Stationary performance evaluation measures in multi …skapodis/phdthesis_eng.pdf · 2009-09-09 ·...

University of Athens

Department of Mathematics

Section of Statistics and Operations Research

Stationary performance evaluationmeasures in multi-dimensional Markov

chains and applications in Queueing Theory

Stella Kapodistria

0 1 2 · · · n ! 2 n ! 1 n!!""!!""

(nj)p

n!jqj!!

""

Athens

2009

Stationary performance evaluation measures inmulti-dimensional Markov chains and applications in

Queueing Theory

Stella Kapodistria

PhD Thesis

g

To Antonis, Vaso,

Espi

and Apostolos

With the completion of my PhD thesis, I would like to express my gratitude to

my supervisor A. Economou for his continuous guidance, his fruitful criticism, his

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 o!ce 118.

Contents

1 Introduction 1

1.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Basic Hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 The q–binomial theorem . . . . . . . . . . . . . . . . . . . . . 9

1.2.2 Transformation formulas for 2!1 series . . . . . . . . . . . . . 10

1.2.3 The q–integral . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.4 Limiting regimes as q " 0+ . . . . . . . . . . . . . . . . . . . 11

1.3 Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Synchronized services in a single server vacation queue 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Model description and notation . . . . . . . . . . . . . . . . . . . . . 18

2.3 The equilibrium state distribution . . . . . . . . . . . . . . . . . . . 19

2.4 Busy period and sojourn time distributions . . . . . . . . . . . . . . 30

2.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Synchronized abandonments in a single server unreliable queue 39

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Model description and notation . . . . . . . . . . . . . . . . . . . . . 42

3.3 The equilibrium state distribution . . . . . . . . . . . . . . . . . . . 43

3.4 Sojourn times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5 System busy period . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4 Synchronized reneging in single server vacation queues – Part I 65

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3 Mean value analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3.1 Mean value analysis of the UAE model . . . . . . . . . . . . . 69

4.3.2 Mean value analysis of the MAE model . . . . . . . . . . . . 70

4.4 Equilibrium distribution . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.4.1 Equilibrium distribution of the UAE model . . . . . . . . . . 73

4.4.2 Equilibrium distribution of the MAE model . . . . . . . . . . 76

4.4.3 Fluid limit of the UAE model . . . . . . . . . . . . . . . . . . 79

4.4.4 Fluid limit of the MAE model . . . . . . . . . . . . . . . . . 83

4.4.5 Limiting regimes of synchronization in the MAE model . . . 88

5 Synchronized reneging in single server vacation queues – Part II 93

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.3 Mean value analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.3.1 Mean value analysis of the UAE model . . . . . . . . . . . . . 97

5.3.2 Mean value analysis of the MAE model . . . . . . . . . . . . 98

5.4 Equilibrium distribution . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.4.1 Equilibrium distribution of the UAE model . . . . . . . . . . 107

5.4.2 Equilibrium distribution of the MAE model . . . . . . . . . . 108

5.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.5.1 Numerical results for the UAE model . . . . . . . . . . . . . 114

5.5.2 Numerical results for the MAE model . . . . . . . . . . . . . 115

Appendix 117

Bibliography 126

Chapter 1

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 e!cient 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.

The queueing discipline. It is the rule followed by the server(s) for choosing cus-

2 Introduction

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 e!cient 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), Ek for 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.

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 e!ciently 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 di"erential 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.

4 Introduction

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 di!culty 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 e!cient 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 " and their sojourn times are identical and independent

random variables exponentially distributed with rate #. This system is the standard

M/M/# queue and its transition rate diagram can be seen in figure 1.1.

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


0"

## 1

!

$$

"## 2

2!

$$

"## 3

3!

$$

"## · · ·

4!

%%

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 ", but now we

suppose that the departures can only occur at the epochs of a Poisson process at

rate #. 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"

## 1(11)p!$$

"## 2

(21)pq!

$$

(22)p

2!

&&

"## 3

(31)pq2!

$$

(32)p

2q!

&&

(33)p

3!

''

"## · · ·

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

6 Introduction

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 models.

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-o" periods, unreliable servers, servers with failures and repairs). Models based


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 e"ect 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

8 Introduction

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=0 un where u0 = 1 and un+1

un

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 ! a1qn)(1 ! a2qn) · · · (1 ! arqn)

(1 ! qn+1)(1 ! b1qn) · · · (1 ! bsqn)(!qn)1+s$rz, (1.1)

we have that every such series assumes the form

r!s

"

a1, a2, . . . , ar

b1, . . . , bs; q, z

#

' r!s(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 ! aqn$1), n = 1, 2, . . . .

It is assumed in the definition of the r!s 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 r!s 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).

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$1)nq$(n2). (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

1!0(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) = 1(z;q)"

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) = 1!0(0;!; q; z) =#$

n=0

zn

(q; q)n=

1

(z; q)#, |z| < 1. (1.6)

If we replace z by ! za in (1.5) and then let a " # to get

Eq(z) = 0!0(!;!; q; z) =#$

n=0

q(n2)

(q; q)nzn = (!z; q)#. (1.7)

10 Introduction

We observe that, as q " 1$ the q–analogues reduce to their standard counterparts.

In particular we have the relationships

limq%1!

eq(z(1 ! q)) = ez (1.8)

limq%1!

Eq(z(1 ! q)) = ez (1.9)

limq%1!

(qaz; q)#(z; q)#

= (1 ! z)$a. (1.10)

1.2.2 Transformation formulas for 2!1 series

Heine (1847,1878) showed that

2!1(a, b; c; q, z) =(b, az; q)#(c, z; q)#

2!1(c/b, z; az; q, b) (1.11)

=(c/b, bz; q)#

(c, z; q)#2!1(abz/c, b; bz; q, c/b) (1.12)

=(abz/c; q)#

(z; q)#2!1(c/a, c/b; c; q, abz/c). (1.13)

Jackson (1910) proved that

2!1(a, b; c; q, z) =(az; q)#(z; q)#

2!2(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

0f(t)dqt = (1 ! q)

#$

n=0

f(qn)qn (1.15)

and Jackson gave the most general definition+ b

af(t)dqt =

+ b

of(t)dqt !

+ a

0f(t)dqt, (1.16)

where+ a

0f(t)dqt = a(1 ! q)

#$

n=0

f(aqn)qn. (1.17)

1.3 Overview of the thesis 11

If f is continuous on [0, a], then it is easily seen that

limq%1!

+ a

0f(t)dqt =

+ a

0f(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+1!r series can be expressed as a q–

integral through the relation

r+1!r

"

a1, . . . , ar+1

b1, . . . , br; q, qz

#

=(a1, . . . , ar+1; q)#

(1 ! q)(q, b1, . . . , br; q)#

)+ 1

0tz$1 (qt, b1t, . . . , brt; q)#

(a1t, . . . , ar+1t; q)#dqt, (1.19)

when Rez > 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

limq%0+

eq(z) =1

1 ! z(1.20)

limq%0+

Eq(z) = 1 + z. (1.21)

We can also prove the limiting relation

limq%0+

r+1!r

"

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

12 Introduction

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 e!ciently 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 e"ect

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 e"ect 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 o". 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

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 station-

14 Introduction

ary distributions by using a generating function approach. In section 5.5 we present

several numerical results that illustrate the e"ect of the various parameters on the

performance measures of the model.

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 e!ciently 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 e"ect of the level

of synchronization to the performance of such systems.

16 Synchronized services in a single server vacation queue

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 e"ective 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 e!cient.

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 di!culties (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 e!cient 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

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

di"erential 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,nn#

-

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


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 e"ect 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 ". 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 $. An alternative interpretation is that the server

takes exponentially distributed multiple vacations with parameter $ 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=o" and 1=on), t $ 0. The corresponding

transition diagram is given in Figure 2.1.

Let (%(n, i) : i = 0, 1 and n $ i) denote the stationary distribution of {(N(t), I(t))}.

2.3 The equilibrium state distribution 19

1, 1(11)pµ

((

"## 2, 1

(21)pqµ))

(22)p2µ

**

"## 3, 1

(31)pq2µ))

(32)p

2qµ

++

(33)p3µ

,,

"## · · ·

0, 0"

## 1, 0

#

--

"## 2, 0

#

--

"## 3, 0

#

--

"## · · ·

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

We also define the partial probability generating functions #0(z) and #1(z) by

#0(z) =#$

n=0

%(n, 0)zn and #1(z) =#$

n=1

%(n, 1)zn .

In section 2.3 we will determine #0(z) and #1(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:

"%(0, 0) = µ#$

n=1

pn%(n, 1) = µ#1(p)

(" + $)%(n, 0) = "%(n ! 1, 0), n $ 1 (2.1)

(" + µ)%(1, 1) = $%(1, 0) + µ#$

j=1

.

j

1

/

pj$1q%(j, 1) (2.2)

(" + µ)%(n, 1) = $%(n, 0) + "%(n ! 1, 1) +

+µ#$

j=n

.

j

n

/

pj$nqn%(j, 1), n $ 2. (2.3)


These equations can be solved e!ciently by employing generating function methods

and we obtain the following.

Theorem 2.1. The equilibrium state probability of an empty system %(0, 0) is given

by

%(0, 0) =

0

" + $

$+

"

" + µEq

.

"

µ

/

3!2

.

!"

$, q, 0;!

"

$q,!

"

µq; q, q

/1$1

. (2.4)

The partial probability generating functions #0(z) and #1(z) are given by

#0(z) =" + $

" + $! "z%(0, 0), |z| < 1 +

$

"(2.5)

#1(z) =

0

1 !" + $

$%(0, 0)

1

eq

.

!"

µ(1 ! z)

/

!"(" + $)(1 ! z)

(" + $! "z)(" + µ ! "z)

)%(0, 0) 3!2

"

!"#(1 ! z), q, 0

!"#q(1 ! z),!"µq(1 ! z); q, q

#

,

|z| < min{1 +$

", 1 +

µ

"}. (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

%(n, 0) =

.

"

" + $

/n

%(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

(" + µ)%(n, 1)zn = $#$

n=1

.

"

" + $

/n

%(0, 0)zn +#$

n=2

"%(n ! 1, 1)zn

+#$

n=1

µ#$

j=n

.

j

n

/

pj$nqn%(j, 1)zn. (2.8)

After some algebraic manipulation in (2.8) we obtain

(" + µ ! "z)#1(z) = µ#1(1 ! q + qz) +"(" + $)(z ! 1)

" + $! "z%(0, 0). (2.9)


Defining

A(z) ="(" + $)(z ! 1)

µ(" + $! "z),

we arrive at

#1(z) =µ

" + µ ! "z#1(1 ! q + qz) +

µ

" + µ ! "zA(z)%(0, 0). (2.10)

From the normalization equation #0(1) + #1(1) = 1 and (2.5) we obtain that

#1(1) = 1 !" + $

$%(0, 0). (2.11)

By iterating (2.10) we can prove inductively that

#1(z) = #1(1 ! qn+1 + qn+1z)n*

k=0

µ

µ + "(1 ! z)qk

+n$

j=0

A(1 ! qj + qjz)j*

k=0

µ

µ + "(1 ! z)qk%(0, 0), n $ 0. (2.12)

Taking limit as n " # in (2.12) and using (2.11) results in

#1(z) =

.

1 !" + $

$%(0, 0)

/ #*

k=0

µ

µ + "(1 ! z)qk

+#$

j=0

A(1 ! qj + qjz)j*

k=0

µ

µ + "(1 ! z)qk%(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 + "(1 ! z)qk, j $ 0.

It is easy to see that aj(z) are well defined for |z| < min{1 + #" , 1 + µ

"}. We have


that the ratio of two successive terms assumes the form

aj+1(z)

aj(z)=

$ + "qj(1 ! z)

$ + "qj+1(1 ! z)

µ

µ + "qj+1(1 ! z)q

=1 + "

#qj(1 ! z)

1 + "#q

j+1(1 ! z)

1

1 + "µqj+1(1 ! z)

q

=(1 ! qqj)(1 ! (!"#(1 ! z))qj)(1 ! 0qj)

(1 ! qj+1)(1 ! (!"#q(1 ! z))qj)(1 ! (!"µq(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) = "µ(1! z)qk and it is clear that in {z % C : |z| < 1+ µ

"} 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 + "(1 ! z)qk=

12

!"µ(1 ! z); q3

#

= eq(!"

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 = #1(0) =

0

1 !" + $

$%(0, 0)

1

eq

.

!"

µ

/

!"

" + µ%(0, 0) 3!2

.

!"

$, q, 0;!

"

$q,!

"

µq; q, q

/

. (2.16)

We multiply (2.16) by Eq("µ), take into account that Eq(

"µ)eq(!"µ) = 1, solve for

%(0, 0) and we obtain (2.4).

The absolute convergence of the series (2.6) is guaranteed for z % {z % C : |z| <

min{1 + #" , 1 + µ

"}}. Indeed, we first observe that the

3!2

.

!"

$(1 ! z), q, 0;!

"

$q(1 ! z),!

"

µq(1 ! z); q, q

/


convergences absolutely (see the comments after the definition (1.2)) for all z such

that !"q(1$z)# (= q$m and !"q(1$z)

µ (= q$m, for m = 0, 1, . . ., consequently it con-

verges for z with |z| < min{1 + #"q , 1 + µ

"q}. Moreover the denominator of (2.6) does

not vanish for z (= 1 + #" , 1 + µ

" . Hence the partial probability generating function

#1(z) as given in equation (2.6) converges for |z| < min{1 + #" , 1 + µ

"}.

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 #0(z) and #1(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) =(" + $)"nn!

$n+1%(0, 0) +

n$

k=1

(" + $)"nn!

$kµn$k+1

(q; q)k$1

(q; q)n%(0, 0)

+"nn!

µn(q; q)n

0

1 !" + $

$%(0, 0)

1

, n $ 1. (2.17)

In particular

E[N ] =(" + $)"

$2%(0, 0) +

"

µ(1 ! q), (2.18)

V ar[N ] =2(" + $)"2

$3%(0, 0) +

2(" + $)"2

$2µ(1 ! q2)%(0, 0) +

2"2

µ2(1 ! q)(1 ! q2)

+(" + $)"

$2%(0, 0) +

"

µ(1 ! q)!.

(" + $)"

$2%(0, 0) +

"

µ(1 ! q)

/2

.

Proof. The factorial moment generating function P (z) =!#

n=0 m(n)zn

n! is given by

P (z) = #0(1 + z) + #1(1 + z). (2.19)


We have already shown in theorem 2.1 that #0(z) and #1(z) converge in a neigh-

borhood of 1, hence P (z) is well defined in a neighborhood of 0. Using the following

relation

3!2

"

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

3!2

"

"#z, q, 0"#qz, "µqz

; q, q

#

= (1 !"

$z)(1 !

"

µz)

#$

n=1

n$1$

k=0

("z)n$1 (q; q)k$kµn$k$1(q; q)n

=($! "z)(µ ! "z)

$µ

#$

n=1

n$

k=1

("z)n$1 (q; q)k$1

$k$1µn$k(q; q)n.

(2.21)

Moreover

eq

.

"

µz

/

=#$

n=0

2

"µz3n

(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. Di!erentiating (2.9) n times and setting z = 1 yields

#(n)1 (1) =

(" + $)"nn!%(0, 0)

µ$n(1 ! qn)+

n"

µ(1 ! qn)#(n$1)

1 (1), n $ 1. (2.23)

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

given by (2.11). The first order scheme given by (2.23) can be used to obtain (2.17).

The exact inversion of #1(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,


we can obtain closed form expressions for some limiting regimes. We present them

in theorems 2.3, 2.4 and 2.5 below.

To emphasize the dependence on the parameters of the model in the rest of this

section, we will denote %(n, i), #0(z) and #1(z) by %(n, i;", µ, p,$), #0(z;", µ, p,$)

and #1(z;", µ, p,$) respectively. Note that µp can be thought of as the e"ective ser-

vice rate per customer. Indeed the overall service 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 " and $ that di"er only in µ

and p, but with µp = µ& fixed, we can think that the models have identical arrival

rates, e"ective service rates per customer and setup rates ", µ& and $ and di"er

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 ", µ& and $ 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 ", e!ective service rate per customer

µ& and setup rate $, the equilibrium state distribution %(1)(n, i) = limq%1! %(n, i;",µ$

1$q , 1 ! q,$) in the limiting case of no synchronization is given by

%(1)(0, 0) =

4

" + $

$+

(" + $)"

µ&

+ 1

0

exp( "µ$ s)

$ + "sds

5$1

(2.24)

%(1)(n, 0) =

.

"

" + $

/n

%(1)(0, 0), n $ 1 (2.25)

%(1)(n, 1) =

.

"

" + $

/n 1

n!

n$

k=1

.

" + $

µ&

/k

(n ! k)!%(1)(0, 0), n $ 1. (2.26)


Proof. We use (1.19) and we obtain that

(1 ! q) 3!2

.

!"

$(1 ! z), q, 0;!

"

$q(1 ! z),!

"

µq(1 ! z); q, q

/

=(!"#(1 ! z), q, 0; q)#

(q,!"#q(1 ! z),!"µq(1 ! z); q)#

+ 1

0

(qt,!"#q(1 ! z)t,!"µq(1 ! z)t; q)#

(!"#(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) 3!2

.

!"

$(1 ! z), q, 0;!

"

$q(1 ! z),!

"

µq(1 ! z); q, q

/

=1 + "

#(1 ! z)

(!"µq(1 ! z); q)#

+ 1

0

(!"µq(1 ! z)t; q)#

1 + "#(1 ! z)t

dqt

= ($ + "(1 ! z))eq

.

!"

µq(1 ! z)

/+ 1

0

Eq("µq(1 ! z)t)

$ + "(1 ! z)tdqt. (2.28)

Replacing µ by µ$

1$q and using (1.8) we have

limq%1!

eq

.

!"

µq(1 ! z)

/

= limq%1! eq

2

! "µ$ q(1 ! z)(1 ! q)

3

= e$!

µ$ (1$z), (2.29)

limq%1!

Eq

.

"

µq(1 ! z)t

/

= limq%1! Eq

2

"µ$ q(1 ! z)t(1 ! q)

3

= e!

µ$ (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

limq%1!

(1 ! q) 3!2

.

!"

$(1 ! z), q, 0;!

"

$q(1 ! z),!

"

µq(1 ! z); q, q

/

= ($ + "(1 ! z))e$!

µ$ (1$z)+ 1

0

e!

µ$ (1$z)t

$ + "(1 ! z)tdt

= ($ + "(1 ! z))e$!

µ$ (1$z)+ 1$z

0

e!

µ$ s

$ + "s

1

1 ! zds. (2.31)

For z = 0 we have in particular

limq%1!

(1 ! q) 3!2

.

!"

$, q, 0;!

"

$q,!

"

µq; q, q

/

= ($ + ")e$!

µ$

+ 1

0

e!

µ$ s

$ + "sds. (2.32)


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

#(1)1 (z) = lim

q%1!#1(z) =

0

1 !" + $

$%(1)(0, 0)

1

e$!

µ$ (1$z)

!"(" + $)

µ&e$

!µ$ (1$z)

+ 1$z

0

e!

µ$ s

$ + "sds %(1)(0, 0). (2.33)

We di"erentiate (2.33) with respect to z and we obtain

d

dz#(1)

1 (z) ="

µ&#(1)

1 (z) +"(" + $)

µ&($ + "(1 ! z))%(1)(0, 0).

We expand ddz#

(1)1 (z), #(1)

1 (z) and "("+#)µ$(#+"(1$z)) in power series and we equate the

coe!cients of zn. We obtain the stable recursive scheme

%(1)(1, 1) ="

µ&%(1)(0, 0) (2.34)

%(1)(n, 1) ="

nµ&%(1)(n ! 1, 1) +

"

nµ&

.

"

" + $

/n$1

%(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 ", e!ective service rate per customer

µ& and setup rate $, the equilibrium state distribution %(2)(n, i) = limq%0+ %(n, i;",µ$

1$q , 1 ! q,$) in the limiting case of full synchronization is given by

%(2)(0, 0) =$µ&

$µ& + "µ& + "$(2.36)

%(2)(n, 0) =

.

"

" + $

/n

%(2)(0, 0), n $ 1 (2.37)

%(2)(n, 1) =

'

(

)

µ$

"+µ$ n2

""+µ$

3n%(2)(0, 0), if µ& = $

#µ$$#

22

""+#

3n!2

""+µ$

3n3

%(2)(0, 0), if µ& (= $, n $ 1..(2.38)

Proof. We take the limit as q " 0+ in (2.4), using (1.21) and (1.22). This yields

%(2)(0, 0) =

0

" + $

$+

"

" + µ&

.

1 +"

µ&

/1$1

, (2.39)


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

#(2)1 (z) = lim

q%0+#1(z)

=

0

1 !" + $

$%(2)(0, 0)

1

1

1 + "µ$ (1 ! z)

!"(" + $)(1 ! z)

(" + $! "z)(" + µ& ! "z)%(2)(0, 0)

=$"z

(" + µ& ! "z)(" + $! "z)%(2)(0, 0). (2.40)

Expanding #(2)1 (z), ("+µ&!"z)$1 and ("+$!"z)$1 in power series and equating

the coe!cients of zn yields

%(2)(n, 1) = $"nn$

k=1

.

1

" + µ&

/k . 1

" + $

/n$k+1

%(2)(0, 0), n $ 1. (2.41)

By computing the geometric sum in (2.41) for the two cases µ& = $ and µ& (= $ we

obtain the two branches of (2.38). !

The last limiting regime that we consider is for fixed ", µ and q, when $ " #.

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 ", service rate µ and service repeat

probability q, the equilibrium state distribution %(3)(n, i) = lim#%# %(n, i;", µ, 1 !q,$) in the limiting case of no deactivation of the server is given by

%(3)(0, 0) = eq

.

