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Applied Mathematical Modelling 29 (2005) 1100–1120

A discrete-time Geo[X]/G/1 retrial queuewith control of admission

J.R. Artalejo a,*, I. Atencia b, P. Moreno c

a Department of Statistics and O.R., Faculty of Mathematics, Complutense University of Madrid, Madrid 28040, Spainb Departamento de Matemática Aplicada, E.T.S.I. de Telecomunicación, Universidad de Málaga, