains and applications in Queueing Theory

Stationary performance evaluation

measures in multi-dimensional Markov

chains and applications in Queueing Theory

Stella Kapodistria

Queueing Theory

Stella Kapodistria

Contents

Chapter

1

Introduction

1.1

Background and motivation

1.2

Basic Hypergeometric series

1.3

Overview of the thesis

Chapter

2

Synchronized services in a

single server vacation queue

2.1

Introduction

2.2

Model description and notation

2.3

The equilibrium state distribution

2.4

Busy period and sojourn time distributions

2.5

Numerical results

Chapter

3

Synchronized abandonments in

a single server unreliable

queue

3.1

Introduction