!"

µ

/

(2.42)

%(3)(n, 0) = 0, n $ 1 (2.43)

%(3)(n, 1) =#$

k=n

(!1)n2

!"µ3k,kn

-

(q; q)k, n $ 1. (2.44)


Proof. Taking limit as $ " # in (2.4) implies that

%(3)(0, 0) =

0

1 +"

" + µEq

.

"

µ

/

3!2

.

0, q, 0; 0,!"

µq; q, q

/1$1

. (2.45)

However, 3!2

2

0, q, 0; 0,!"µ q; q, q3

is simplified to 2!1

2

q, 0;!"µq; q, q3

. Setting a = 0

in (2.20), we obtain

2!1(q, 0; bq; q, q) =1 ! b

b(eq(b) ! 1). (2.46)

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

%(3)(0, 0) =

4

1 +"

" + µEq

.

"

µ

/

1 + "µ

!"µ

.

eq

.

!"

µ

/

! 1

/

5$1

, (2.47)

which gives (2.42) after some simplifications.

Taking $ " # in (2.5) and using (2.47) yields

#(3)0 (z) = %(3)(0, 0) = eq

.

!"

µ

/

, (2.48)

so we have immediately (2.43). We now take $ " # in (2.6) and we obtain

#(3)1 (z) =

%

1 ! %(3)(0, 0)&

eq

.

!"

µ(1 ! z)

/

!"(1 ! z)

" + µ ! "z%(3)(0, 0) 3!2

.

0, q, 0; 0,!"

µq(1 ! z); q, q

/

. (2.49)

But by (2.46), we have

3!2

.

0, q, 0; 0,!"

µq(1 ! z); q, q

/

= 2!1

.

q, 0;!"

µq(1 ! z); q, q

/

=1 + "(1$z)

µ

!"(1$z)µ

.

eq

.

!"

µ

/

! 1

/

. (2.50)

Plugging (2.50) in (2.49) results after some simplification to

#(3)1 (z) = eq

.

!"

µ(1 ! z)

/

! eq

.

!"

µ

/

. (2.51)


Expanding (2.51) in power series of (1 ! z) using (1.6) yields

#(3)1 (z) =

#$

k=0

2

!"µ3k

(q; q)k

k$

n=0

.

k

n

/

(!z)n ! eq

.

!"

µ

/

=#$

n=1

#$

k=n

2

!"µ3k

(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 &(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 &(n,i)(s) satisfy the system

&(0,0)(s) = 1 (2.52)

&(n,0)(s) ="

" + $ + s&(n+1,0)(s) +

$

" + $ + s&(n,1)(s), n $ 1 (2.53)

&(n,1)(s) ="

" + µ + s&(n+1,1)(s) +

µpn

" + µ + s

+µ

" + µ + s

n$

j=1

.

n

j

/

pn$jqj&(j,1)(s), n $ 1. (2.54)

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

$i(s, z) =#$

n=i

&(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.

2.4 Busy period and sojourn time distributions 31

Theorem 2.6. The Laplace-Stieltjes transforms &(1,1)(s), &(1,0)(s) and the mixed

transforms $1(s, z), $0(s, z) are given by the equations

&(1,1)(s) =µ(1 ! q) 1!1(! "

µ+s ;!"q2

µ+s ; q,µq2

µ+s)

("q + µ + s) 1!1(! "µ+s ;!

"qµ+s ; q,

µqµ+s)

(2.56)

$1(s, z) =µ(1 ! q)z2

2!2

2

q,! z1$z ; (µ+s)zq

"(1$z) ,! zq2

1$z ; q, µzq2

"(1$z)

3

((" + µ + s)z ! ")(1 ! (1 ! q)z)

!"z&(1,1)(s) 2!2

2

q,! z1$z ; (µ+s)zq

"(1$z) ,! zq1$z ; q, µzq

"(1$z)

3

(" + µ + s)z ! "(2.57)

&(1,0)(s) =$

"$1

.

s,"

" + $ + s

/

(2.58)

$0(s, z) = 1 +$z

(" + $ + s)z ! "

.

$1 (s, z) ! $1

.

s,"

" + $ + 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) = qz1$(1$q)z 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

"(1 ! z)

/k q(k2)

6ki=1(1 ! (µ+s)z

"(1$z)qi)

. (2.62)

Multiplying (2.54) with (" + µ + s)zn and adding for all n $ 1 results after some

manipulations to

[(" + µ + s)z ! "]$1(s, z) =µpz2

1 ! pz! "z&(1,1)(s) +

µz

1 ! pz$1(s, T (z)). (2.63)


By iterating (2.63) we obtain

[(" + µ + s)z ! "]$1(s, z) =µpz2

1 ! pz! "z&(1,1)(s)

+n$

k=1

4

k*

i=1

µTi(z)

q[(" + µ + s)Ti(z) ! "]

5

0

µpT 2k (z)

1 ! pTk(z)

1

!n$

k=1

4

k*

i=1

µTi(z)

q[(" + µ + s)Ti(z) ! "]

5

"Tk(z)&(1,1)(s)

+n+1*

i=1

µTi(z)

q[(" + µ + s)Ti(z) ! "][(" + µ + s)z ! "]$1(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

[(" + µ + s)z ! "]$1(s, z) =µpz2

1 ! pz! "z&(1,1)(s)

+#$

k=1

4

k*

i=1

µTi(z)

q[(" + µ + s)Ti(z) ! "]

5

0

µpT 2k (z)

1 ! pTk(z)

1

!#$

k=1

4

k*

i=1

µTi(z)

q[(" + µ + s)Ti(z) ! "]

5

"Tk(z)&(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)

(" + µ + s)Ti(s) ! "=

(!1)qiz

"(1 ! z)(1 ! (µ+s)z"(1$z)q

i), i $ 1

sok*

i=1

µTi(z)

q[(" + µ + s)Ti(s) ! "]= ak(z). (2.66)

We plug (2.66) into (2.65) and we obtain

[(" + µ + s)z ! "]$1(s, z) =#$

k=0

ak(z)

0

µpT 2k (z)

1 ! pTk(z)

1

!#$

k=0

ak(z)"Tk(z)&(1,1)(s). (2.67)

2.4 Busy period and sojourn time distributions 33

By setting z = ""+µ+s in equation (2.67) we obtain

&(1,1)(s) =

!#k=0 ak

2

""+µ+s

3

4

µpT 2k

“

!!+µ+s

”

1$pTk

“

!!+µ+s

”

5

!#k=0 ak

2

""+µ+s

3

"Tk

2

""+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 !-notation. Then (2.68) yields (2.56).

We then solve (2.67) for $1(s, z) and express the sums involved in the standard

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

We now multiply the equations (2.52) and (2.53) with z0 and zn respectively and

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

[(" + $ + s)z ! "]$0(s, z) = (" + $ + s)z ! "! z"&(1,0)(s) + $z$1(s, z). (2.69)

We set z = ""+#+s in (2.69) and solve for &(1,0)(s) so we obtain (2.58). Solving (2.69)

for $0(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 $ (in the case where

µ(1!q) (= $) or a mixture of an exponential distribution and an Erlang-2 distribution

with parameter $ (in the case where µ(1 ! q) = $). 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$#x), if $ (= µ(1 ! q)

(1 ! p2)(1 ! e$#x) + p2(1 ! e$#x ! xe$#x), if $ = µ(1 ! q), x $ 0,(2.70)

where

p1 =(" + $)µ(1 ! q)

$(µ(1 ! q) ! $)%(0, 0), p2 =

" + $

$%(0, 0). (2.71)


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 #1(1)

given by (2.11) while the probability that the server is deactivated is #0(1). Then

the sojourn time S has the representation

S =

7

Y +!J

j=1 Xj , with probability "+## %(0, 0)

!Jj=1 Xj, with probability 1! "+#

# %(0, 0),(2.72)

where Y and X1,X2, . . . are independent exponentially distributed random variables

with rates $ 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) =" + $

$%(0, 0)

$

$ + s

µ(1 ! q)

µ(1 ! q) + s+

.

1 !" + $

$%(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)

µ(1$q)+s + p1##+s , if $ (= µ(1 ! q)

(1 ! p2)##+s + p2

2

##+s

32, if $ = 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 e"ect of each single parameter

p, $ and " 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 theorem 2.2. In figure 2.2 we present the graph of E[N ] with respect to p

for " = 3.5 and $ = 1.5. The function is decreasing convex as p varies from 0 to 1.

2.5 Numerical results 35

0 0.2 0 .4 0 .6 0 .8 10

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 ;", µ, p,$] versus p.

0 2 4 6 8 10 123.5

3 .55

3.6

3 .65

3.7

3 .75

3.8

θ

E[N

;θ]

λ=3.5

µ=1

p=0.5

Fig. 2.3: E[N ;", µ, p,$] versus $.

0 0.5 1 1.5 20

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 ;", µ, p,$] versus ".

Figure 2.3 shows the influence of the setup rate $ in the mean number of customers

in the system, when " = 3.5 and p = 0.5. The function is decreasing convex with

a horizontal asymptote as $ " #. This agrees with our intuitive expectation and

the result of theorem 2.5. Figure 2.4 shows the behavior of E[N ] as the arrival rate

" varies, when $ = 1.5 and p = 0.5. As expected, the mean number of customers in

the system increases when " increases.

The next figures 2.5 and 2.6 demonstrate the e"ect of the level of synchronization

in the mean and the variance of the number of customers in the system. In these


0 0.2 0 .4 0 .6 0 .8 13.6

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 ;", µp$, p,$] versus p.

0 0.2 0 .4 0 .6 0 .8 12

4

6

8

10

12

14

16

18

20

22

p

Var

[N;p

]

λ=3.5

θ=1.5

µp=1

Fig. 2.6: V ar[N ;", µp$, p,$] versus p.

two figures, the arrival rate ", the e"ective service rate per customer µ$ = µp and

the setup rate $ are kept fixed, " = 3.5, $ = 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 70

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=0.65

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,


&(1,0)(s), given in theorem 2.6. Although there is no symbolic inversion of &(1,0)(s),

the use of the r!s notation 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 " = 3.5, $ = 1.5, µp = 1 and we consider three cases

for p: 0.4, 0.65 and 0.9. Again, the numerical findings in figure 2.7 are in absolute

accordance with the fact that the mean number of customers increases as the level

of synchronization increases.

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 o" periods, where customers’ impatience is due

to the absence of the server(s). Such systems can model satisfactorily the reneging

40 Synchronized abandonments in a single server unreliable queue

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 o" 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 di"erential

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

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 ', 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 ',nn#

-

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.


3.2 Model description and notation

We consider an M/M/1 queueing system in which customers arrive according to a

Poisson process at rate ". The service is provided by a single server, who serves the

customers on a FCFS basis. The successive service times are independent exponen-

tially distributed random variables with rate µ. The system is subject to failures

that occur when the server is at a functioning state, according to a Poisson process

at rate (. At a failure epoch, the server is turned o" and all present customers are

forced to leave the system. Then, the repair process starts immediately. The repair

times are exponentially distributed random variables with rate ). While the server

is o", the stream of new arrivals continues. However, since the server is down, these

new customers become impatient and perform synchronized abandonments in the

following way: A transportation facility is set on and it arrives at the system accord-

ing to a Poisson process at rate '. Every arrival epoch of the transportation facility

constitutes an abandonment opportunity for the present customers. We suppose

that each one of them decides to abandon the system with probability p or remains

in the system with probability q, independently of the others.

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=o" (under repair) and 1=on (functioning)), t $ 0.

The corresponding transition rate diagram is given in figure 3.1.

Let (%(n, i) : n $ 0 and i = 0, 1) denote the equilibrium distribution of

{(N(t), I(t))}. We also define the partial probability generating functions #0(z)

and #1(z) by

#0(z) =#$

n=0

%(n, 0)zn and #1(z) =#$

n=0

%(n, 1)zn .

In section 3.3 we will determine #0(z) and #1(z) in terms of q–hypergeometric

series. Moreover, in sections 3.4 and 3.5 we will also see that the study of the sojourn

times and the busy period distribution of this system is also facilitated using the

theory of q–hypergeometric series.


0, 1" ##

%

..

1, 1

%!!!

!

//!!!!!!!

!!!!!!

µ00

" ##2, 1

%"""""""""

11""""""""""""""""""""""""

" ##

µ00 3, 1

%##############

22####################################

" ##

µ00 · · ·

0, 0

&

--

" ## 1, 0

&

--

(10)p'

""" ## 2, 0

&

--

(21)pq'

""

(20)p

2'

33" ## 3, 0

&

--

(32)pq2'

""

(31)p

2q'

33

(30)p3'

33" ## · · ·

Fig. 3.1: Transition rate diagram of {(N(t), I(t))}.

3.3 The equilibrium state distribution

The balance equations of the model are given as follows:

(" + ) + ')%(0, 0) = (#$

j=0

%(j, 1) + '#$

j=0

pj%(j, 0) (3.1)

(" + ) + ')%(n, 0) = "%(n ! 1, 0) + '#$

j=n

.

j

n

/

pj$nqn%(j, 0), n $ 1 (3.2)

(" + ()%(0, 1) = µ%(1, 1) + )%(0, 0) (3.3)

(" + µ + ()%(n, 1) = "%(n ! 1, 1) + µ%(n + 1, 1) + )%(n, 0), n $ 1. (3.4)

These equations can be solved e!ciently by employing generating function methods

and we obtain the following.

Theorem 3.1. The partial probability generating functions #0(z) and #1(z) are

given by

#0(z) =()

() + ()("(1 ! z) + ) + ')2!1

.

0, q;!"q(1 ! z)

) + '; q,

'

) + '

/

,

|z| < 1 +) + '

"(3.5)


#1(z) =)(1 ! z0)z#0(z) ! )(1 ! z)z0#0(z0)

((" + µ + ()z ! "z2 ! µ)(1 ! z0),

|z| < min

.

1 +) + '

", z1

/

, (3.6)

where

z0 =" + µ + ( !

8

(" + µ + ()2 ! 4"µ

2"% (0, 1), (3.7)

z1 =" + µ + ( +

8

(" + µ + ()2 ! 4"µ

2"% (1,#). (3.8)

The convergence of the series is absolute in the corresponding open disks and uniform

in every compact subset of them.

Proof. Summing equations (3.1) and (3.2) over all n = 0, 1, . . . we obtain that

)#0(1) = (#1(1). (3.9)

Equation (3.9) together with the normalization equation #0(1) + #1(1) = 1 yields

that

#0(1) =(

) + (, #1(1) =

)

) + (. (3.10)

Multiplying equations (3.1) and (3.2) by z0 and zn respectively and summing for all

n = 0, 1, . . ., we obtain that

(" + ) + ')#0(z) = "z#0(z) + '#$

n=0

#$

j=n

.

j

n

/

pj$nqn%(j, 0)zn + (#$

j=0

%(j, 1)

= "z#0(z) + '#$

j=0

%(j, 0)(p + qz)j + (#1(1)

= "z#0(z) + '#0(p + qz) + (#1(1).

Hence

#0(z) ='

"(1 ! z) + ) + '#0(1 ! q + qz) +

(#1(1)

"(1 ! z) + ) + '. (3.11)


Iterating equation (3.11) yields

#0(z) = #0(1 ! qn+1 + qn+1z)n*

i=0

'

"qi(1 ! z) + ) + '

+(#1(1)n$

k=0

1

"qk(1 ! z) + ) + '

k$1*

i=0

'

"qi(1 ! z) + ) + ', (3.12)

where we assume that for k = 0 the empty product6k$1

i=0 is by definition equal to

1. Taking the limit as n " # in (3.12) we obtain

#0(z) = (#1(1)#$

k=0

1

"qk(1 ! z) + ) + '

k$1*

i=0

'

"qi(1 ! z) + ) + ', (3.13)

whenever the series converges.

Expressing equation (3.13) in terms of the canonical form of q–series we obtain that

#0(z) =(

"(1 ! z) + ) + '#1(1) 2!1

.

0, q;!"q(1 ! z)

) + '; q,

'

) + '

/

. (3.14)

Then by substituting #1(1) from (3.10) yields equation (3.5). The absolute conver-

gence of the series (3.5) is guaranteed for z % {z % C : |z| < 1 + &+'" }. Indeed,

we first observe that the 2!1

2

0, q;!"q(1$z)&+' ; q, '

&+'

3

series converges absolutely (see

the comments after the definition (1.2)) for all z such that !"q(1$z)&+' (= q$m, for

m = 0, 1, 2, . . ., consequently it converges for z with |z| < 1 + &+'"q . Moreover, the

denominator in (3.5) does not vanish for z (= 1 + &+'" . Hence the partial probability

generating function #0(z) as given in equation (3.5) converges for |z| < 1 + &+'" .

Multiplying equations (3.3) and (3.4) by z0 and zn respectively and summing for

all n = 0, 1, . . . we obtain

(" + µ + ()#1(z) ! µ%(0, 1) = "z#1(z) +µ

z(#1(z) ! %(0, 1)) + )#0(z),

or equivalently

((" + µ + ()z ! "z2 ! µ)#1(z) = !µ(1 ! z)%(0, 1) + )z#0(z). (3.15)


Equation (3.15) is identical to Yechiali (2007) equation (2.6). We observe that the

quadratic polynomial

f(z) = (" + µ + ()z ! "z2 ! µ (3.16)

has two real roots z0 and z1 given by (3.7) and (3.8). Setting z = z0 in equation

(3.15) (note that #1(z0) converges since z0 < 1) yields

%(0, 1) =)z0#0(z0)

µ(1 ! z0). (3.17)

Substituting (3.17) into (3.15) and solving for #1(z) yields (3.6). Having established

the radius of convergence for #0(z), it can be easily checked that the partial prob-

ability generating function #1(z) given by (3.6) converges for z % {z % C : |z| <

min(1 + &+'" , z1)}. !

We can now use (3.5)-(3.6) to derive explicit expressions for some important

performance measures of the system. More specifically, we will derive the factorial

moments of the number of customers in the system and the stationary probabilities

in some limiting regimes.

We can now proceed to the calculation of the moments for the distribution of

the number of customers in the system. Note that the moments of all orders exist

since the partial probability generating functions #0(z) and #1(z) converge inside

an open disk with radius of convergence strictly greater than 1.

Remark 3.1. Di!erentiating (3.11) n times and setting z = 1 yields

#(n)0 (1) =

n"

) + '(1 ! qn)#(n$1)

0 (1), n $ 1. (3.18)

Di!erentiating (3.15) n times and setting z = 1 yields

)#(1)0 (1) + )#(0)

0 (1) + µ%(0, 1) = (#(1)1 (1) + (µ + ( ! ")#(0)

1 (1)

)#(n)0 (1) + n)#(n$1)

0 (1) = (#(n)1 (1) + n(µ + ( ! ")#(n$1)

1 (1)

!n(n ! 1)"#(n$2)1 (1), n $ 2. (3.19)


Equations (3.18) and (3.19) form an iterative scheme with initial conditions for

#(0)0 (1) = #0(1) and #(0)

1 (1) = #1(1) given by (3.10). The first order scheme given

by (3.18) gives easily #(n)0 (1) in product closed form for any desired n. However, the

second order scheme given by (3.19) has coe"cients and constant term that depend

on n, so its direct solution seems impossible. Nevertheless, a closed form for #(n)1 (1)

can be obtained by applying results from the theory of q–series, in particular using

Heine’s transformation formulas (1.11)-(1.13).

We have the following theorem.

Theorem 3.2. The factorial moments m(n) = E[N(N ! 1)(N ! 2) · · · (N ! n + 1)]


m(n) =)

) + (

n!

z1 ! z0

n$

k=0

"n

(k6n$k

i=1 () + '(1 ! qi))

%

z1(1 ! z0)k+1 ! z0(1 ! z1)

k+1&

+)z0#0(z0)

1 ! z0

n!("/()n$1

((z1 ! z0)[(1 ! z0)

n ! (1 ! z1)n]

+(

) + (

n!"n

6ni=1() + '(1 ! qi))

, n $ 1, (3.20)

where z0 and z1 are given in equations (3.7) and (3.8) respectively. In particular

E[N ] =)

(

z0#0(z0)

1 ! z0+

)

) + (

"! µ

(+

"

) + '(1 ! q). (3.21)

Proof. The factorial moments exponential generating function P (z) is given by

P (z) =#$

n=0

m(n)zn

n!= E

4

#$

n=0

.

N

n

/

zn

5

= E[(1+z)N ] = #0(1+z)+#1(1+z). (3.22)

We have already shown in theorem 3.1 that #0(z) and #1(z) converge in a neigh-

borhood of 1, hence P (z) is well defined in a neighborhood of 0. Using Heine’s

transformation formula (1.12), equation (3.5) assumes the form

#0(z) =(

) + ( 2!1

.

0, q;'q

) + '; q,!

"(1 ! z)

) + '

/

=(

) + (

#$

n=0

"n

6ni=1() + '(1 ! qi))

(z ! 1)n,


that gives

#0(1 + z) =(

) + (

#$

n=0

"n

6ni=1() + '(1 ! qi))

zn. (3.23)

Moreover from equation (3.6) we obtain that

#1(1 + z) = )(1 + z)#0(1 + z)

f(1 + z)+ )

z0#0(z0)

1 ! z0

z

f(1 + z), (3.24)

where f(z) = !"(z0 ! z)(z1 ! z) is given in equation (3.16). By partial fraction

expansion and elementary algebra we obtain that

1

f(1 + z)=

1

"(z1 ! z0)(1 ! z0)

1

1 + z1$z0

+1

"(z1 ! z0)(z1 ! 1)

1

1 ! zz1$1

=#$

n=0

1

"(z1 ! z0)

0

(!1)n

(1 ! z0)n+1+

1

(z1 ! 1)n+1

1

zn

=#$

n=0

("/()n

((z1 ! z0)

9

(!1)n(z1 ! 1)n+1 + (1 ! z0)n+1:

zn, (3.25)

and then easily

1 + z

f(1 + z)=

#$

n=0

("/()n

((z1 ! z0)

9

(!1)nz0(z1 ! 1)n+1 + z1(1 ! z0)n+1:

zn. (3.26)

Now (3.23) and (3.26) imply easily that

(1 + z)#0(1 + z)

f(1 + z)=

1

() + ()(z1 ! z0)

#$

n=0

n$

k=0

"n

(k6n$k

i=1 () + '(1 ! qi))

)%

(!1)kz0(z1 ! 1)k+1 + z1(1 ! z0)k+1&

zn , (3.27)

where z0 and z1 are given in equations (3.7) and (3.8), respectively. Plugging (3.25)

and (3.27) into (3.24) and using (3.23), we can expand (3.22) in powers of z. After

some straightforward algebra we obtain (3.20) and in particular, for n = 1, we de-

duce (3.21). !

In the model we have two types of lost customers, those that abandon the system

due to impatience during the repair phase of the server and those that are forced


to leave the system at the failure epochs of the server. We will now calculate the

proportion of customers who leave the system at the failure epochs, the proportion

of customers who abandon the system because of impatience and the proportion of

served customers. We begin by computing the corresponding rates.

When the system is in state (n, 1), n $ 0, the failure rate of the server is ( and

when a failure occurs all present customers leave the system, therefore the rate of

lost customers due to failures, Rfailures, is given by

Rfailures =#$

n=0

(n%(n, 1) = (#"1(1). (3.28)

Similarly, the rate of served customers, Rserved, is given by

Rserved =#$

n=1

µ%(n, 1) = µ(#1(1) ! %(0, 1)). (3.29)

Finally, the rate of abandonments due to impatience when the server is down,

Rabandonments, is given by

Rabandonments = "! Rfailures ! Rserved = '(1 ! q)#"0(1). (3.30)

Clearly, #"0(1) can be directly obtained from equation (3.18), that is #"

0(1) =%&+%

"&+'(1$q) , while #"

1(1) can be obtained as #"1(1) = E[N ] ! #"

0(1) = &z0!0(z0)%(1$z0) +

"$µ%

&&+% + &

&+%"

&+'(1$q) , since the mean number of customers in the system, E[N ], is

given in equation (3.21). The fractions of lost customers due to failures, served cus-

tomers and lost customers due to abandonments are readily computed by dividing

(3.28)-(3.30) by the overall rate ".

We now turn our attention to the behavior of the model under certain limiting

regimes. To emphasize the dependence on the parameters of the model in the rest

of this section, we will denote %(n, i), #0(z) and #1(z) by %(n, i;", µ, ', p, ), (),

#0(z;", µ, ', p, ), () and #1(z;", µ, ', p, ), (), respectively. Note that 'p can be

thought of as the e"ective abandonment rate per customer. Indeed the overall

abandonment time of a customer is a geometric sum of exponentially distributed


random variables with rate ' and so we can easily see that it is also exponentially

distributed with parameter 'p. Under this perspective, if we have two models with

the same parameters ", µ, ) and ( that di"er only in ' and p, but with 'p = '& fixed,

we can think that the models have identical arrival rates ", service rates µ, e"ec-

tive abandonment rates per customer '&, setup rates ) and catastrophe rates ( and

di"er only in the ‘level of synchronization’ p. Indeed, the case p " 0+ corresponds

to no synchronization since the customers depart almost singly at the abandonment

epochs. On the contrary, the case p " 1$ corresponds to full synchronization since

almost all present customers depart simultaneously from the system when an aban-

donment opportunity occurs.


where ", µ, '&, ) and ( are kept fixed in the two limiting cases p " 0+ (q " 1$)

and p " 1$ (q " 0+). The case p " 0+ corresponds exactly to the model studied

by Yechiali (2007) where the customers perform independent abandonments. In

theorem 3.3 we derive the equilibrium state probability generating functions while

in theorem 3.4 we obtain the factorial moments of the state distribution.

Theorem 3.3. For a system with arrival rate ", service rate µ, e!ective aban-

donment rate per customer '&, setup rate ) and catastrophe rate (, the generating

functions

#(1)i (z) = lim

q%1!#i(n, i;", µ,

'&

1 ! q, 1 ! q, ), (), i = 0, 1,

in the limiting case of no synchronization are given by

#(1)0 (z) =

()

( + )

1

'&

+ 1

z

(1 ! t)"#$$1

(1 ! z)"#$

e$!#$

(t$z)dt (3.31)

#(1)1 (z) =

)(1 ! z0)z#(1)0 (z) ! )(1 ! z)z0#

(1)0 (z0)

((" + µ + ()z ! "z2 ! µ)(1 ! z0), (3.32)

where z0 ="+µ+%$

*("+µ+%)2$4"µ2" .

Proof. Using Heine’s transformation formula (1.11), equation (3.5) assumes the


following form

#0(z) =(

) + (

)

) + '

(q; q)#

(!"(1$z)&+' , '

&+' ; q)#2!1

.

!"(1 ! z)

) + ',

'

) + '; 0; q, q

/

.

(3.33)

We use (1.19) and we obtain that

(1 ! q) 2!1

.

!"(1 ! z)

) + ',

'

) + '; 0; q, q

/

=(!"(1$z)

&+' , '&+' ; q)#

(q; q)#

+ 1

0

(qt; q)#

(!"(1$z)t&+' , 't&+' ; q)#

dqt. (3.34)

Plugging (3.34) into (3.33) and simplifying several terms we have that (3.33) assumes

the form

#0(z) =(

) + (

)

)(1 ! q) + '&

+ 1

0

(qt; q)#

(!"(1$z)t&+' , 't&+' ; q)#

dqt

=(

) + (

)

)(1 ! q) + '&

+ 1

0

(qt; q)#

( 't&+' ; q)#eq(!

"(1 ! z)t

) + ')dqt. (3.35)

Replacing ' by '$

1$q and using (1.8) and (1.10) we have

limq%1!

eq

.

!"(1 ! z)t

) + '

/

= limq%1! eq

2

! "(1$z)t&(1$q)+'$ (1 ! q)

3

= e$!#$ (1$z)t. (3.36)

limq%1!

(qt; q)#

( 't&+' ; q)#= limq%1!

(q(1+ "(1!q)#$ ) #$t

"(1!q)+#$ ;q)"

( #$t"(1!q)+#$ ;q)"

= (1 ! t)"#$ $1. (3.37)

Taking the limit as q " 1$ in (3.35), taking into account equations (3.36) and (3.37)

and setting t = s$z1$z yields (3.31), which is Yechiali (2007) formula (2.7). Equation

(3.32) results immediately from (3.6). !

Theorem 3.4. The factorial moments m(1)(n) = E[N(N ! 1)(N ! 2) · · · (N ! n + 1)]



m(1)(n) =

)

) + (

n!

z1 ! z0

n$

k=0

"n

(k6n$k

i=1 () + '&i)

%

z1(1 ! z0)k+1 ! z0(1 ! z1)

k+1&

+)z0#

(1)0 (z0)

1 ! z0

n!("/()n$1

((z1 ! z0)[(1 ! z0)

n ! (1 ! z1)n]

+(

) + (

n!"n

6ni=1() + '&i)

, n $ 1, (3.38)

where z0 and z1 are given in equations (3.7) and (3.8) respectively and #(1)0 (z0) is

given by (3.31). In particular

E(1)[N ] =)

(

z0#(1)0 (z0)

1 ! z0+

)

) + (

"! µ

(+

"

) + '&. (3.39)

Proof. We replace ' by '&/(1 ! q) in (3.20) and (3.21) and we take the limit as

q " 1$. We obtain (3.38) and (3.39) respectively. !

The expression (3.39) gives the mean number of customers in Yechiali (2007)

model. Indeed, Yechiali (2007) equations (2.10) and (2.13) can be used to derive

independently our equation (3.39).

We now derive the corresponding results for the other extreme case of full syn-

chronization, i.e. when p " 1$. In theorem 3.5 we derive the equilibrium state

probability generating functions while in theorem 3.6 we obtain the factorial mo-

ments of the state distribution.

Theorem 3.5. For a system with arrival rate ", service rate µ, e!ective abandon-

ment rate per customer '&, setup rate ) and catastrophe rate (, the equilibrium state

distribution %(2)(n, i) = limq%0+ %(n, i;", µ, '$

1$q , 1 ! q, ), () in the limiting case of

full synchronization is given by

%(2)(n, 0) = c1

.

"

) + '& + "

/n

, n $ 0 (3.40)

%(2)(n, 1) =

'

(

)

c2[n(1 ! z0) + 1]2

""+&+'$

3n, when z1 = "+&+'$

"

c2

%

c3

2

1z1

3n+ c4

2

""+&+'$

3n&

, when z1 (= "+&+'$

" , n $ 0,(3.41)


where

c1 =(

) + (

) + '&

) + '& + ",

c2 =()

) + (

z0

µ(1 ! z0)

) + '&

"(1 ! z0) + ) + '&,

c3 =" + ) + '& ! µ

" + ) + '& ! "z1,

c4 =µ(1 ! z0)

µ ! z0(" + ) + '&),

and z0, z1 are given in equations (3.7) and (3.8), respectively.

Proof. We take the limit as q " 0+ in (3.33), using (1.20) and (1.22). This yields

#(2)0 (z) = lim

q%0+

(

) + (

)

) + '

(q; q)#

(!"(1$z)&+' , '

&+' ; q)#2!1

.

!"(1 ! z)

) + ',

'

) + '; 0; q, q

/

=(

) + (

)

) + '&1

(1 + "(1$z)&+'$ )(1 ! '$

&+'$ )

=(

) + (

) + '&

) + '& + "! "z. (3.42)

Expanding () + '& + " ! "z)$1 in power series and equating the coe!cients of zn

yields (3.40). Taking the limit as q " 0$ in equation (3.6) and plugging equation

(3.42) yields after some calculations

#(2)1 (z) =

()

) + (

) + '&

" + ) + '&z0

µ(1 ! z0)(1 ! ""+&+'$ z0)

1 ! "z0"+&+'$ z

(1 ! 1z1

z)(1 ! ""+&+'$ z)

.

(3.43)

Using partial fraction expansion, we express #(2)1 (z) in power series and equating

the coe!cients of zn for the two cases z1 = "+&+'$

" and z1 (= "+&+'$

" we obtain the

two branches of (3.41). !

Theorem 3.6. The factorial moments m(2)(n) = E[N(N ! 1)(N ! 2) · · · (N ! n + 1)]



m(2)(n) =

)

) + (

n!

z1 ! z0

n$

k=0

"n

(k() + '&)n$k

%

z1(1 ! z0)k+1 ! z0(1 ! z1)

k+1&

+)z0#

(2)0 (z0)

1 ! z0

n!("/()n$1

((z1 ! z0)[(1 ! z0)

n ! (1 ! z1)n]

+(

) + (

n!"n

() + '&)n, n $ 1, (3.44)

where z0 and z1 are given in equations (3.7) and (3.8) respectively and #(2)0 (z0) is

given by (3.42). In particular

E(2)[N ] =)

(

z0#(2)0 (z0)

1 ! z0+

)

) + (

"! µ

(+

"

) + '&. (3.45)

Proof. We replace ' by '&/(1 ! q) in (3.20) and (3.21) and we take the limit as

q " 0+. We obtain (3.44) and (3.45) respectively. !

3.4 Sojourn times

Let S denote the unconditional total sojourn time of an arbitrary customer in the

system, regardless of whether he completes service or not. Moreover, let S(n,i) de-

note the conditional total sojourn time of a tagged customer in the system, given

that upon arrival he finds the system in state (n, i).

We employ first-step analysis excluding arrivals, because future arrivals do not in-

fluence the tagged customer. Indeed, by conditioning on whether the next transition

is a service completion or a failure when the system is up we obtain the equations

E[S(0,1)] =1

µ + ((3.46)

E[S(n,1)] =1

µ + (+

µ

µ + (E[S(n$1,1)], n $ 1. (3.47)

Similarly, by conditioning on whether the next transition corresponds to a repair

3.4 Sojourn times 55

completion or to an abandonment opportunity, we obtain the equations

E[S(n,0)] =1

) + '+

)

) + 'E[S(n,1)] +

'q

) + '

n$

i=0

.

n

i

/

pn$iqiE[S(i,0)], n $ 0. (3.48)

The system of recursive relations (3.46)-(3.48) can be solved explicitly employing

a generating function approach and using the theory of q–hypergeometric series.

The solution is summarized in the following theorem.

Theorem 3.7. The conditional expected total sojourn time of a tagged customer in

the system S(n,i), given that upon his arrival he finds the system in state (n, i), is

given as follows

E[S(n,0)] =1

() + '(1 ! q))

) + (

(!

)µ

((µ + ()

n$

k=0

.

n

k

/

2

! %%+µ

3k

) + '(1 ! qk+1)(3.49)

E[S(n,1)] =1

(

"

1 !.

µ

µ + (

/n+1#

, n $ 0. (3.50)

Proof. Iterating equation (3.47), using (3.46), yields immediately (3.50). We now

define the generating functions of the mean conditional expected total sojourn times

Si(z), i = 0, 1, given as

Si(z) =#$

n=0

E[S(n,i)]zn, |z| < 1, i = 0, 1. (3.51)

The convergence of these series in the open unit disk will be proved below, as they

appear in the calculations. Indeed, multiplying equation (3.50) with zn and adding

for all n $ 0 results to

S1(z) =1

(

.

1

1 ! z!

µ

µ(1 ! z) + (

/

, (3.52)

which is readily seen to converge in the open unit disk. Regarding the series S0(z),

multiplying equations (3.48) with () + ')zn and adding for all n $ 0 yields

() + ')S0(z) =1

1 ! z+ )S1(z) + 'q

#$

n=0

n$

i=0

.

n

i

/

pn$iqiE[S(i,0)]zn

=1

1 ! z+ )S1(z) + 'q

#$

i=0

E[S(i,0)]

.

q

p

/i #$

n=i

.

n

i

/

(pz)n.(3.53)


Using now the ‘upper’ binomial coe!cients generating function

#$

n=i

.

n

i

/

xn =xi

(1 ! x)i+1, |x| < 1, (3.54)

we have that (3.53) assumes the form

() + ')S0(z) =1

1 ! z+ )S1(z) +

'q

1 ! (1 ! q)zS0(

qz

1 ! pz). (3.55)

We observe that equation (3.55) can be put in the form

S0(z) =H(z)

G(z)S0(T (z)) +

K(z)

G(z), (3.56)

where

T (z) =qz

1 ! (1 ! q)z(3.57)

and

G(z) = ) + ', H(z) = ' T (z)z , K(z) =

1

1 ! z+ )S1(z). (3.58)

The solution of (3.56) can be done by iteration. To this end it seems convenient

to introduce here an operator notation: The transformation T (z) defined by (3.57)

is a linear fractional transformation and therefore its k-th compositions defined by

T0(z) = z and Tk(z) = T (Tk$1(z)), k $ 1, can be computed in closed form. Indeed,

it can be proved inductively that

Tk(z) =qkz

1 ! (1 ! qk)z, k $ 0. (3.59)

By iterating (3.56) n times we obtain

S0(z) =n$

k=0

K(Tk(z))

H(Tk(z))

k*

i=0

H(Ti(z))

G(Ti(z))+ S0(Tn+1(z))

n*

i=0

H(Ti(z))

G(Ti(z)). (3.60)

However, note thatH(Ti(z))

G(Ti(z))=

'Ti+1(z)

() + ')Ti(z)

3.4 Sojourn times 57

andK(Tk(z))

H(Tk(z))=

.

1

1 ! Tk(z)+ )S1(Tk(z))

/

Tk(z)

'Tk+1(z)

so (3.60) assumes the form

S0(z) =n$

k=0

0

1

1 ! Tk(z)+ )S1(Tk(z))

1

Tk(z)

'Tk+1(z)

.

'

) + '

/k+1 Tk+1(z)

z

+S0(Tn+1(z))

.

'

) + '

/n+1 Tn+1(z)

z, (3.61)

and by taking the limit as n " # we obtain

S0(z) =#$

k=0

0

1

1 ! Tk(z)+ )S1(Tk(z))

1

Tk(z)

'Tk+1(z)

.

'

) + '

/k+1 Tk+1(z)

z. (3.62)

Observing that

1

1 ! Tk(z)=

1 ! (1 ! qk)z

1 ! z

S1(Tk(z)) =1 ! (1 ! qk)z

(

.

1

1 ! z!

µ

((1 ! (1 ! qk)z) + µ(1 ! z)

/

,

we have that (3.62) is written in the form

S0(z) =1

) + '

#$

k=0

0

1

1 ! z+

)

(

.

1

1 ! z!

µ

µ(1 ! z) + ((1 ! (1 ! qk)z)

/1.

'q

) + '

/k

=1

() + ')(1 ! z)

#$

k=0

4

) + (

(!

)µ

((µ + ()

1

1 + %%+µ

z1$z qk

5

.

'q

) + '

/k

=1

() + '(1 ! q))(1 ! z)

) + (

(!

1

() + ')(1 ! z)

)µ

((µ + ()

#$

k=0

2

'q&+'

3k

1 + %%+µ

z1$z qk

,

which can be put in the standard q–series notation as

S0(z) =1

() + '(1 ! q))(1 ! z)

) + (

(

!)µ

(() + ')(µ(1 ! z) + ()2!1

"

q,! %%+µ

z1$z

! %q%+µ

z1$z

; q,'q

) + '

#

, (3.63)


which is easily seen to converge in the open unit disk (indeed the values of z where the

denominators vanish are z = 1 and z = µ+%µ that lie outside the open unit disk while

the singularities of the 2!1 series for z such that ! %q%+µ

z1$z = q$m, m = 0, 1, 2, . . . lie

also outside the open unit disk). Using Heine’s transformation formula (1.12) yields

S0(z) =1

() + '(1 ! q))(1 ! z)

) + (

(

!)µ

(() + '(1 ! q))(µ + ()(1 ! z)2!1

"

q, 'q&+''q2

&+'

; q,!(

( + µ

z

1 ! z

#

and by expanding the q–series we obtain

S0(z) =1

() + '(1 ! q))(1 ! z)

) + (

(

!)µ

((µ + ()(1 ! z)

#$

k=0

2

! %%+µ

3k

) + '(1 ! qk+1)

.

z

1 ! z

/k

. (3.64)

Using (3.54) to expand the term zk

(1$z)k+1 in the right side of (3.64) in powers of z

yields

S0(z) =1

() + '(1 ! q))

) + (

(

#$

n=0

zn

!)µ

((µ + ()

#$

k=0

2

! %%+µ

3k

) + '(1 ! qk+1)

#$

n=k

.

n

k

/

zn

=#$

n=0

;

<

=

1

() + '(1 ! q))

) + (

(!

)µ

((µ + ()

n$

k=0

.

n

k

/

2

! %%+µ

3k

) + '(1 ! qk+1)

>

?

@zn.

(3.65)

Now (3.65) implies readily (3.49). !

In the two limiting regimes that we have considered in the previous section, where

", µ, ), ( and '& are kept fixed, we can proceed a bit further and give the results

for the conditional expected total sojourn times in the case of no synchronization

3.5 System busy period 59

(p " 0+) and full synchronization (p " 1$). The results are immediate by taking

the appropriate limits in (3.49). More specifically we have the following theorems.

Theorem 3.8. Consider a system with arrival rate ", service rate µ, e!ective aban-

donment rate per customer '&, setup rate ) and catastrophe rate (. In the limiting

case of no synchronization (' = '&/(1 ! q), q " 1$), the conditional expected total

sojourn time of a tagged customer in the system S(1)(n,i), given that upon his arrival

he finds the system in state (n, i), is given as follows

E[S(1)(n,0)] =

1

() + '&)

) + (

(!

)µ

((µ + ()

n$

k=0

.

n

k

/

2

! %%+µ

3k

) + '&(k + 1), n $ 0, (3.66)

E[S(1)(n,1)] =

1

(

"

1 !.

µ

µ + (

/n+1#

, n $ 0. (3.67)

Theorem 3.9. Consider a system with arrival rate ", service rate µ, e!ective aban-

donment rate per customer '&, setup rate ) and catastrophe rate (. In the limiting

case of full synchronization (' = '&/(1 ! q), q " 0+), the conditional expected total

sojourn time of a tagged customer in the system S(2)(n,i), given that upon his arrival

he finds the system in state (n, i), is given as follows

E[S(2)(n,0)] =

) + (

() + '&)(

"

1 !)

) + (

.

µ

( + µ

/n+1#

, n $ 0, (3.68)

E[S(2)(n,1)] =

1

(

"

1 !.

µ

µ + (

/n+1#

, n $ 0. (3.69)

3.5 System busy period

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.

Let L(n,i), i = 0, 1 and n $ 0 be a generic random variable representing a first

passage time to one of the states in {(0, 0), (0, 1)} starting from (n, i) and denote by

&(n,i)(s) = E[e$sL(n,i) ] its Laplace Stieltjes transform. The busy period L is equal to


L(1,0) with probability ((0,0)((0,0)+((0,1) and equal to L(1,1) with probability ((0,1)

((0,0)+((0,1) .

Therefore the Laplace Stieltjes transform of the busy period &L(s) is given by

&L(s) =%(0, 0)

%(0, 0) + %(0, 1)&(1,0) +

%(0, 1)

%(0, 0) + %(0, 1)&(1,1).

The probabilities of an empty system %(0, 0) and %(0, 1) are calculated by setting

z = 0 in equations (3.5) and (3.6), respectively. Moreover, by conditioning on the

time of the next event (first-step analysis) we obtain that &(n,i)(s) satisfy the system

&(0,0)(s) = &(0,1)(s) = 1 (3.70)

&(n,0)(s) ="

" + ) + ' + s&(n+1,0)(s) +

)

" + ) + ' + s&(n,1)(s)

+'

" + ) + ' + s

n$

j=0

.

n

j

/

pn$jqj&(j,0)(s), n $ 1 (3.71)

&(n,1)(s) ="

" + µ + ( + s&(n+1,1)(s) +

µ

" + µ + ( + s&(n$1,1)(s)

+(

" + µ + ( + s, n $ 1. (3.72)

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

$i(s, z) =#$

n=0

&(n,i)(s)zn, i = 0, 1, s $ 0, |z| < 1. (3.73)

These mixed transforms $i(s, z) =!#

n=0 &(n,i)(s)zn, i = 0, 1 do converge for s $ 0

and |z| < 1. Indeed the LST &(n,i)(s) = E[e$sL(n,i) ] are well-defined for s $ 0.

Moreover, for s $ 0 we have that &(n,i)(s) = E[e$sL(n,i) ] & 1, hence |$i(s, z)| &!#

n=0 |&(n,i)(s)||z|n &!#

n=0 |z|n < #.

The mixed transforms $i(s, z) carry information for all first passage times distri-

butions from an arbitrary state to one of the states in {(0, 0), (0, 1)} and in particular

for the busy period of the system. We use the $i(s, z) transforms, i = 0, 1 to calcu-

late &(1,i)(s), i = 0, 1.


Multiplying (3.72) with (" + µ + ( + s)zn and adding for all n $ 1 results after

some manipulations to

9

(" + µ + ( + s)z ! "! µz2:

$1(s, z) = (" + µ + ( + s)z ! "

!"z&(1,1)(s) + (z2

1 ! z. (3.74)

We observe that the quadratic polynomial

g(s, z) = (" + µ + ( + s)z ! µz2 ! " (3.75)

has two roots

z0(s) =" + µ + ( + s !

8

(" + µ + ( + s)2 ! 4"µ

2µ% (0, 1), (3.76)

z"0(s) =" + µ + ( + s +

8

(" + µ + ( + s)2 ! 4"µ

2µ% (1,#). (3.77)

Setting z = z0(s) in equation (3.74) (note that $1(s, z0(s)) converges since z0(s) %(0, 1)) we obtain

&(1,1)(s) =z0(s) [µ + ( ! µz0(s)]

"(1 ! z0(s)). (3.78)

Substituting (3.78) into (3.74) and taking into account (3.75)–(3.77) we obtain

µ(z ! z0(s))(z"0(s) ! z)$1(s, z) = (z ! z0(s))

0

µz"0(s) + (z

(1 ! z)(1 ! z0(s))

1

(3.79)

which gives that

$1(s, z) =1 ! µz#0(s)(1$z0(s))$%

µz#0(s)(1$z0(s)) z

(1 ! z)(1 ! 1z#0(s)

z)=

1 ! s+µ(1$z0(s))%+s+µ(1$z0(s))z

(1 ! z)(1 ! µz0(s)" z)

. (3.80)

Multiplying now (3.71) with ("+ ) + ' + s)zn and adding for all n $ 1 results after

some manipulations to

[(" + ) + ' + s)z ! "]$0(s, z) = (" + s)z ! "! "z&(1,0)(s)

+)z$1(s, z) +'z

1 ! pz$0(s, T (z)), (3.81)


where T (z) is given by (3.57). We observe that equation (3.81) is of the form

A(s, z)$0(s, z) = B(z)$0(s, T (z)) + C(s, z), (3.82)

where

A(s, z) = (" + ) + ' + s)z ! ",

B(z) ='z

1 ! pz,

C(s, z) = (" + s)z ! "! "z&(1,0)(s) + )z$1(s, z). (3.83)

Therefore, it can be solved by iteration, following the technique that we described

in section 3.4 for the sojourn times. Hereafter, we will suppress the details of the

method, as it has been described in detail above. By iterating equation (3.82) we

can prove inductively that

A(s, z)$0(s, z) = $0(s, Tn+1(z))B(z)n*

i=1

B(Ti(z))

A(s, Ti(z))

+B(z)n$

k=0

C(s, Tk(z))

B(Tk(z))

k*

i=1

B(Ti(z))

A(s, Ti(z))

= $0(s, Tn+1(z))B(z)n*

i=1

B(Ti(z))

A(s, Ti(z))

+n$

k=0

C(s, Tk(z))k*

i=1

B(Ti$1(z))

A(s, Ti(z)). (3.84)

Taking the limit as the number of iterations n " # yields

[(" + ) + ' + s)z ! "]$0(s, z) =#$

k=0

[(" + s)Tk(z) ! " + )Tk(z)$1(s, Tk(z))]k*

i=1

'Ti(z)

q[(" + ) + ' + s)Ti(z) ! "]

!"&(1,0)(s)#$

k=0

Tk(z)k*

i=1

'Ti(z)

q[(" + ) + ' + s)Ti(z) ! "]. (3.85)


The absolute convergence of the series can be proved straightforward by applying

the ratio test. We can now verify that

ak(s, z) =k*

i=1

'Ti(z)

q[(" + ) + ' + s)Ti(z) ! "]

= (!1)k.

'z

"(1 ! z)

/k q(k2)

( (&+'+s)q"

z1$z ; q)k

, k $ 0. (3.86)

Moreover, we observe that (3.80) yields

(" + s)z ! " + )z$1(s, z) =a(s)z3 + b(s)z2 + c(s)z + d

(1 ! z)(1 ! µz0(s)" z)

,

where

a(s) =µz0(s)

"(" + s),

b(s) = !(" + s)(1 +µz0(s)

") ! µz0(s) ! )

s + µ(1 ! z0(s))

( + s + µ(1 ! z0(s)),

c(s) = 2" + s + µz0(s) + ),

d = !".

We aim to factor the numerator in the rational form of (" + s)z ! " + )z$1(s, z)

in first order terms of the form 1 ! *z as its denominator. To this end, let zi(s),

i = 1, 2, 3 be the roots of the cubic polynomial a(s)z3 + b(s)z2 + c(s)z + d. Then

a(s)z3 + b(s)z2 + c(s)z + d = !"(1 !1

z1(s)z)(1 !

1

z2(s)z)(1 !

1

z3(s)z).

Let

c0(s, z) = (µz0(s)

"! 1)

z

1 ! z,

ci(s, z) = (1

zi(s)! 1)

z

1 ! z, i = 1, 2, 3,

c4(z) = !z

1 ! z.


We then have that

(" + s)Tk(z) ! " + )Tk(z)$1(s, Tk(z)) =

!"(1 ! c1(s, z)qk)(1 ! c2(s, z)qk)(1 ! c3(s, z)qk)

(1 ! c0(s, z)qk)(1 ! c4(z)qk),

a rational function of qk. Then, we can easily see that

(" + s)Tk(z) ! " + )Tk(z)$1(s, Tk(z))

can be expressed using q–factorials in the form

(" + s)Tk(z) ! " + )Tk(z)$1(s, Tk(z)) =

!"(1 ! c1(s, z))(1 ! c2(s, z))(1 ! c3(s, z))

(1 ! c0(s, z))(1 ! c4(z))bk(s, z), (3.87)

where bk(s, z) are given by

bk(s, z) =(c0(s, z), c1(s, z)q, c2(s, z)q, c3(s, z)q, c4(z); q)k(c0(s, z)q, c1(s, z), c2(s, z), c3(s, z), c4(z)q; q)k

, k $ 0. (3.88)

Then equation (3.85) can be written as

[(" + ) + ' + s)z ! "]$0(s, z) =

!"(1 ! c1(s, z))(1 ! c2(s, z))(1 ! c3(s, z))

(1 ! c0(s, z))(1 ! c4(z))

#$

k=0

ak(s, z)bk(s, z)

!"&(1,0)(s)#$

k=0

ak(s, z)Tk(z). (3.89)

We set z = ""+&+'+s in (3.89) and define ci(s) = ci(s,

""+&+'+s), i = 0, 1, 2, 3 and

c4(s) = c4("

"+&+'+s). Then, after reducing the sums in the canonical form of the

q–series, (3.89) yields

&(1,0)(s) = !() + ' + s + ")(1 ! c1(s))(1 ! c2(s))(1 ! c3(s))

"(1 ! c0(s))(1 ! c4(s)) 1!1(c4(s); c4(s)q; q,'q

&+'+s)

) 5!5

"

c0(s), c1(s)q, c2(s)q, c3(s)q, c4(s)

c0(s)q, c1(s), c2(s), c3(s), c4(s)q; q,

'

) + ' + s

#

. (3.90)


We can now use (3.78) and (3.90) to obtain &L(s). Although the symbolic inversion

of &L(s) is not possible, one can perform numerical inversion to obtain the distri-

bution of the busy period L or other associated measures (see e.g. Abate et al.

(2000)).

Chapter 4

Synchronized reneging in

single server vacation queues

Part I

In chapters 4 and 5 we present a detailed analysis of two fundamental queueing

models with vacations and impatient customers, where the source of impatience is

the absence of the server. Instead of the standard assumption that customers per-

form independent abandonments, we consider situations where customers abandon

the system simultaneously. This is, for example, the case in remote systems where

customers may decide to abandon the system, when a transport facility becomes

available. In chapter 4 we study the Markovian case, while in chapter 5 we study

the non-Markovian counterparts.

4.1 Introduction

Queueing systems with reneging (i.e., impatient customers) have been studied exten-

sively. The main assumption in the literature is that customers perform independent

abandonments, that is, each one of them sets an impatience clock and abandons the

system as soon as the clock expires. For Markovian models, this type of abandon-

68 Synchronized reneging in single server vacation queues – Part I

ments introduces state-inhomogeneous transition rate matrices, a fact that compli-

cates 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

intractable.

The study of queueing systems with impatient customers goes back at least to

the pioneering papers of Palm (1953, 1957) who studied the M/M/c queue, where

the customers have independent exponentially distributed impatience times. Subse-

quently, Daley (1965), Takacs (1974) and Baccelli et al. (1984) considered various

queueing models with general service and/or inter-arrival times and more involved

abandonment schemes.

More recently, Boxma and de Waal (1994) studied the M/M/c queue with gen-

erally distributed impatience times, while Altman and Borovkov (1997) investigated

the stability issue in a retrial queue with impatient customers. In all the afore-

mentioned works, customers become impatient due to the long waiting time already

experienced, although the server provides continuously service. The study of reneg-

ing within the class of queueing systems with vacations is a new endeavor. Although,

there exists a significant number of papers and books on vacation queueing systems

(see, e.g., Takagi (1991) and Tian and Zhang (2006)), the reneging feature has not

yet received much attention. Only recently, Altman and Yechiali (2006) and Yechiali

(2007) considered systems with vacations, where the cause of the impatience is the

absence of the server. The authors assume that the customers perform independent

abandonments, whenever the server is unavailable.

In chapters 4 and 5, we study two models with vacations, where the customers

are impatient but they perform synchronized abandonments. These models are mo-

tivated 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 decide whether to leave the system or not. Therefore, we have

synchronized departures for some of the customers. In chapter 4 we deal with the

4.2 Model description 69

Markovian cases, while in chapter 5 we treat the general times cases.

The first model is the single-server queue with multiple vacations, where cus-

tomers decide whether to abandon the system or not when the vacation periods

finish. In the second model, we suppose that the abandonments opportunities occur

according to a Poisson process during vacation periods. At the abandonment oppor-

tunities, every present customer remains in the system with probability q or aban-

dons the system with probability p = 1!q, independently of the others. The analysis

of this model extends the analysis of Altman and Yechiali (2006), in the framework of

synchronized abandonments. The new feature of these models with synchronization

is the presence of binomial type jumps at the abandonment epochs. Similar models

with binomial type transitions have been recently studied by Economou (2004), Ar-

talejo et al. (2007), Economou and Fakinos (2008).

Chapter 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 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

parameters of the models.

4.2 Model description

We consider a queueing system where customers arrive one by one according to a

Poisson process at rate ". Service is provided by a single server who can be in one

of two modes: on (active) or o" (non-active - on vacation). Customers are served

singly when the server is on, while no service is provided when the server is o".

The service times, B, are exponentially distributed with rate µ, where the random

variable B represents the service time. There is infinite waiting room. Whenever the

system becomes empty, the server begins a vacation, V , where the random variable

V represents the vacation time. We assume multiple vacations, i.e., if the system

is still empty at the end of a vacation, the server takes another one. If, on the

contrary, there is at least one waiting customer at the end of a vacation, the server


starts again to provide service. The vacation times are exponentially distributed

with rate ). Regarding the abandonments we consider two models:

• Unique Abandonment Epoch (UAE) : Every time the server finishes a vacation,

every present customer decides whether to stay in the system with probability

q or to abandon it with probability p = 1 ! q, independently of the others.

• Multiple Abandonment Epochs (MAE) : During server vacations, abandon-

ment opportunities occur according to a Poisson process with rate +. At these

epochs, every present customer remains in the system with probability q or

abandons the system with probability p = 1 ! q, independently of the others.

Hence, in either model, the number of customers is reduced according to a bino-

mial distribution at every abandonment epoch. However, the analysis of the UAE

model turns out to be much easier than the one of the MAE model. For this reason,

in what follows, we describe briefly the results for the UAE model and we provide

more details for the analysis of the MAE model.

We are interested in the equilibrium behavior of the model, so we need to establish

first the stability condition. For the pure vacation model (p = 0) the stability

condition is (see, e.g., Takagi (1991))

, ="

µ< 1. (4.1)

Hence, the above condition is su!cient for the stability of the UAE and MAE models.

It is also necessary, since the system behaves as a standard M/M/1 queue while the

server is active. Clearly, the only exception is the degenerate case p = 1 for the UAE

model. Then condition (4.1) is not required for stability. Throughout the chapter

we assume the validity of condition (4.1).

4.3 Mean value analysis 71

4.3 Mean value analysis

4.3.1 Mean value analysis of the UAE model

We suppose that the system is in equilibrium and we define the random variable

L to be the number of customers in the system and S to be the sojourn time of a

customer. Let also Li be the conditional number of customers in the system, given

that the server is in state i, i = 0, 1. Further we denote by pi the probability (or

fraction of time) that the server is in state i, i = 0, 1.

Let us consider a tagged arriving customer. Then, by PASTA property, the

probability that this customer finds the server in state i is pi. If he finds the server

providing service, then his mean sojourn time is E(L1)1µ + 1

µ . If he finds the server on

vacation, then he first has to wait for the vacation time to expire; the mean residual

vacation time is 1& . Then, with probability p, he will abandon and, with probability

q, he will remain for service, in which case the mean number of customers that he

will find in front of him is qE(L0). Indeed, by PASTA, he sees at his arrival epoch on

average E(L0) customers in the system and each of them will remain for service with

probability q. So, in this case, his mean sojourn time is 1& + p · 0+ q · (qE(L0)

1µ + 1

µ).

Hence,

E(S) = p1

.

E(L1)1

µ+

1

µ

/

+ p0

.

1

)+ q2E(L0)

1

µ+ q

1

µ

/

. (4.2)

Further, Little’s law states that

E(L) = "E(S), (4.3)

where the unconditional E(L) is related to the conditional ones as

E(L) = p0E(L0) + p1E(L1). (4.4)

Conservation of work gives the relation

p1 = ("p0q + "p1)1

µ, (4.5)

and clearly,

p0 + p1 = 1. (4.6)


Finally, by gluing the periods during which the server is on vacation, we observe that

the vacation completion epochs constitute a Poisson process. Hence, by PASTA, we

have that E(L0) coincides with the mean number of customers in the system just

before a vacation time finishes, which is equal to the mean number of Poisson (")

arrivals in a vacation time. Thus,

E(L0) = "1

). (4.7)

Now we have su!ciently many equations for the unknown mean values. Solution of

(4.2)-(4.7) yields the following result.

Theorem 4.1. The mean sojourn time is given by

E(S) =1

1 ! ,p

.

1

)+ (q2 ! 1)

,

)

/

+q

1 ! ,p·

1

µ(1 ! ,), (4.8)

and the fraction of time the server is inactive and active, respectively,

p0 =1 ! ,

1 ! ,p, p1 =

,q

1 ! ,p. (4.9)

4.3.2 Mean value analysis of the MAE model

Let us, again, consider a tagged arriving customer. Then, by PASTA, the probability

that this customer finds the server in state i is pi. If he finds the server providing

service, then his mean sojourn time is E(L1)1µ + 1

µ . If he finds the server on vacation,

then he first has to wait for the vacation time to expire before servicing starts and,

while waiting, he may decide to abandon at one of the abandonment opportunities.

Let E(V &) be his mean time in the system till the end of the vacation; note that

E(V &) will be less than the mean residual vacation time 1& . If the tagged customer

decides to stay till the end of the vacation, then his sojourn time after return of

the server depends on the number of customers (still) in front of him. Define %

as the probability that the tagged customer stays in the system till the end of the

vacation period and define %& as the probability that the tagged customer and a

customer, who was already present at his arrival, both stay in the system. Then


%&E(L0)1µ + % 1

µ is his mean sojourn time, from the moment the server returns from

vacation. Hence,

E(S) = p1

.

E(L1)1

µ+

1

µ

/

+ p0

.

E(V &) + %&E(L0)1

µ+ %

1

µ

/

, (4.10)

and Little’s law yields

E(L) = "E(S), (4.11)

where the unconditional E(L) is related to the conditional ones as

E(L) = p0E(L0) + p1E(L1). (4.12)

Also, if we would act as if the customers arriving during a vacation are waiting in

a “vacation area” and transferred to the queue as soon as the server returns, then

application of Little’s law to the vacation area yields

E(L0) = "E(V &). (4.13)

Analogous to (4.5), conservation of work gives the relation

p1 = ("p0% + "p1)1

µ, (4.14)

and clearly,

p0 + p1 = 1. (4.15)

Now we need additional relations for the quantities %, %& and E(V &). By condi-

tioning on the next event after the arrival of the tagged customer, whether it is the

end of the vacation (with probability &)+& ) or an opportunity of abandonment (with

probability ))+& ), we have that

% =)

+ + )· 1 +

+

+ + )q%.

Hence

% =)

+p + ). (4.16)

Along the same lines,

%& =)

+ + )· 1 +

+

+ + )q2%&,


so

%& =)

+(1 ! q2) + ). (4.17)

Finally, again conditioning on the next event after the arrival of the tagged customer,

we obtain

E(V &) =1

+ + )+

)

+ + )· 0 +

+

+ + )qE(V &) +

+

+ + )p · 0,

yielding

E(V &) =1

+p + ). (4.18)

This completes the formulation of the mean value relations. By solving (4.10)-(4.15),

taking into account (4.16)-(4.18) we get the following result.


E(S) =1

1 ! , + ,%

.

1

+p + )+ (q2 ! 1)

+%

+(1 ! q2) + )·,

)

/

+%

1 ! , + ,%·

1

µ(1 ! ,)(4.19)


p0 =1 ! ,

1 ! , + ,%, p1 =

,%

1 ! , + ,%. (4.20)

4.4 Equilibrium distribution

We consider the two models described in Section 4.2. Then, each system can be

described by a continuous-time Markov chain {(L(t), I(t)), t $ 0}, with state space

{(0, 0)} + {(n, i) : i = 0, 1, n = 1, 2, . . .}, where L(t) is the number of customers

in the system at time t and I(t) expresses the mode of the server at time t (more

explicitly, it is equal to 1 if the server is on at that time t and 0 otherwise). Below,

we focus on the determination of the equilibrium distribution of the Markov chain

{(L(t), I(t)), t $ 0}. The two models are treated separately.

To this end, let (%(n, i) : i = 0, 1 and n $ i), denote the equilibrium distribution.

We define the (partial) probability generating functions (PGFs) #0(z) and #1(z) of

4.4 Equilibrium distribution 75

the equilibrium distribution by

#0(z) =#$

n=0

%(n, 0)zn and #1(z) =#$

n=1

%(n, 1)zn, |z| & 1.

We also set L, I and S to be random variables representing the number of cus-

tomers in the system, the state of the server and the sojourn time of a customer,

when the system is in equilibrium. We also denote by pi = P (I = i) the equilibrium

probability that the server is in state i, i = 0, 1.

1, 1

µ

44!!!!!!!!!!!

!!!!!

" ##2, 1

" ##

µ00 3, 1

" ##

µ00 · · ·

0, 0 " ## 1, 0

(10)q&

--

(11)p&

""" ## 2, 0

(20)q

2&

--

(21)pq&

$$$$$$$

55$$$$$$$

(22)p

2&

33" ## 3, 0

(30)q

3&

--

(31)pq2&

$$$$$$$

55$$$$$$$

(32)p

2q&%%%%%%

!!%%%%%%%%%%%%%%%%%%%%

(33)p3&

66" ## · · ·

Fig. 4.1: State-transition diagram for the UAE model.

4.4.1 Equilibrium distribution of the UAE model

Figure 4.1 shows the state-transition diagram for the UAE model. The set of balance

equations is given as follows,

"%(0, 0) = µ%(1, 1) + )#$

j=1

.

j

0

/

pj%(j, 0) (4.21)

(" + ))%(n, 0) = "%(n ! 1, 0), n $ 1 (4.22)

(" + µ)%(1, 1) = µ%(2, 1) + )#$

j=1

.

j

1

/

qpj$1%(j, 0) (4.23)

(" + µ)%(n, 1) = µ%(n + 1, 1) + "%(n ! 1, 1)

+)#$

j=n

.

j

n

/

qnpj$n%(j, 0), n $ 2. (4.24)


Provided (4.1), this set of equations, together with the normalization equation

%(0, 0) +#$

n=1

(%(n, 0) + %(n, 1)) = 1,

has a unique solution. This solution is presented in the next theorem.

Theorem 4.3. Provided (4.1), the equilibrium state distribution %(n, i) is given by

%(0, 0) =)

) + "·

1 ! ,

1 ! ,p(4.25)

%(n, 0) = %(0, 0)

.

"

" + )

/n

, n $ 0 (4.26)

%(n, 1) =

'

(

)

%(0, 0) ("+&)q&+("$µ)q

2

,n !2

"q&+"q

3n3

, if ) (= (µ ! ")q

%(0, 0)(,p + q)n,n, if ) = (µ ! ")q, n $ 1.(4.27)

Proof. By iterating (4.22) we obtain (4.26) yielding

#0(z) =" + )

" + ) ! "z%(0, 0). (4.28)

By multiplying (4.23) by z and (4.24) by zn and adding for all n = 1, 2, . . . we obtain

(" + µ)#1(z) = "z#1(z) +µ

z(#1(z) ! z%(1, 1)) + )#0(p + qz) ! )#0(p). (4.29)

Solving (4.29) for #1(z) and plugging (4.28), while using (4.21), yields

#1(z) =q,z() + ")

() + "q(1 ! z))(1 ! ,z)%(0, 0). (4.30)

Expanding (4.30), for the case ) (= (µ ! ")q, in partial fractions and using the

geometric series leads to the upper branch of (4.27). For the case ) = (µ ! ")q,

expanding (1 ! ,z)$2 in power series and equating the coe!cients of zn yields the

other branch of (4.27). Finally, (4.25) follows from the normalization equation. !


Remark 4.1. From equations (4.28) and (4.30) we obtain the equilibrium mean

number of customers in the system given as

E(L) =#$

n=1

n(%(n, 0) + %(n, 1))

= #"0(1) + #"

1(1)

= %(0, 0)" + )

)·"

)+ %(0, 0)

(" + ))q,

)(1 ! ,)·)(1 ! ,) + ,) + "q(1 ! ,)

)(1 ! ,)

= p0"

)+ p1

) + "q(1 ! ,)

)(1 ! ,).

This result can be alternatively derived using the mean value approach, as showed

in Section 4.3.

Remark 4.2. Let X(*) denote an exponential random variable with rate * and

Y (j,*) denote an Erlang random variable consisting of j phases with rate *. Let

us consider a tagged arriving customer. Then, by the PASTA property, he finds the

system in state (n, i) with probability %(n, i). If he finds the system in state (n, 1),

then his sojourn time is Y (n + 1, µ). If he finds the system in state (n, 0), then with

probability p his sojourn time will be X()), since the tagged customer abandons the

system, and with probability,n

j

-

pn$jqj+1 his sojourn time will be X())+Y (j+1, µ),

for j = 0, 1, . . . , n, where the random variables X()) and Y (j+1, µ) are independent.

Hence, by using the geometric form of the equilibrium distribution (4.25)-(4.27), we

have that the LST of the sojourn time S(s) = E(e$sS) can be represented as

S(s) = p0p)

) + s+ p0q

)

) + s·

)µ

)µ + () + q")s+ p1

µ ! "

µ ! " + s·

)µ

)µ + () + q")s.

This shows that the sojourn time S is a mixture of X()), X()) + X( &µ&+q" ), X(µ !

") + X( &µ&+q" ) with mixing probabilities p0p, p0q and p1, respectively.


4.4.2 Equilibrium distribution of the MAE model

In this case the state-transition diagram is given in figure 4.2. The set of balance

equations for this model is given as follows:

(" + +)%(0, 0) = µ%(1, 1) + +#$

j=0

.

j

0

/

pjq0%(j, 0) (4.31)

(" + ) + +)%(n, 0) = "%(n ! 1, 0) + +#$

j=n

.

j

n

/

pj$nqn%(j, 0), n $ 1 (4.32)

(" + µ)%(1, 1) = )%(1, 0) + µ%(2, 1) (4.33)

(" + µ)%(n, 1) = )%(n, 0) + "%(n ! 1, 1) + µ%(n + 1, 1), n $ 2. (4.34)

Note that in the balance equations (4.31) and (4.32) we included the pseudo-transitions

(n, 0) " (n, 0) with rates +,n0

-

pn$nqn = +qn, which correspond to epochs in the Pois-

son abandonment opportunities process where all customers remain in the system,

i.e., no abandonments occur. This simplifies the presentation of the balance equa-

tions.

1, 1µ

%%&&&&&&&&&&&&

" ##2, 1

" ##

µ00 3, 1

" ##

µ00 · · ·

0, 0 " ## 1, 0

&

--

(11)p)

33" ## 2, 0

&

--

(21)pq)

33

(22)p

2)

66" ## 3, 0

&

--

(31)pq2)

33

(32)p

2q)

66

(33)p

3)

77" ## · · ·

Fig. 4.2: State-transition diagram for the MAE model.

In Theorem 4.4 of this section we will determine the equilibrium probability

%(0, 0) and the equilibrium PGF #i(z), i = 0, 1, in the form of infinite series of finite

products. These series can be expressed compactly in terms of q-hypergeometric

series. Moreover, we will see that the theory of q-hypergeometric series easily yields

interesting results for some limiting regimes.

We are now in position to state the main result of this section.


Theorem 4.4. Provided (4.1), the equilibrium state probability of an empty system

%(0, 0) is given by

%(0, 0) =A

+

#$

j=0

j*

k=0

+

) + + + "qk

=A

) + +

(q; q)#

(! "&+) ,

)&+) ; q)#

2!1

.

!"

) + +,

+

) + +; 0; q, q

/

. (4.35)

The partial PGFs #0(z) and #1(z) are given by

#0(z) =A

+

#$

j=0

j*

k=0

+

) + + + "qk(1 ! z)(4.36)

=A

) + +

(q; q)#

(!"(1$z)&+) , )

&+) ; q)#2!1

.

!"(1 ! z)

) + +,

+

) + +; 0; q, q

/

(4.37)

#1(z) = !Az

"z + µz ! "z2 ! µ+

)z

"z + µz ! "z2 ! µ#0(z), (4.38)

where

A =)(µ ! ")() + +(1 ! q))

µ) + (µ ! ")+(1 ! q). (4.39)

The convergence of the series is absolute in {z % C : |z| & 1} and uniform in every

compact subset of {z % C : |z| < 1}.

Proof. Multiplying both sides of equations (4.31) and (4.32) by z0 and zn, respec-

tively, and summing them for all n = 0, 1, 2, . . . we obtain

(" + ) + +)#0(z) ! )%(0, 0) = µ%(1, 1) + "z#0(z) + +#$

n=0

#$

j=n

.

j

n

/

pj$nqn%(j, 0)zn

or

(" + ) + + ! "z)#0(z) = )%(0, 0) + µ%(1, 1) + +#0(1 ! q + qz), (4.40)

which leads to

#0(z) =)%(0, 0) + µ%(1, 1)

) + + + "(1 ! z)+

+

) + + + "(1 ! z)#0(1 ! q + qz). (4.41)


Furthermore, by multiplying both sides of equations (4.33) and (4.34) by z and

zn, respectively, and summing them for all n = 1, 2, 3, . . . we obtain after some

rearrangements that

#1(z) = !()%(0, 0) + µ%(1, 1))z

"z + µz ! "z2 ! µ+

)z

"z + µz ! "z2 ! µ#0(z). (4.42)

By iterating equation (4.41) and setting

A = )%(0, 0) + µ%(1, 1) (4.43)

we obtain

#0(z) =A

+

n$

j=0

j*

k=0

+

) + + + "qk(1 ! z)

+#0(1 ! qn+1 + qn+1z)n*

k=0

+

) + + + "qk(1 ! z), n $ 0. (4.44)

By letting n " # we get

#0(z) =A

+

#$

j=0

j*

k=0

+

) + + + "qk(1 ! z), (4.45)

which is expressed as a q-hypergeometric series in the form (4.37). This shows also

that the infinite series does converge. We set z = 0 in (4.45), yielding

%(0, 0) =A

+

#$

j=0

j*

k=0

+

) + + + "qk, (4.46)

which can be put in the form (4.35). We set z = 1 in (4.45), which leads to

#0(1) =A

+

#$

j=0

j*

k=0

+

) + +=

A

). (4.47)

Note also that by (4.42), (4.43) and (4.47) we obtain

#1(1) =)

µ ! "#"

0(1). (4.48)


To obtain #"0(1) multiply (4.41) by ) + + + "(1 ! z), di"erentiate and take z " 1.

We then have

#"0(1) =

"A

)() + +(1 ! q)). (4.49)

Equations (4.48) and (4.49) yield

#1(1) ="A

(µ ! ")() + +(1 ! q)). (4.50)

We have now expressed the various quantities of interest and the PGFs #0(z) and

#1(z) in terms of the parameters of the model and the parameter A. Using (4.47) and

(4.50) and the normalization equation we obtain (4.39) which concludes the proof. !

Remark 4.3. By di"erentiating twice (4.42) and once (4.41) and taking z " 1

we obtain, after some long calculations, the mean number of customers in system.

However, as we have showed in Section 4.3, the mean value approach gives the result

much more easily.

4.4.3 Fluid limit of the UAE model

In this section we study the scaled queue length process in the UAE model as both

the arrival rate " and service rate µ tend to infinity, while keeping the tra!c intensity

, = "µ fixed. That is, we consider the sequence of queue length processes Lm(t) with

arrival rates "m = m", service rates µm = mµ and constant vacation rate )m = ),

and then we are interested in obtaining the equilibrium distribution of the fluid

scaled limit

L(t) = limm%#

Lm(t)

m.

The fluid scaled sample path of Lm(t) is shown in figure 4.3. Clearly, a cycle first

starts with an exponential vacation V , at the start of which there is no fluid and

during which the amount of fluid increases with rate ". At the end of the vacation

an exponential amount X = "V has been accumulated in the queue; a fraction p

of X will then immediately abandon and the remainder qX will be drained at rate

µ! ", which takes an exponential time P = qX/(µ ! "). When the queue is empty

again, the cycle repeats.


V P

" µ ! "

X

qX

Fig. 4.3: The fluid scaled sample path of Lm(t) in the UAE model as m " #.

During a cycle we can hit a vacation time with probability E(V )E(V )+E(P ) or hit a

service period time with probability E(P )E(V )+E(P ) . If we take the consecutive vacation

times and glue them together we form a renewal process, with interevent times in-

dependent and identical exponential random variables at rate ). At the beginning

of each interrenewal time the fluid level is 0 and it is accumulated at rate " until the

end of the vacation time. Then at any given time the level of the fluid is equal to

"Va, where Va is the age of the vacation time V . However, the age of the vacation

time is stochastically identical with the remaining vacation time. Since the random

variable V is exponentially distributed at rate ), we obtain by the memoryless prop-

erty that the remaining vacation time is also exponentially distributed at rate ).

Therefore, at an arbitrary moment during a vacation time the level of fluid is equal

to X = "V . If we now consider the consecutive service periods and also glue them

together we again form a renewal process, with interevent times independent and

identical exponential random variables at rate )(µ!")/q. At the beginning of each

interrenewal time the fluid level is qX and until the end of the service period time

it is drained to 0 at rate µ ! ". At any given moment during a service period the

fluid level is equal to qX ! (µ!")Pa, where Pa is the age of the service period time.

However qX = (µ ! ")P , so the fluid level at an arbitrary moment during a service

period is distributed as (µ! ")(P !Pa). Moreover, P !Pa is the remaining service

period time in a cycle and due to the memoryless property of the exponential distri-


bution is stochastically identical to P , therefore the amount of fluid at an arbitrary

time of the service period is equal to (µ ! ")P = qX.

Let the random variable L denote the amount of fluid in the queue, when the

system is in equilibrium. From the memoryless property of exponentials, we imme-

diately obtain

Ld=

7

X w.p. E(V )E(V )+E(P ) ,

qX w.p. E(P )E(V )+E(P ) ,

(4.51)

where

E(V ) =1

), E(P ) =

q

µ ! ""E(V ) =

q,

1 ! ,

1

).

In addition to the above intuitive derivation of equation (4.51), we prove the follow-

ing theorem by directly employing (4.25), (4.28) and (4.30).

Theorem 4.5. The LST of L is given by

E(e$sL) =1 ! ,

1 ! , + q,

&"

&" + s

+q,

1 ! , + q,

&q"&q" + s

.

Proof . We can determine the LST of L as the limit of E(e$sLm/m) as m " #,

E(e$sL) = limm%#

E(e$sLm/m)

= limm%#

[#(m)0 (e$s/m) + #(m)

1 (e$s/m)]

= limm%#

0

m" + )

m" + ) ! m"e$s/m

)

) + m"

1 ! ,

1 ! p,

+q,e$s/m() + m")

() + m"q(1 ! e$s/m))(1 ! ,e$s/m)

)

) + m"

1 ! ,

1 ! p,

5

= limm%#

0

)

) + m"(1 ! e$s/m)

1 ! ,

1 ! p,

+q,e$s/m)

() + m"q(1 ! e$s/m))(1 ! ,e$s/m)

1 ! ,

1 ! p,

5

.

Since

limm%#

m(1 ! e$s/m) = s, (4.52)


we obtain

E(e$sL) =)

) + "s

1 ! ,

1 ! p,+

)

) + "qs

q,

1 ! p,.

!

Convergence of the fluid limit for the UAE model

For the UAE model, we plot the graph of the convergence of the mean number

of customers E(Lm/m) for the sequences of models with arrival rates "m = m",

service rates µm = mµ and constant vacation rates )m = ) to the mean of the fluid

model E(L). We assume that the time unit has been re-scaled to agree with the

mean service time, i.e. we set E[B] = 1 and have also taken " = 0.5, E[V ] = 1 and

p = 0.5. As can be seen in figure 4.4 we can obtain a good approximation of the

number of customers in the system for a quite small n.

0 10 20 30 40 50 600

0.1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1

m

E[L

m/m

]

U AE model: λ=0.5 , B is E xponentia lly distributed with E [B]=1,V is E xponentia lly distributed with E [V ]=1

p=0.5

E [Lm/m]E [L]

Fig. 4.4: Convergence of E(Lm/m) to E(L) of the fluid model.

4.4.4 Fluid limit of the MAE model

Now we study the scaled queue length process in the MAE model as both the

arrival rate " and service rate µ tend to infinity, while keeping the tra!c intensity


, = "µ fixed. So we consider the sequence of queue length processes Lm(t) with

arrival rates "m = m", service rates µm = mµ and constant vacation rates )m = )

and abandonment rates +m = +. We are interested in obtaining the equilibrium

distribution of the fluid scaled limit

L(t) = limm%#

Lm(t)

m.

The fluid scaled sample path of Lm(t) is shown in figure 4.5. The cycle starts

with an empty queue and a random number of exponential vacations V1, . . . , VN ,

each with mean 1&+) and during which the amount of fluid increases with rate ".

The amount of fluid that is added during Vn is Xn = "Vn. The number of vacations

is geometrically distributed with success probability &)+& ,

P (N = n) =

.

+

+ + )

/n$1 )

+ + ), n = 1, 2, . . . .

At the end of each vacation Vn, a fraction p of the total amount of fluid in the queue

will instantly abandon. The only exception is the last vacation VN . At the end of

VN no fluid is removed, since this is not an abandonment epoch, but the end of the

vacation period. The total amount of fluid in the queue at the end of VN is denoted

by the random variable Z; so

Z =N$

n=1

qN$nXn.

Subsequently the service period starts, during which the queue is drained at rate

µ ! ". The time required to empty the queue is P = Z/(µ ! "). Finally, when the

queue is empty, the cycle repeats.

As before, let the random variable L denote the amount of fluid in the queue, when

the system is in equilibrium. Further, let Li be the amount of fluid in the queue,

given that the server is in state i, i = 0, 1. During a cycle we can hit a vacation

period V , where V =!N

i=1 Vi, with probability E(V )E(V )+E(P ) or hit a service period

time with probability E(P )E(V )+E(P ) . We observe that V is exponentially distributed

at rate ), since Viiid, Exp() + +) and N is independent of Vi and is geometrically


V

V1 V2 ... VNP

"

"

µ ! "Z

Fig. 4.5: The fluid scaled sample path of Lm(t) in the MAE model as m " #.

distributed with success probability &)+& . If we then take the consecutive vacation

periods and glue them together we form a Poisson process, with interevent times

independent and identical exponential random variables at rate ). The distribution

L0 of the fluid level at an arbitrary moment is identical to the distribution of the fluid

level Z just before the end of the vacation period because of the PASTA property.

If we now consider the consecutive service periods and also glue them together we

again form a renewal process. Proceeding in the same manner as in the case of the

fluid limit of the UAE model we can easily see that the fluid level at an arbitrary

moment during the service period is distributed as (µ!")Pe, where Pe is the residual

service period time. We, then, have that

L0d= Z =

N$

n=1

qN$nXn. (4.53)

Let L1 be the amount of fluid that is drained during the residual service period Pe,

so

L1 = Pe(µ ! ") = Ze. (4.54)

Hence, by (4.53) and (4.54) we get

E(e$sL0) =)

+

#$

k=1

k$1*

n=0

+

) + + + "sqn

=)

) + + + s"2!1

.

0, q;!s"q

) + +; q;

+

) + +

/

,


and

E(e$sL1) =1 ! E(e$sL0)

E(L0)s

=) + +(1 ! q)

"·1 ! E(e$sL0)

s,

where we used that

E(L0) = E(Z) ="

) + +(1 ! q).

Finally, we have

E(e$sL) =E(V )

E(V ) + E(P )E(e$sL0) +

E(P )

E(V ) + E(P )E(e$sL1),

where V is the total vacation period, i.e., V = V1 + · · ·+ VN (and so E(V ) = 1& ). In

addition to this intuitive derivation of the LST of L, we prove the following theorem

by directly employing the expressions (4.36) and (4.38).

Theorem 4.6. The LST of L is given by

E(e$sL) =A

)2!1

.

0, q;+q

) + +; q;!

s"

) + +

/

+A

) + +(1 ! q)

,

1 ! , 2!1

.

0, q;+q2

) + +; q;!

s"

) + +

/

=A

)+

"() ! (µ ! ")s)

µ) + (µ ! ")+(1 ! q) 2!1

.

0, q;+q2

) + +; q;!

s"

) + +

/

,

where the constant A is given by (4.39). Moreover,

E(L) ="(µ ! ")

µ) + (µ ! ")+p+

"2)

(µ) + (µ ! ")+p)() + +(1 ! q2)).

Proof . We determine the LST of L as

E(e$sL) = limm%#

E(e$s Lm

m )

= limm%#

%

#(m)0 (e$s/m) + #(m)

1 (e$s/m)&

.


From (4.36) we get

#(m)0 (e$s/m) =

A

+

#$

j=0

j*

k=0

+

) + + + m"qk(1 ! e$s/m).

Taking the limit as m " # and using (4.52) yields

limm%#

#(m)0 (e$s/m) =

A

+

#$

j=0

j*

k=0

+

) + + + s"qk

=A

) + + + s" 2!1

.

0, q;!s"q

) + +; q;

+

) + +

/

=A

)2!1

.

0, q;+q

) + +; q;!

s"

) + +

/

.

where the last equality follows from formula (1.12). Similarly, from (4.38),

limm%#

#(m)1 (e$s/m) = lim

m%#[!

Ae$s/m

m("e$s/m + µe$s/m ! "e$2s/m ! µ)

+)e$s/m

m("e$s/m + µe$s/m ! "e$2s/m ! µ)#(m)

0 (e$s/m)]

=A

s(µ ! ")!

)

s(µ ! ")lim

m%##(m)

0 (e$s/m)

=A

s(µ ! ")!

A

s(µ ! ")2!1

.

0, q;+q

) + +; q;!

s"

) + +

/

=A

µ ! "

#$

n=1

2

"&+)

3n(!s)n$1

( )q&+) ; q)n

=A

µ ! "

#$

n=1

"&+)

2

"&+)

3n$1(!s)n$1

(1 ! )q&+) )(

)q2

&+) ; q)n$1

=A

µ ! "

"

) + +(1 ! q)2!1

.

0, q;+q2

) + +; q;!

s"

) + +

/

.


Finally

E(e$sL) =A

) 2!1

.

0, q;+q

) + +; q;!

s"

) + +

/

+A

) + +(1 ! q)

,

1 ! , 2!1

.

0, q;+q2

) + +; q;!

s"

) + +

/

=A

)!

A

)

s"

) + +(1 ! q) 2!1

.

0, q;+q2

) + +; q;!

s"

) + +

/

+A

) + +(1 ! q)

,

1 ! , 2!1

.

0, q;+q2

) + +; q;!

s"

) + +

/

=A

)+

A

) + +(1 ! q)

.

,

1 ! ,!

s"

)

/

2!1

.

0, q;+q2

) + +; q;!

s"

) + +

/

=A

)+

"() ! (µ ! ")s)

µ) + (µ ! ")+(1 ! q)2!1

.

0, q;+q2

) + +; q;!

s"

) + +

/

.

To obtain the E(L), we can first easily verify that basic results,

2!1

.

0, q;+q2

) + +; q; 0

/

= 1

d

ds 2!1

.

0, q;+q2

) + +; q;!

s"

) + +

/A

A

A

A

s=0

= !"

) + + ! +q2.

Then, taking the derivative of the LST of L, substituting s = 0 and multiplying by

!1 yields

E(L) = !d

ds

.

"() ! (µ ! ")s)

µ) + (µ ! ")+(1 ! q) 2!1

.

0, q;+q2

) + +; q;!

s"

) + +

//A

A

A

A

s=0

="(µ ! ")

µ) + (µ ! ")+(1 ! q)· 1 +

")

µ) + (µ ! ")+(1 ! q)·

"

) + + ! +q2,

which completes the proof of the theorem. !

Convergence of the fluid limit for the MAE model

For the MAE model, in figure 4.6 we present the convergence of the mean number

of customers E(Lm/m) for the sequences of models with arrival rates "m = m",

service rates µm = mµ, constant vacation rates )m = ) and constant abandonment


0 10 20 30 40 50 600

0.1

0 .2

0 .3

0 .4

0 .5

0 .6

m

E[L

m/m

]

M AE model: λ=0.5 , B is E xponentia lly distributed with E [B]=1,V is E xponentia lly distributed with E [V ]=2

ζ*=1 p=0.6

E [Lm/m]E [L]

Fig. 4.6: Convergence of E(Lm/m) to E(L) of the fluid model.

rates +m = + to the mean of the fluid model E(L). We assume that the time unit

has been re-scaled to agree with the mean service time, i.e. we set E[B] = 1 and

have also taken " = 0.5, E[V ] = 2, +& = 1 and p = 0.6. As can be seen in figure 4.6

we can obtain a good approximation of the number of customers in the system for

a quite small n.

4.4.5 Limiting regimes of synchronization in the MAE model

To emphasize the dependence on the parameters of the MAE model in the rest of this

section, we will denote %(n, i), #0(z) and #1(z) by %(n, i;", µ, +, p, )), #0(z;", µ, +,

p, )) and #1(z;", µ, +, p, )) respectively. Note that +p can be thought of as the ef-

fective abandonment rate per customer. Indeed the overall abandonment time of

a customer is a geometric sum of exponentially distributed random variables with

rate +; hence it is also exponentially distributed with parameter +p. Under this

perspective, if we have two models with the same parameters ", µ and ) that di"er

only in + and p, but with +p = +& fixed, we can think that the models have identical

arrival rates, service rates, e"ective abandonment rates per customer and vacation

rates and di"er only in the ‘level of synchronization’ p. Indeed, the case p " 0+

corresponds to no synchronization since the customers abandon almost singly the


system. On the contrary, the case p " 1$ corresponds to full synchronization since

almost all present customers abandon simultaneously.


where ", µ, +& and ) are kept fixed in the two limiting cases p " 0+ (q " 1$) and

p " 1$ (q " 0+). For the limiting case of no synchronization we introduce

%(1)(0, 0) = limq%1!

%(0, 0;", µ,+&

1 ! q, 1 ! q, )) (4.55)

#(1)i (z) = lim

q%1!#i(z;", µ,

+&

1 ! q, 1 ! q, )), i = 0, 1 (4.56)

and for the limiting case of full synchronization,

%(2)(0, 0) = limq%0+

%(0, 0;", µ,+&

1 ! q, 1 ! q, )) (4.57)

#(2)i (z) = lim

q%0+#i(z;", µ,

+&

1 ! q, 1 ! q, )), i = 0, 1. (4.58)

The corresponding results for (4.55)-(4.58) are presented in Theorems 4.7 and 4.8.

Using results of the q–theory we will study the case of no synchronization (i.e.,

independent abandonments) that has been investigated by Altman and Yechiali

(2006). The following theorem corresponds to their results for the M/M/1 type

model (see their Section 2, in particular their equations (2.9), (2.8) and (2.3)).

Theorem 4.7. In case q " 1$ and +(1 ! q) = +& fixed, we have

%(1)(0, 0) =A&

+&

+ 1

0(1 ! s)

"$$

$1e$!$$

sds (4.59)

#(1)0 (z) =

A&

+&e

!$$ z(1 ! z)$

"$$

+ 1

z(1 ! s)

"$$ $1e$

!$$ sds (4.60)

#(1)1 (z) = !

A&z

"z + µz ! "z2 ! µ+

)z

"z + µz ! "z2 ! µ#(1)

0 (z), (4.61)

where

A& =)(µ ! ")() + +&)

µ) + (µ ! ")+&. (4.62)


Proof. Using (1.19) we express (4.37) as a q-integral and we obtain that

#0(z) =A

) + +

(q; q)#

(!"(1$z)&+) , )

&+) ; q)#2!1

.

!"(1 ! z)

) + +,

+

) + +; 0; q, q

/

=A

() + +)(1 ! q)

+ 1

0

(qs; q)#

(!"(1$z)s&+) , )s&+) ; q)#

dqs. (4.63)

Taking the limit as q " 1$ and using (1.8), (1.10) and (1.18) yields

limq%1!

#0(z) =A&

+&

+ 1

0(1 ! s)

"$$

$1e$!$$

(1$z)sds, (4.64)

where A& = limq%1! A. After a change of variable in (4.64) we arrive at (4.60) which

is Yechiali and Altman (2006) equation (2.8). Equations (4.59) and (4.61) are now

obvious by taking limits as q " 1$ in (4.35) and (4.38). !

For the case of full synchronization we have the following theorem.

Theorem 4.8. In case q " 0+ and +(1 ! q) = +& fixed, we have

%(2)(0, 0) =A&() + +&)

)() + +& + ")(4.65)

%(2)(n, 0) =

.

"

) + +& + "

/n

%(2)(0, 0), n $ 1 (4.66)

%(2)(n, 1) =

'

(

)

A$

&+)$+"$µ

%2

"µ

3n!2

"&+)$+"

3n&

, if µ (= ) + +& + "

nA$

µ

2

"µ

3n, if µ = ) + +& + ", n $ 1,

(4.67)

where

A& =)(µ ! ")() + +&)

µ) + (µ ! ")+&. (4.68)

Proof. We take the limit as q " 0+ in (4.37). This yields

#(2)0 (z) =

A&() + +&)

)() + +& + "(1 ! z)), (4.69)

where A& is given by (4.68). By expanding (4.69) in power series of z we obtain

easily (4.65) and (4.66). Taking q " 0+ in (4.38) implies, after some simplifications,


that

#(2)1 (z) =

A&"z

µ() + +& + ")(1 ! "µz)(1 ! "

&+)$+"z). (4.70)

By analyzing2

(1 ! "µz)(1 ! "

&+)$+"z)3$1

in partial fractions for the two cases µ (=) + +& + " and µ = ) + +& + ", and expanding in power series of z we obtain (4.67).

!

Chapter 5

Synchronized reneging in

single server vacation queues

Part II

In this chapter we extend the analysis of the two queueing models with vacations and

impatient customers introduced in chapter 4, in a framework with general service

and vacation times.

5.1 Introduction

In this chapter, we study two models with vacations, where the customers are im-

patient but they perform synchronized abandonments. These models are 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 cus-

tomers decide whether to leave the system or not. Therefore, we have synchronized

departures for some of the customers.

The first model is the single-server queue with multiple vacations, where cus-

tomers decide whether to abandon the system or not when the vacation periods

96 Synchronized reneging in single server vacation queues – Part II

finish. In the second model, we suppose that the abandonments epochs occur ac-

cording to a Poisson process during vacation periods. At the abandonment epochs,

every present customer remains in the system with probability q or abandons the

system with probability p = 1 ! q, independently of the others.

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 stationary distributions by using a generating

function approach. In Section 5.5 we present several numerical results that illustrate

the e"ect of the various parameters on the performance measures of the model.

5.2 Model description

We consider a queueing system where customers arrive one by one according to a

Poisson process at rate ". Service is provided by a single server who can be in

one of two modes: on (active) or o" (non-active - on vacation). Customers are

served singly when the server is on, while no service is provided when the server

is o". The service times are generally distributed according to a distribution B(t),

having Laplace-Stieltjes transform (LST) B(s) = E(e$sB) and finite first and second

moments E(B) and E(B2), where the random variable B represents the service time.

The residual (or equilibrium) service time is denoted by Be, the distribution Be(t)

of which is given by

Be(t) =

B t0 (1 ! B(u))du

E(B).

There is infinite waiting room. Whenever the system becomes empty, the server

begins a vacation. We assume multiple vacations, i.e., if the system is still empty

at the end of a vacation, the server takes another one. If, on the contrary, there

is at least one waiting customer at the end of a vacation, the server starts again

to provide service. The vacation times are generally distributed according to a

distribution V (t), having LST V (s) = E(e$sV ) and finite first and second moments

E(V ) and E(V 2), where the random variable V represents the vacation time. The

5.2 Model description 97

residual vacation time is denoted by Ve with distribution

Ve(t) =

B t0 (1 ! V (u))du

E(V ),

and LST

Ve(s) =1 ! V (s)

E(V )s. (5.1)

Regarding the abandonments we consider the two models that we defined in chapter

4, i.e.:

• Unique Abandonment Epoch (UAE) : Every time the server finishes a vacation,

every present customer decides whether to stay in the system with probability

q or to abandon it with probability p = 1 ! q, independently of the others.

• Multiple Abandonment Epochs (MAE) : During server vacations, abandon-

ment opportunities occur according to a Poisson process with rate +. At these

epochs, every present customer remains in the system with probability q or

abandons the system with probability p = 1 ! q, independently of the others.

Hence, in either model, the number of customers is reduced according to a bino-

mial distribution at every abandonment epoch. However, the analysis of the UAE

model turns out to be much easier than the one of the MAE model. For this reason,

in what follows, we describe briefly the results for the UAE model and we provide

more details for the analysis of the MAE model.

We are interested in the equilibrium behavior of the model, so we need to establish

first the stability condition. For the pure vacation model (p = 0) the stability

condition is (see, e.g., Takagi (1991))

, = "E(B) < 1. (5.2)

Hence, the above condition is su!cient for the stability of the UAE and MAE models.

It is also necessary, since the system behaves as a standard M/G/1 queue while the

server is active. Clearly, the only exception is the degenerate case p = 1 for the case

of the UAE model. Then condition (5.2) is not required for stability. Throughout

the paper we assume the validity of condition (5.2).


5.3 Mean value analysis

We now assume the general framework introduced in Section 5.2, i.e., the service

and vacation times are both generally distributed. The analysis is similar for the two

models, so we treat them simultaneously up to the point where the abandonment

mechanism enters.

We suppose that the system is in equilibrium and we consider as before the

random variables L representing the number of customers in the system and S rep-

resenting the sojourn time of a customer. Let also Li be the conditional number of

customers in the system, given that the server is in state i, i = 0, 1. Further we

denote by pi the probability (or fraction of time) that the server is in state i, i = 0, 1.

Let us consider a tagged arriving customer. Then, by PASTA, the probability that

this customer finds the server in state i is pi and in this case, he finds on average

E(Li) customers in front of him.

If he finds the server providing service, then his mean sojourn time is equal to

the residual service time of the customer in service plus the service times of all cus-

tomers waiting in the queue plus his own service time. Hence, his mean sojourn

time is E(Be) + (E(L1) ! 1)E(B) + E(B).

If he finds the server on vacation, then he has to wait for the vacation time to

expire before servicing starts and he may decide to abandon at one of the abandon-

ment opportunities (there is just one in case of the UAE model, but possibly many

in case of the MAE model). Let E(V &) be his mean time in the system till the end

of the vacation. If the tagged customer is still in the system just after the end of

the vacation, then his sojourn time after the return of the server depends on the

number of customers (still) in front of him. Define % as the probability that the

tagged customer stays in the system at the end of the vacation period and define

%& as the probability that the tagged customer and a customer, who was already

present at his arrival, both stay in the system. Then, we have that the mean number


of customers in front of him at the end of the vacation period is %&E(L0). Hence

E(S) = p1 (E(Be) + E(L1)E(B)) + p0 (E(V &) + %&E(L0)E(B) + %E(B)) . (5.3)

Further, Little’s law states that

E(L) = "E(S). (5.4)

Also, if we would act as if the customers arriving during a vacation are waiting in

a “vacation area” and transferred to the queue as soon as the server returns, then

application of Little’s law to the vacation area yields

E(L0) = "E(V &). (5.5)

The unconditional E(L) is related to the conditional ones as

E(L) = p0E(L0) + p1E(L1). (5.6)

Conservation of work gives

p1 = ("p0% + "p1)E(B), (5.7)

and,

p0 + p1 = 1. (5.8)

If we determine %, then the equations (5.7)-(5.8) yield immediately the probabilities

p0 and p1. If we also determine %& and E(V &), then the equations (5.3)-(5.6) su!ce

for the computation of the unknown mean values E(L), E(L0), E(L1) and E(S). The

computations of %, %& and E(V &) depend on the specific abandonment mechanism,

so we treat the UAE and the MAE models separately.

5.3.1 Mean value analysis of the UAE model

In the UAE model a customer that arrives when the server is on vacation has a

unique opportunity to abandon the system at the end of the vacation. Therefore

he will stay in the system for the residual vacation time and then he will decide


whether to leave or not, so E(V &) = E(Ve). The probability % is clearly equal to q,

while %& is equal to q2. Solution of (5.3)-(5.8) yields:


E(S) =1

1 ! ,p

,

E(Ve) + (q2 ! 1),E(Ve)-

+q

1 ! ,p

.

,E(Be)

1 ! ,+ E(B)

/

,


p0 =1 ! ,

1 ! ,p, p1 =

,q

1 ! ,p. (5.9)

5.3.2 Mean value analysis of the MAE model

In the MAE model a customer that arrives when the server is on vacation may

have several opportunities to abandon the system. The computation of %, %& and

E(V &) is not immediate as in case of the UAE model. However, we observe that,

if the tagged customer arrives during a vacation, then the time till abandonment

is exponential with rate +p. By denoting the time till abandonment by T , we can

write

V & = min(Ve, T ) and % = P (Ve < T ).

Hence, by conditioning on the length of Ve,

% =

+ #

0P (t < T )dVe(t)

=

+ #

0e$)ptdVe(t)

= Ve(+p), (5.10)

and

E(V &) = E(Ve + T ) ! E(max(Ve, T ))

= E(Ve) +1

+p!.

E(Ve) + %1

+p

/

= (1 ! %)1

+p. (5.11)


To compute %& we condition on the length of Ve and on the number of abandonment

epochs during Ve,

%& =

+ #

0

#$

n=0

e$)t(+t)n

n!(q2)ndP (Ve & t)

=

+ #

0e$)te)tq

2dVe(t)

=

+ #

0e$)(1$q2)tdVe(t)

= Ve(+(1 ! q2)). (5.12)

By solving (5.3)-(5.8), and taking into account (5.10)-(5.12), we finally get:

Theorem 5.2. The mean sojourn time is equal to

E(S) =1

1 ! , + ,%(E(V &) + (%& ! 1),E(V &)) +

%

1 ! , + ,%

.

,E(Be)

1 ! ,+ E(B)

/

,

where %, %& and E(V &) are given by (5.10), (5.12) and (5.11), respectively, and

p0 =1 ! ,

1 ! , + ,%, p1 =

,%

1 ! , + ,%. (5.13)

5.4 Equilibrium distribution

The aim of this section is to determine the PGF of the number of customers in the

system. By conditioning on the state of the server, we obtain

E(zL) = p0E(zL0) + p1E(zL1). (5.14)

On what the PGF of L1 is concerned we have the following theorem.

Theorem 5.3. The PGF of L1 can be obtained as

E(zL1) =(1 ! ,)

,·z(1 ! B("(1 ! z)))

B("(1 ! z)) ! z·

1 ! E(zLv )

E(Lv)(1 ! z), (5.15)

where Lv denotes the number of customers in the system just after the end of the

vacation.


We will provide three separate proofs for theorem 5.3 because each one of them

reveals a new probabilistic viewpoint for studying the system under consideration.

The first one is more intuitive while the other two use strict probabilistic arguments

from the theory of regenerative processes.

Proof I. To find the PGF of L1 we first need the number of customers in the system

just after the end of a vacation which we have denoted by Lv. We now proceed as in

Fuhrmann (1984). Define the primary customers to be the ones just after the start

of the busy period and the secondary customers to be the ones who arrive during

the busy period. Further, we change the service discipline to non-preemptive LCFS;

this does not a"ect the number of customers in the system. So, after servicing a

primary customer, the server will serve any secondary customer until there is none

present. Each primary customer generates a standard M/G/1 busy period, at the

end of which the server either begins servicing the next primary customer or, if the

system is empty, takes a vacation. Let Qp be the number of primary customers

waiting for service in the queue (excluding the one possibly in service). If we remove

server vacations from the time axis and glue together the service periods, then we

readily obtain from the renewal reward theorem (see, e.g., Ross (2003)) that the

fraction of time the queue contains n primary customers is equal to

P (Qp = n) =P (Lv > n)

E(Lv), n $ 0.

Hence,

E(zQp) =1

E(Lv)

#$

n=0

P (Lv > n)zn =1 ! E(zLv )

E(Lv)(1 ! z). (5.16)

Let L1|M/G/1 denote the conditional number of customers in the corresponding stan-

dard M/G/1 with arrival rate " and service time distribution B(t), given that the

server is in state 1 (i.e., active); so, according to the Pollaczek-Khinchin formula,

E(zL1|M/G/1) =(1 ! ,)

,·z(1 ! B("(1 ! z)))

B("(1 ! z)) ! z. (5.17)

Since L1 = L1|M/G/1 + Qp, where L1|M/G/1 and Qp are independent, we obtain, by

(5.16) and (5.17),

E(zL1) = E(zL1|M/G/1)E(zQp), (5.18)


which concludes the proof of theorem 5.3. !

Proof II. Let %j be the fraction of time spent in state j during a busy period.

Then %j = E["j]E["] , where %j denotes the time spent in state j during a busy period

and % the length of the busy period. The r.v. Lv denotes the number of customers

just after the end of the vacation time, with corresponding PGF E[zLv ]. The server

starts to serve when Lv customers have been accumulated in the system. Therefore

the mean length of the busy period for this system is equal to E[Lv ] ·E[&], where &

is the duration of the busy period for the M/G/1 queue. More explicitly

E[%] = E[Lv ]E[&] = E[Lv]E[B]

1 ! ,

= E[Lv ]1

"

,

1 ! ,. (5.19)

Let &j denote the time spent in state j during a busy period for the M/G/1 queue.

Given Lv = n, we can easily obtain that

E[%j |Lv = n] =j$

k=1

E[&k] , j < n, (5.20)

since the time spent in state j given that Lv = n > j is equal to the time spent in

j starting from state j until the first time that the system visits j ! 1 + the time

spent in j from j ! 1 to j ! 2 + · · · + the time spent in j from 1 to 0 which is equal

to the time spent in state 1 in the M/G/1 queue in one cycle + the time spent in

state 2 in the M/G/1 queue in one cycle + · · · + the time spent in state j in the

M/G/1 queue in one cycle. Similarly, when Lv = n & j we obtain,

E[%j |Lv = n] =n$

k=1

E[&j$n+k] , j $ n. (5.21)

Therefore

E[%j |Lv = n] = 1{j<n}

j$

k=1

E[&k] + 1{j'n}

n$

k=1

E[&j$n+k] . (5.22)


. . . . . . . . .

!

!n

!j!

n ! 1 !n ! 2!1

!0

Fig. 5.1: Schematic representation for the case where j < n.

Moreover we obtain that

%j =E[%j]

E[%]

=E[%j]

E[Lv]E[&]

=1

E[Lv]E[&]

#$

n=1

E[%j |Lv = n] Pr[Lv = n] . (5.23)

. . . . . . . . .

!

!n

!j

!

n ! 1 !n ! 2!1

!0

Fig. 5.2: Schematic representation for the case where j $ n.


By plugging (5.22) in (5.23) we obtain

%j =1

E[Lv]

#$

n=1

j$

k=1

E[&k]

E[&]Pr[Lv = n]1{j<n}

+1

E[Lv]

#$

n=1

n$

k=1

E[&j$n+k]

E[&]Pr[Lv = n]1{j'n} . (5.24)

We define pk to be the steady state probabilities of the M/G/1 queue and E[zLM/G/1 ]

the corresponding PGF. But E[#j ]E[#] = pj

1$p0- E[&j ] = pj

E[#]* . Equation (5.24),

then, assumes the form

%j =1

,E[Lv ]

#$

n=1

j$

k=1

pk Pr[Lv = n]1{j<n}+1

,E[Lv]

#$

n=1

n$

k=1

pj$n+k Pr[Lv = n]1{j'n} .

(5.25)

Then the PGF E[zL1 ] takes the following form

E[zL1 ] =#$

j=1

%jzj

(5.25)=

1

,E[Lv]

#$

j=1

zj#$

n=1

j$

k=1

pk Pr[Lv = n]1{j<n}

+1

,E[Lv]

#$

j=1

zj#$

n=1

n$

k=1

pj$n+k Pr[Lv = n]1{j'n}

=1

,E[Lv]

#$

n=1

n$1$

j=1

zjj$

k=1

pk Pr[Lv = n]

+1

,E[Lv]

#$

n=1

#$

j=n

zjn$

k=1

pj$n+k Pr[Lv = n]

=1

,E[Lv]

#$

n=1

;

=

#$

j=1

zjj$

k=1

pk !#$

j=n

zjj$

k=1

pk

+#$

j=n

zjj$

k=j$n+1

pk

>

@Pr[Lv = n]


=1

,E[Lv]

#$

n=1

;

=

#$

j=1

zjj$

k=1

pk !#$

j=n

zjj$n$

k=1

pk

>

@Pr[Lv = n]

=1

,E[Lv]

#$

n=1

;

=

#$

j=1

zjj$

k=1

pk ! zn#$

j=1

zjj$

k=1

pk

>

@Pr[Lv = n]

=1

,E[Lv]

#$

n=1

Pr[Lv = n](1 ! zn)#$

k=1

pk

#$

j=k

zj

=1

,E[Lv]

#$

n=1

Pr[Lv = n](1 ! zn)#$

k=1

pkzk

1 ! z

=1

,E[Lv]

#$

n=1

Pr[Lv = n]1 ! zn

1 ! z(E[zLM/G/1 ] ! p0)

=(E[zLM/G/1 ] ! p0)

,E[Lv ]

1 ! E[zLv ]

1 ! z. (5.26)

Using that E[zLM/G/1 ] = p0(1$z)B("(1$z))

B("(1$z))$zand p0 = 1 ! , in equation (5.26) yields

(5.15). !

Proof III. Following the notation that was introduced in proof II we will now

divide a cycle into a random number of disjoint intervals separated by the service

completion epochs and calculate E[%j ] as the sum of the contributions from the

disjoint intervals to the expected sojourn time in state j. Thus we define the random

variables Nj by

Nj = the number of service completion epochs in one cycle at which j cus-

tomers are left behind, j = 0, 1, . . ..

Moreover, using the lack of memory of the Poisson arrival process, we define the

quantity Akj by

Akj = the expected amount of time that j customers are present during a

service time starting when k customers are present.

Then, given that Lv = n, we note that the first service in a cycle starts with n


customers, it follows that

E[%j |Lv = n] = Anj +j$

k=1

E[Nk|Lv = n]Akj , j = n, n + 1, . . . (5.27)

while for all states j < n we observe that they can’t be reached during the first

service and so

E[%j |Lv = n] =j$

k=1

E[Nk|Lv = n]Akj , j = 1, 2, . . . , n ! 1 . (5.28)

In general

E[%j |Lv = n] = Anj1{j'n} +j$

k=1

E[Nk|Lv = n]Akj , j = 1, 2, . . . . (5.29)

We can obtain a relation between E[%j |Lv = n] and E[Nj |Lv = n], using a simple

observation that given Lv = n

the number of downcrossings from state j + 1 to j in one cycle

= the number of upcrossings from state j to j + 1 in one cycle, for all j =

n, n + 1, . . . ,

the number of downcrossings from state j + 1 to j in one cycle

= the number of upcrossings from state j to j + 1 in one cycle + 1, for all

j = 1, 2, . . . , n ! 1 .

The expected number of downcrossings from state j +1 to j in one cycle, given that

Lv = n, is by definition equal to E[Nj |Lv = n]. On the other hand, since the arrival

process is a Poisson process, we have that the expected number of upcrossing from

state j to j + 1 in one cycle, given that Lv = n, is equal to "E[%j |Lv = n]. Thus

we find

E[Nj |Lv = n] = 1{j<n} + "E[%j|Lv = n], j = 1, 2, . . . . (5.30)

Substituting equation (5.30) into equation (5.29) we obtain that

E[%j |Lv = n] = Anj1{j'n}+j$

k=1

Akj1{k<n}+"j$

k=1

E[%k|Lv = n]Akj , j = 1, 2, . . . .

(5.31)


Then

E[%j ] =#$

n=1

E[%j |Lv = n] Pr[Lv = n]

=#$

n=1

Anj1{j'n} Pr[Lv = n] +#$

n=1

j$

k=1

Akj1{k<n} Pr[Lv = n]

+"#$

n=1

j$

k=1

E[%k|Lv = n]Akj Pr[Lv = n]

=#$

n=1

Anj1{j'n} Pr[Lv = n] +#$

n=1

j$

k=1


+"j$

k=1

E[%k]Akj . (5.32)

Dividing both sides of equation (5.32) by E[%] we obtain

%j =1

E[%]

#$

n=1

Anj1{j'n} Pr[Lv = n] +1

E[%]

#$

n=1

j$

k=1


+"j$

k=1

%kAkj . (5.33)

To specify the constants Akj, suppose that at epoch 0 a service starts when k

customers are present. Define the random variable Ij(t) = 1 if at time t the service

is still in progress and j customers are present and let Ij(t) = 0 otherwise. Then for

j $ k,

Akj = E[

+ #

0Ij(t)dt]

=

+ #

0Pr[Ij(t) = 1]dt]

=

+ #

0Pr[B $ t]e$"t

("t)j$k

(j ! k)!dt

=

+ #

0(1 ! B(t))e$"t

("t)j$k

(j ! k)!dt . (5.34)


Let Akj = aj$k and let A(z) =!#

n=0 anzn be the corresponding PGF, then

A(z) =#$

n=0

zn+ #

0(1 ! B(t))e$"t

("t)n

n!dt

=

+ #

0(1 ! B(t))e$"t

#$

n=0

("tz)n

n!dt

=

+ #

0(1 ! B(t))e$"(1$z)tdt

=1 ! B("(1 ! z))

"(1 ! z). (5.35)

We define the generating function E[zL1 ] by

E[zL1 ] =#$

j=1

%jzj , |z| & 1 .

Multiplying both sides of (5.33) by zj and summing over j we derive that

E[zL1 ] =1

E[%]

,

A(z)E[zLv ] + A(z)z1 ! E[zLv ]1z

1 ! z

-

+ "E[zL1 ]A(z) . (5.36)

Solving for E[zL1 ] and substituting equations (5.19) and (5.35) yields (5.15). !

It is now clear that the determination of the PGF of the number of customers

in the system reduces to the computation of E(zL0), E(zLv ) and E(Lv). These

computations depend on the specific abandonment mechanism, so we treat the UAE

and the MAE model separately.

5.4.1 Equilibrium distribution of the UAE model

In this case the probabilities p0 and p1 are given by (5.9). The number of customers

during a vacation, L0, are exactly the ones who arrived during the age of the va-

cation, and the age is in distribution the same as the residual vacation. Hence, by

conditioning on Ve = t, the number of arrivals is Poisson with parameter "t, and


thus we get

E(zL0) =

+ #

0e$"t(1$z)dVe(t)

= Ve("(1 ! z)). (5.37)

The number of arrivals during a vacation of length t, who decide to stay at the end

of the vacation, is Poisson with parameter q"t. Hence, the PGF of the number of

customers in the system, just after the end of the vacation, is

E(zLv ) = V (q"(1 ! z)). (5.38)

We can now combine (5.14), (5.15), (5.37), (5.38) and (5.1) to obtain the PGF

of the number of customers in the system. We have the following theorem.

Theorem 5.4. The PGF of the number of customers in the system is given by

E(zL) =1 ! ,

1 ! ,pVe("(1 ! z)) +

q(1 ! ,)

1 ! ,p

z(1 ! B("(1 ! z)))

B("(1 ! z)) ! zVe(q"(1 ! z)).

Remark 5.1. In case q = 1 ! p = 1, the equation above reduces to the well-known

Fuhrmann-Cooper decomposition for the number of customers in the M/G/1 with

server vacations (and no abandonments),

E(zL) = Ve("(1 ! z))E(zLM/G/1), (5.39)

where LM/G/1 is the (unconditional) number of customers in the corresponding

standard M/G/1.

5.4.2 Equilibrium distribution of the MAE model

In this case the probabilities p0 and p1 are given by (5.13). We need also to ob-

tain E(zL0), E(zLv ) and E(Lv). We start with the latter. Conditioning on the

event that V = t, the number of abandonment epochs is Poisson with parameter +t.

Given the number of abandonment epochs is n(> 0), the event times (s1, s2, . . . , sn)

of these epochs will be distributed as the order statistics (U1:n, U2:n, . . . , Un:n) of a

random sample (U1, U2, . . . , Un) from the uniform distribution in (0, t]. The number


of arrivals in each of the intervals (0, s1], (s1, s2], . . ., (sn$1, sn], (sn, t] are Poisson

with parameters "s1, "(s2 ! s1), . . ., "(sn ! sn$1), "(t! sn) respectively. Moreover,

the individuals that arrive during these intervals with remain till time t with prob-

abilities qn, qn$1, . . ., q, 1 respectively. Since the sum of Poisson random variables

is again Poisson, we can conclude that the number of customers at the end of the

vacation is Poisson with parameter

'(t, n, s1, . . . , sn) = "s1qn + "(s2 ! s1)q

n$1 + · · · + "(sn ! sn$1)q + "(t ! sn)

= !"qn$1(1 ! q)s1 ! "qn$2(1 ! q)s2 ! · · ·! "(1 ! q)sn + "t,

valid for n > 0, and if n = 0, this number is Poisson with parameter "t. Hence,

E(zLv |V = t) = e$)te$"t(1$z)

+#$

n=1

+ t

0

+ t

s1

· · ·+ t

sn!1

e$)t(+t)n

n!e$$(t,n,s1,...,sn)(1$z) n!

tndsn · · · ds1 .

(5.40)

To put E(zLv |V = t) in a more compact form, we use the auxiliary identity

In(t, n,*1,*2, . . . ,*n) =

+ t

0

+ t

s1

· · ·+ t

sn!1

e+1s1++2s2+···++nsndsn · · · ds2ds1

=n+1$

k=1

(!1)k+1etPn

i=k +i

6n$ki=0

!k+ij=k *j ·

6k$1i=1

!k$1j=k$i *j

, (5.41)

which can be easily established by induction. In order to use (5.41) to simplify (5.40)

we substitute

*j = "(1 ! q)(1 ! z)qn$j , j = 1, 2, . . . , n (5.42)


andn$

i=k

*i = "(1 ! z)(1 ! qn$k+1) (5.43)

k+i$

j=k

*j = "(1 ! z)(1 ! qi+1)qn$k$i (5.44)

k$1$

j=k$i

*j = "(1 ! z)qn$k+1(1 ! qi) (5.45)

n$k*

i=0

k+i$

j=k

*j = ["(1 ! z)]n$k+1q(n!k+1

2 )(q; q)n$k+1 (5.46)

k$1*

i=1

k$1$

j=k$i

*j = ["(1 ! z)]k$1q(n$k+1)(k$1)(q; q)k$1. (5.47)

in (5.41), yielding (after some algebra)

In(t, n,*1,*2, . . . ,*n) =et"(1$z)

["(1 ! z)]n

n$

k=0

(!1)ke$t"(1$z)qn!k

(q; q)k(q; q)n$kq(n!k

2 )+(n$k)k. (5.48)

Using (5.40), (5.41) and (5.48) we obtain

E(zLv |V = t) = e$()+"(1$z))t#$

n=0

+nn$

k=0

et"(1$z)(1$qn!k)

("(1 ! z))n(!1)kq$(n+k$1)(n$k)/2

(q; q)k(q; q)n$k

= e$)t#$

n=0

.

+

"(1 ! z)

/n n$

k=0

(!1)kq$(n+k$1)(n$k)/2

(q; q)k(q; q)n$ke$t"(1$z)qn!k

= e$)t#$

k=0

(!1)k

(q; q)k

#$

n=k

.

+

"(1 ! z)

/n q$(n+k$1)(n$k)/2

(q; q)n$ke$t"(1$z)qn!k

= e$)t#$

k=0

(!1)k

(q; q)k

#$

n=0

.

+

"(1 ! z)

/n+k q$(n+2k$1)n/2

(q; q)ne$t"(1$z)qn

= e$)t#$

n=0

.

+

"(1 ! z)

/n q$(n2)

(q; q)ne$t"(1$z)qn

#$

k=0

1

(q; q)k

.

!+q$n

"(1 ! z)

/k

= e$)t#$

n=0

.

+

"(1 ! z)

/n q$(n2)

(q; q)ne$t"(1$z)qn 1

(! )q!n

"(1$z) ; q)#


= e$)t#$

n=0

.

+

"(1 ! z)

/n q$(n2)

(q; q)ne$t"(1$z)qn

)1

2

)"(1$z)

3nq$(n

2)$n(!"(1$z)q) ; q)n(! )

"(1$z) ; q)#

=e$)t

(! )"(1$z) ; q)#

#$

n=0

qne$t"(1$z)qn

(q; q)n(!"(1$z)q) ; q)n

. (5.49)

Hence, after unconditioning, we conclude that the PGF of Lv assumes the form

E(zLv ) =

+ #

0E(zLv |V = t)dV (t)

=

+ #

0

e$)t

(! )"(1$z) ; q)#

#$

n=0

qn


(!"(1$z)q) ; q)n

dV (t)

=1

(! )"(1$z) ; q)#

#$

n=0

qnV (+ + "(1 ! z)qn)

(q; q)n(!"(1$z)q) ; q)n

. (5.50)

Moreover,

E(Lv|V = t) = e$)t"t

+#$

n=1

+ t

0

+ t

s1

· · ·+ t

sn!1

e$)t(+t)n

n!'(t, n, s1, . . . , sn)

n!

tndsn · · · ds1.

(5.51)

To put E(Lv |V = t) in a more compact form we use the auxiliary identity

Jn(t, n,*1,*2, . . . ,*n) =

+ t

0

+ t

s1

· · ·+ t

sn!1

[*0 + *1s1 + · · · + *nsn] dsn · · · ds2ds1

= *0tn

n!+

n$

k=1

*kk tn+1

(n + 1)!, (5.52)

which can be easily established by induction. In order to use (5.52) to simplify (5.51)

we substitute

*0 = "t

*j = !"(1 ! q)qn$j, j = 1, 2, . . . , n


in (5.52), yielding

Jn(t, n,*1,*2, . . . ,*n) = "tn+1

(n + 1)!

1 ! qn+1

1 ! q. (5.53)

Using (5.51) and (5.53) we obtain

E(Lv|V = t) ="

+p(1 ! e$)pt). (5.54)

Note that, after unconditioning, the mean value of Lv assumes the form

E(Lv) ="

+p(1 ! V (+p)). (5.55)

To determine the PGF of the number of customers during a vacation, we can copy

the approach above, where the vacation V should be replaced by its age Ve. This

leads to

E(zL0) =

+ #

0E(zL0 |Ve = t)dVe(t)

=

+ #

0

e$)t

(! )"(1$z) ; q)#

#$

n=0

qn


(!"(1$z)q) ; q)n

dVe(t)

=1

(! )"(1$z) ; q)#

#$

n=0

qnVe(+ + "(1 ! z)qn)

(q; q)n(!"(1$z)q) ; q)n

. (5.56)

Based on the PGFs of L0, Lv and the mean value E(Lv) we can now use (5.14),

(5.15), (5.56) and (5.50) to obtain the PGF of the number of customers in the

system. Therefore, we immediately obtain the following result.

Theorem 5.5. The PGF of the number of customers in the system is given by

E(zL) =1 ! ,

1 ! , + ,%E(zL0)+

%(1 ! ,)

1 ! , + ,%

z(1 ! B("(1 ! z)))

B("(1 ! z)) ! z

1 ! E(zLv )

E(Lv)(1 ! z), (5.57)

where %, E(zL0), E(zLv ) and E(Lv) are given by (5.10), (5.56), (5.50) and (5.55),

respectively.


Remark 5.2. As the rate + of the abandonment epochs tends to zero (i.e., no

abandonments), then equation (5.57) again reduces to the standard decomposition

(5.39). To show this, we first use the identity

(a; q)n = (a$1q1$n; q)n(!a)nq(n2)

to rewrite (5.50) in the form

E(zLv ) =#$

n=0

+n

("(1 ! z))nV (+ + "(1 ! z)qn)

(q; q)n(! )q!n

"(1$z) ; q)#q(n2)

.

For + = 0 this equation reduces to E(zLv ) = V ("(1! z)). Similarly, it can be shown

that E(zL0) = Ve("(1 ! z)) for + = 0. Substitution of these expressions into (5.57)

yields (5.39).

5.5 Numerical results

This section is devoted to several numerical results that shed further light on the

e"ect of the various model parameters and distributions on the system performance.

To this end we perform several numerical experiments by keeping all but one pa-

rameter fixed and study the mean number of customers in the system as a function

of the varying parameter. The e"ect of ", E(B) or E(V ) on the mean number

of customers in the system, E(L), appears to be what is normally expected, i.e.,

E(L) is increasing in ", E(B) and E(V ). Much more interesting is the e"ect of the

abandonment probability p on E(L). In the numerical scenarios presented below

we assume that B is exponentially distributed with E(B) = 1, and to show the

e"ect of the dispersion of the vacation times, we consider Erlang, exponential and

Hyper-exponential vacation time distributions.

We present the results for the UAE and the MAE model separately, as they are

qualitatively di"erent.


5.5.1 Numerical results for the UAE model

For the UAE model, we plot the graph of E(L) as a function of p, while keep-

ing all other parameters fixed. We consider two cases regarding the mean vacation

time E(V ) that correspond to figures 5.3 and 5.4. Figure 5.3, where E(V ) = 1,

corresponds to ‘small’ vacation times, while figure 5.4, where E(V ) = 10, corre-

sponds to ‘large’ vacation times. Moreover, in every figure we plot 3 curves, each

corresponding to a di"erent vacation time distribution (Erlang, exponential and

Hyper-exponential). We observe that E(L) increases as the coe!cient of variation

of the vacation time, cV , increases, while keeping E(V ) fixed. This agrees with the

usual observation that congestion increases with variability.

0 0.2 0 .4 0 .6 0 .8 10.2

0 .4

0 .6

0 .8

1

1.2

1 .4

1 .6

1 .8

p

E[L

]

U AE model: λ=0.5 and B is E xponentia lly distributed with E [B]=1

V∼ E rlang E (V )=1 c

V=0.7

V∼ E xp E (V )=1V∼ H

2 E (V )=1 c

V=1.3

Fig. 5.3: E(L) versus p.

0 0.2 0 .4 0 .6 0 .8 13.5

4

4.5

5

5.5

6

6.5

7

7.5

8

p

E[L

]

U AE model: λ=0.5 and B is E xponentia lly distributed with E [B]=1

V∼ E rlang E (V )=10 c

V=0.7

V∼ E xp E (V )=10V∼ H

2 E (V )=10 c

V=1.3

Fig. 5.4: E(L) versus p.

In figure 5.3 the mean number of customers E(L) is decreasing with respect to

the abandonment probability p. This agrees with our intuition that ‘the greater the

abandonment probability, the less the congestion of the system’. However, figure

5.4 shows that this is not always the case, i.e., the mean number of customers E(L)

may exhibit non-monotonic behavior with respect to the abandonment probability

p. Thus, an increase in the abandonment probability may lead to an increase of the

mean number of customers in the system!

This finding may be intuitively justified as follows. Indeed, for large vacation


times, the mean number of customers E(L) depends primarily on what happens

when the server is on vacation. However, the abandonment probability p does not

influence the mean number of customers given that the server is on vacation, since

the unique abandonment epoch occurs at the end of the vacation. Moreover, a

large number of customers will accumulate during a large vacation time. Thus, in

this case, a high abandonment probability p implies that the busy period will start

with just a few customers, so the next vacation period which is responsible for the

accumulation of many customers in the system will start soon. So, in this case, an

increase of the abandonment probability implies an increase of the congestion of the

system.

5.5.2 Numerical results for the MAE model

For the MAE model the most important numerical finding concerns the e"ect of

synchronization. Figures 5.5-5.7 demonstrate the e"ect of the level of synchroniza-

tion on E(L). In these figures, the arrival rate ", the e"ective abandonment rate

per customer +& = +p and the mean service time E(B) are kept fixed, i.e., " = 0.9,

+& = 0.6 and E(B) = 1. In every graph we consider a di"erent value for E(V ), i.e.

E(V ) = 0.8, E(V ) = 0.4 and E(V ) = 0.25 in figures 5.5, 5.6 and 5.7, respectively.

In each graph we consider three di"erent cases for the distribution of V (x) being

Erlang, exponential or Hyper-exponential with coe!cients of variation cV = 0.7,

cV = 1 and cV = 1.3. For all cases we observe that E(L) is an increasing convex

function of p, i.e., the more the synchronization the more the congestion. On the

other hand, the e"ect of the variance of the vacation times is not clear. Figure

5.5 shows that for systems with large mean vacation times, the more variable the

vacation times, the less congested the system. In contrast, figure 5.7 shows that for

systems with small mean vacation times, the mean number of customers increases

with the variation of the vacation times. Moreover, for moderate mean vacation

times, the situation is mixed (see figure 5.6). In summary, as the mean vacation

time increases, the e"ect of the vacation variability on the congestion of the system

turns from negative to positive!

This finding can be intuitively justified as follows. For large vacation times, the


0 0.1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 18.92

8.94

8.96

8.98

9

9.02

9.04

9.06

9.08

9.1

p

E[L

]

M AE model: λ=0.9 , ζ*=0 .6 , B is E xponentia lly distributed with E [B]=1

V∼ E rlang E (V )=0 .8 cV

=0.7

V∼ E xp E (V )=0 .8V∼ H

2 E (V )=0 .8 c

V=1.3

Fig. 5.5: E(L) versus p, +& fixed.

0 0.1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 19.02

9.03

9.04

9.05

9.06

9.07

9.08

9.09

pE

[L]



=0.7

V∼ E xp E (V )=0 .4V∼ H

2 E (V )=0 .4 c

V=1.3


0 0.1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 19.035

9.04

9.045

9.05

9.055

9.06

9.065

9.07

9.075

p

E[L

]



=0.7

V∼ E xp E (V )=0 .25V∼ H

2 E (V )=0 .25 c

V=1.3


0 0.1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 10.8

0 .9

1

1.1

1 .2

1 .3

1 .4

1 .5

p

E[L

]

M AE model: λ=0.5 , ζ=1.5 and B is E xponentia lly distributed with E [B]=1

V∼ E rlang E (V )=0 .7 c

V=0.7

V∼ E xponentia l E (V )=0 .7V∼ H

2 E (V )=0 .7 c

V=1.3

Fig. 5.8: E(L) versus p, + fixed.


number of customers at the end of a vacation period tends to a fixed number (see

the formula for E(Lv)). Hence big and very big vacation times are then followed

by busy periods that start from practically the same mean number of customers.

So a more irregular (more variable) distribution for the vacation times leaves the

possibility for some small vacation times and then the subsequent busy periods will

start with significantly fewer customers. Thus the more variable the vacation time

distribution, the more likely there exist busy periods starting with few customers.

In contrast, vacation times with a large mean but a small variance imply that al-

most all busy periods start with the same big number of customers. Hence, for large

mean vacation times, the variability of the vacation time has a positive e"ect on the

performance of the system, in the sense that it reduces congestion. For small mean

vacation times this e"ect is reversed.

In figure 5.8 we present the graph of E(L) with respect to p. In this figure, the

arrival rate ", the mean vacation time E(V ), the abandonment rate + and the mean

service time E(B) are kept fixed, " = 0.5, E(V ) = 1, + = 1.5 and E(B) = 1. The

function is decreasing convex as p varies from 0 to 1. In this case, the e"ect of the

variance of vacation time is that the more regular the distribution the milder the

e"ect of the abandonment probability p on the mean number of customers in the

system.

Appendix

Proof of summation formula (2.20)

3!2

"

a, q, 0

aq, bq; q, q

#

=#$

n=0

(a, q; q)n(q, aq, bq; q)n

qn

=#$

n=0

qn

(bq; q)n

1 ! a

1 ! aqn

= (1 ! a)#$

n=0

qn

(bq; q)n

#$

k=0

(aqn)k

= (1 ! a)#$

k=0

ak#$

n=0

(qk+1)n

(bq; q)n

= (1 ! a)#$

k=0

ak2!1

"

q, 0

bq; q, qk+1

#

(1.12)= (1 ! a)

#$

k=0

ak (b, qk+2; q)#(bq, qk+1; q)#

2!1

"

q, 0

qk+2; q, b

#

= (1 ! a)(1 ! b)#$

k=0

ak (qk+2; q)#(qk+1; q)#

#$

n=0

bn

(qk+2; q)n

(1.3)= (1 ! a)(1 ! b)

#$

k=0

ak (qk+2; q)#(qk+1; q)#

#$

n=0

bn (qk+n+2; q)#(qk+2; q)#

= (1 ! a)(1 ! b)#$

k=0

ak#$

n=0

bn (qk+n+2; q)#(qk+1; q)#

= (1 ! a)(1 ! b)#$

k=0

ak#$

n=0

bn (q; q)k+n+1

(q; q)k

= (1 ! a)(1 ! b)#$

k=0

ak#$

n=k+1

bn$k$1 (q; q)n(q; q)k

= (1 ! a)(1 ! b)#$

n=1

n$1$

k=0

akbn$k$1 (q; q)n(q; q)k

.

Proof of auxiliary identity (5.41)

We will prove by induction equation (5.41).

In(t, n,*1,*2, . . . ,*n) =

+ t

0

+ t

s1

· · ·+ t

sn!1

e+1s1++2s2+···++nsndsn · · · ds2ds1

=n+1$

k=1

(!1)k+1etPn

i=k +i1

6n$ki=0

!k+ij=k *j ·

6k$1i=1

!k$1j=k$i *j

. (5.41)

We can easily establish that the auxiliary identity (5.41) stands n = 1. Suppose that

the auxiliary identity (5.41) stands for all n & m we will prove that it also stands

for n = m + 1, i.e.

Im+1(t,m + 1,*1,*2, . . . ,*m+1)

=

+ t

0· · ·+ t

sm!1

+ t

sm

e+1s1++2s2+···++m+1sm+1dsm+1dsm · · · ds1

=

+ t

0· · ·+ t

sm!1

e+1s1++2s2+···++msm

0

e+m+1sm+1

*m+1

1t

sm

dsm · · · ds1

=

+ t

0· · ·+ t

sm!1

e+1s1++2s2+···++msm

0

e+m+1t ! e+m+1sm

*m+1

1

dsm · · · ds1

=e+m+1t

*m+1Im(t,m,*1,*2, . . . ,*m)

!1

*m+1Im(t,m,-1,-2, . . . ,-m),

where -i = *i for all i = 0, 1, . . . ,m ! 1 and -m = *m + *m+1. Then

Im+1(t,m + 1,*1,*2, . . . ,*m+1)

=e+m+1t

*m+1

m+1$

k=1

(!1)k+1etPm

i=k +i1

6m$ki=0

!k+ij=k *j ·

6k$1i=1

!k$1j=k$i *j

!1

*m+1

m+1$

k=1

(!1)k+1etPm

i=k ,i1

6m$ki=0

!k+ij=k -j ·

6k$1i=1

!k$1j=k$i -j

=e+m+1t

*m+1

m+1$

k=1

(!1)k+1etPm

i=k +i1

6m$ki=0

!k+ij=k *j ·

6k$1i=1

!k$1j=k$i *j

!1

*m+1

m$

k=1

(!1)k+1etPm+1

i=k +i1

6m$1$ki=0

!k+ij=k *j · (

!m+1j=k *j) ·

6k$1i=1

!k$1j=k$i *j

!1

*m+1(!1)m+2 1

6mi=1

!mj=m+1$i -j

=e+m+1t

*m+1

m+1$

k=1

(!1)k+1etPm

i=k +i1

6m$ki=0

!k+ij=k *j ·

6k$1i=1

!k$1j=k$i *j

!1

*m+1

m$

k=1

(!1)k+1etPm+1

i=k +i

!mj=k *j

6m+1$ki=0

!k+ij=k *j ·

6k$1i=1

!k$1j=k$i *j

!1

*m+1(!1)m+2 1

6mi=1

!mj=m+1$i -j

=1

*m+1

m+1$

k=1

(!1)k+1etPm+1

i=k +i

!m+1j=k *j

6m+1$ki=0

!k+ij=k *j ·

6k$1i=1

!k$1j=k$i *j

!1

*m+1

m$

k=1

(!1)k+1etPm+1

i=k +i

!mj=k *j

6m+1$ki=0

!k+ij=k *j ·

6k$1i=1

!k$1j=k$i *j

!1

*m+1(!1)m+2 1

6mi=1

!m+1j=m+1$i -j

=1

*m+1

m+1$

k=1

(!1)k+1etPm+1

i=k +i*m+1

6m+1$ki=0

!k+ij=k *j ·

6k$1i=1

!k$1j=k$i *j

!1

*m+1(!1)m+2 1

6mi=1

!mj=m+1$i -j

=m+1$

k=1

(!1)k+1etPm+1

i=k +i1

6m+1$ki=0

!k+ij=k *j ·

6k$1i=1

!k$1j=k$i *j

+(!1)m+3 16m+1

i=1

!m+1j=m+1$i *j

=m+2$

k=1

(!1)k+1etPm+1

i=k +i1

6m+1$ki=0

!k+ij=k *j ·

6k$1i=1

!k$1j=k$i *j

which concludes the proof.

Proof of auxiliary identity (5.52)

We will prove by induction equation (5.52).

Jn(t, n,*1,*2, . . . ,*n) =

+ t

0

+ t

s1

· · ·+ t

sn!1

[*0 + *1s1 + · · · + *nsn] dsn · · · ds2ds1

= *0tn

n!+

n$

k=1

*kk tn+1

(n + 1)!. (5.52)

Equivalently we must prove that

+ t

0

+ t

s1

· · ·+ t

sn!1

dsn · · · ds2ds1 =tn

n!(ap.I)

and+ t

0

+ t

s1

· · ·+ t

sn!1

si dsn · · · ds2ds1 = itn+1

(n + 1)!, i = 1, . . . , n. (ap.II)

We will provide two proofs of formulas (ap.I) and (ap.II). The first one is analytic

while the second one is probabilistic.

Proof I. In order to prove formulas (ap.I) and (ap.II) we will first prove that

+ t

s0

+ t

s1

· · ·+ t

sn!1

(t ! sn)k dsn · · · ds2ds1 = k!(t ! s0)n+k

(n + k)!, k % N. (ap.III)

We can easily establish that the auxiliary identity (ap.III) stands for n = 1 and for

all k % N. Suppose that the auxiliary identity (ap.III) stands for some n = m and

for all k % N we will prove that it also stands for n = m + 1, i.e.+ t

s0

· · ·+ t

sn!1

+ t

sn

(t ! sn+1)k dsn+1dsn · · · ds1 =

+ t

0· · ·+ t

sn!1

(t ! sn)k+1

k + 1dsn · · · ds1

=1

k + 1(k + 1)!

(t ! s0)n+k+1

(n + 1 + k)!

= k!(t ! s0)n+1+k

(n + 1 + k)!.

We can now easily establish that the auxiliary identity (ap.I) stands for some arbi-

trary n, i.e.+ t

0· · ·+ t

sn!2

+ t

sn!1

dsndsn$1 · · · ds1 =

+ t

0· · ·+ t

sn!2

(t ! sn$1) dsn$1 · · · ds1

(ap.III)=

tn

n!.

We will now proceed with the proof of (ap.II). We must discriminate on whether

i (= n and i = n, we then have that

for i (= n+ t

0· · ·+ t

sn!2

+ t

sn!1

si dsndsn$1 · · · ds1

=

+ t

0· · ·+ t

si!1

si

+ t

si

· · ·+ t

sn!2

(t ! sn$1) dsn$1 · · · dsi+1dsi · · · ds1

(ap.III)=

+ t

0· · ·+ t

si!1

si(t ! si)n$i

(n ! i)!dsi · · · ds1

= !1

(n ! i)!

+ t

0· · ·+ t

si!1

(t ! si)n$i+1dsi · · · ds1

+t

(n ! i)!

+ t

0· · ·+ t

si!1

(t ! si)n$idsi · · · ds1

(ap.III)= !

1

(n ! i)!(n ! i + 1)!

tn+1

(n + 1)!

+t

(n ! i)!(n ! i)!

tn

n!

= itn+1

(n + 1)!

for i = n+ t

0· · ·+ t

sn!2

+ t

sn!1

sn dsndsn$1 · · · ds1

= !+ t

0· · ·+ t

sn!2

+ t

sn!1

(t ! sn) dsndsn$1 · · · ds1

+t

+ t

0· · ·+ t

sn!2

+ t

sn!1

dsndsn$1 · · · ds1

(ap.III)= !

tn+1

(n + 1)!+ t

tn

n!

= ntn+1

(n + 1)!.

Proof II. Let U1, U2, . . . , Un be n independent and identical random variables

uniformly distributed over [0, t]. Let U1:n, U2:n, . . . , Un:n be the order statistics of

U1, U2, . . . , Un so that

U1:n & U2:n & . . . & Un:n .

Let f(s1, s2, . . . , sn) be the joint density of U1:n, U2:n, . . . , Un:n. Then it can be shown

that

f(s1, s2, . . . , sn) =

'

(

)

n!tn if 0 & s1 & . . . & sn & t

0 otherwise.(ap.IV)

Then equation (ap.I) can be obtained by

+ t

0

+ t

s1

· · ·+ t

sn!1

f(s1, s2, . . . , sn) dsn · · · ds2ds1 = 1 .

From equation (ap.IV) we derive the marginal density of Ui:n, i = 1, 2, . . . , n, as

fi(u) =n!

(i ! 1)!(n ! i)!

2u

t

3i$1 1

t

2

1 !u

t

3n$i, u % [0, t]. (ap.V)

Then the expected value of Ui:n is given as

E[Ui:n] =it

n + 1. (ap.VI)

To prove equation (ap.II) we rewrite it in the following form

+ t

0

+ t

s1

· · ·+ t

sn!1

sif(s1, s2, . . . , sn) dsn · · · ds2ds1 =it

n + 1, i = 1, . . . , n. (ap.VII)

It is easy to see that the left hand side of equation (ap.VII) is equal to E[Ui:n], which

concludes the proof.

Bibliography

[1] Abate, J., Choudhury, G.L. and Whitt, W. (2000) An introduction to numerical

transform inversion and its application to probability models, in Grassmann,

W.K., ed., Computational Probability, pp.257–323, Kluwer Academic Publish-

ers, Boston.

[2] Adan, I. (1991) A Compensation Approach for Queueing Problems, Doctoral

Thesis, Technische Universiteit Eindhoven, Eindhoven.

[3] Adan, I., Economou, A. and Kapodistria, S. (2009) Synchronized reneging in

queueing systems with vacations, Queueing Systems 62, 1–33.

[4] Adan, I. and van der Wal, J. (1998) Di!erence and Di!erential

Equations in Stochastic Operations Research. Online Notes, URL:

http://www.win.tue.nl/ iadan/

[5] Adan, I. and Weiss, G. (2006) Analysis of a simple Markovian re–entrant line

with infinite supply of work under the LBFS policy. Queueing Systems 54,

169–183.

[6] Ahlfors, L.V. (1979) Complex Analysis, 3rd Edition, McGraw-Hill, Singapore.

[7] Altman, E. and Borovkov, A.A. (1997) On the stability of retrial queues. Queue-

ing Systems 26, 343–363.

[8] Altman, E., and Yechiali, U. (2006) Analysis of customers’ impatience in queues

with server vacations, Queueing Systems 52, 261–279.

[9] Altman, E. and Yechiali, U. (2008) Infinite-server queues with system’s addi-

tional tasks and impatient customers, Probability in the Engineering and Infor-

mational Sciences 22, 477-493.

[10] Altiok, T. (1987) Queues with group arrivals and exhaustive service discipline.

Queueing Systems 2(4), 307-320.

[11] Artalejo, J.R., Economou, A. and Lopez-Herrero, M.J. (2007) Evaluating

growth measures in populations subject to binomial and geometric catastro-

phes, Mathematical Biosciences and Engineering 4, 573–594.

[12] Artalejo, J.R. and Gomez-Corral, A. (1997) Steady state solution of a single-

server queue with linear repeated requests, Journal of Applied Probability 34,

223–233.

[13] Artalejo, J.R. and Pozo, M. (2002) Numerical calculation of the stationary dis-

tribution of the main multiserver retrial queue, Annals of Operations Research

116, 41–56.

[14] Baccelli, F., Boyer, P. and Hebuterne, G. (1984) Single-server queues with

impatient customers. Advances in Applied Probability 16, 887–905.

[15] Baykal-Gursoy M. and Xiao, W. (2004) Stochastic decomposition in M/M/#queues with Markov modulated service rates, Queueing Systems 48, 75–88.

[16] Bini, D.A., Latouche, G. and Meini, B. (2005) Numerical Methods for Structured

Markov Chains, Oxford Science Publications, New York.

[17] Boxma, O.J. and de Waal, P.R. (1994) Multiserver queues with impatient cus-

tomers. ITC 14, 743–756.

[18] Brockwell, P.J., Gani, J. and Resnick, S.I. (1982) Birth, immigration and catas-

trophe processes, Advances in Applied Probability 14, 709–731.

[19] Cohen, J.W. and Boxma, O.J. (1983) Boundary Value Problems in Queueing

System Analysis, North-Holland Mathematics Studies, 79, North-Holland Pub-

lishing Co., Amsterdam.

[20] Cooper, R.B. (1970) Queues served in cyclic order: waiting times. The Bell

System Technical Journal 49, 399-413.

[21] Daley, D.J. (1965) General customer impatience in the queue GI/G/1. Journal

of Applied Probability 2, 186–205.

[22] Denteneer, D., van Leeuwaarden, J.S.H. and Adan, I.J.B.F. (2007) The acqui-

sition queue. Queueing Systems 56(3), 229–240.

[23] Economou, A. (2004) The compound Poisson immigration process subject to

binomial catastrophes, Journal of Applied Probability 41, 508–523.

[24] Economou, A. and Fakinos, D. (2008) Alternative approaches for the transient

analysis of Markov chains with catastrophes, Journal of Statistical Theory and

Practice 2, 183–197.

[25] Economou, A. and Kapodistria, S. (2009) q-series in Markov chains with bi-

nomial transitions: Studying a queue with synchronization, Probability in the

Engineering and Informational Science 23, 75–99.

[26] Economou, A. and Kapodistria, S. (2009) Synchronized abandonments in

a single server unreliable queue, European Journal of Operational Research,

doi:10.1016/j.ejor.2009.07.014.

[27] Fuhrmann, S.W. (1984) A note on the M/G/1 queue with server vacations.

Operations Research 32, 1368–1373.

[28] Gaver, D.P. (1962) A waiting time with interrupted service, including priorities.

Journal of the Royal Statistical Socieity 24, 698-720.

[29] Gail, H.R., Hantler, S.L. and Taylor, B.A. (2000) Use of characteristics roots for

solving infinite state Markov chains, in Grassmann, W.K., ed., Computational

Probability, pp.205–255, Kluwer Academic Publishers, Boston.

[30] Gasper, G. and Rahman, M. (2004) Basic Hypergeometric Series, 2nd Edition,

Cambridge University Press.

[31] Grassmann, W.K. (2002) Real eigenvalues of certain tridiagonal matrix poly-

nomials, with queueing applications, Journal of Linear Algebra and its Appli-

cations 342, 93–106.

[32] Hanschke (1987) Explicit formulas for the characteristics of the M/M/2/2 queue

with repeated attempts. Journal of Applied Probability 24, 486–494.

[33] Harris, C.M. and Marchal, W.G. (1988) State dependence in M/G/1 queue

with generalized vacations. Operations Research, 36(4), 560-565.

[34] Ismail, M.E.H. (1985) A queueing model and a set of orthogonal polynomials,

Journal of Mathematical Analysis and Applications 108, 575–594.

[35] Kao, E.P.C. (1997) An Introduction to Stochastic Processes, Duxbury Press,

Belmont.

[36] Keilson, J. (1962) Queues sub ject to service interruption. Annals Mathematical

Statistics, 33, 1314-1322.

[37] Keilson, J. and Servi, L.D. (1993) The matrix M/M/# system: retrial models

and Markov modulated sources, Advances in Applied Probability 25, 453–471.

[38] Kemp, A.W. and Newton, J. (1990) Certain state-dependent processes for di-

chotomised parasite populations, Journal of Applied Probability 27, 251–258.

[39] Kemp, A.W. (1992) Steady-state Markov chain models for the Heine and Euler

distributions, Journal of Applied Probability 29, 869–876.

[40] Kemp, A.W. (1998) Absorption sampling and absorption distribution, Journal


[41] Kemp, A.W. (2005) Steady-state Markov chain models for certain q-confluent

hypergeometric distributions, Journal of Statistical Planning and Inference

135, 107–120.

[42] Krantz, S.G. (1999) Handbook of Complex Variables, Birkhanser, Boston.

[43] Krishnamoorthy, A., Deepak, T.G. and Joshua, V.C. (2005) An M/G/1 retrial

queue with nonpersistent customers and orbital search, Stochastic Analysis and

Applications 23, 975–997.

[44] Kroese, D.P., Scheinhardt, W.R.W. and Taylor, P.G. (2004) Spectral properties

of the tandem Jackson network, seen as a quasi-birth-and-death process, Annals


[45] Kulkarni, V.G. (1995) Modeling and Analysis of Stochastic Systems, Chapman

and Hall, London, UK.

[46] Latouche, G. and Ramaswami, V. (1999) Introduction to Matrix-Analytic Meth-

ods in Stochastic Modeling, ASA-SIAM Series on Statistics and Applied Prob-

ability, SIAM, Philadelphia.

[47] Levy, Y. and Yechiali, U. (1975) Utilization for the idle time in an M/G/1

queue with server vacations. Management Science,22, 202-211.

[48] Levy, Y. and Yechiali, U. (1976) An M/M/s queue with server vacations. Cana-

dian Journal of Operational Research and Information Processing, 14, 153-163.

[49] Mitrani, I. and Chakka, R. (1995) Spectral expansion solution for a class

of Markov models: Application and comparison with the matrix-geometric

method, Performance Evaluation 23, 241–260.

[50] Mitrany, I. L. and Avi-Itzhak, B. (1968) A many server queue with service

interruptions. Operational Research, 16(3), 628-638.

[51] Neuts, M.F. (1981) Matrix-geometric Solutions in Stochastic Models: An Algo-

rithmic Approach, Johns Hopkins University Press, Baltimore, MD.

[52] Neuts, M.F. (1989) Structured Stochastic Matrices of M/G/1 Type and Their

Applications, Marcel Dekker Inc., New York.

[53] Neuts, M.F. (1994) An interesting random walk on the nonnegative integers,

Journal of Applied Probability 31, 48–58.

[54] Palm, C. (1953) Methods of judging the annoyance caused by congestion. Tele

4, 189–208.

[55] Palm, C. (1957) Research on telephone tra!c carried by full availability groups.

Tele 1, 107. (English translation of results first published in 1946 in Swedish in

the same journal, which was then entitled Tekniska Meddelanden fran Kungl.

Telegrfstyrelsen).

[56] Perel, N. and Yechiali, U. (2009) Queues with slow servers and impatient cus-

tomers, European Journal of Operational Research.

[57] Ross, S.M. (2003) Introduction to Probability Models (8th ed.), Academic press,

Londen.

[58] Shaked, M. and Shanthikumar, J.G. (1986), Multivariate imperfect repair, Op-

erations Research 34(3), 437-448.

[59] Takacs, L. (1974) A single-server queue with limited virtual waiting time. Jour-

nal of Applied Probability 11, 612–617.

[60] Takagi, H. (1991) Queueing Analysis: A Foundation of Performance Evalu-

ation. Vol. 1. Vacation and Priority Systems, North-Holland Publishing Co.,

Amsterdam.

[61] Tian, N. and Zhang, Z.G. (2006) Vacation Queueing Models. Theory and Ap-

plications, Springer, New York.

[62] Tijms, H.C. (1994) Stochastic Models: An Algorithmic Approach, Wiley, New

York.

[63] Wilf, H. S. (1990) Generatingfunctionology (1st ed.), Academic Press Profes-

sional, Inc., San Diego, CA.

[64] Yadin, M. and Naor, P. (1963) Queueing systems with a removable service

station. Operations Research 14, 393-405.

[65] Yechiali, U. (2007) Queues with system disasters and impatient customers when

system is down, Queueing Systems 56, 195–202.

Stationary performance evaluation measures in multi …skapodis/phdthesis_eng.pdf · 2009-09-09 ·...

Documents

Transcript of Stationary performance evaluation measures in multi …skapodis/phdthesis_eng.pdf · 2009-09-09 ·...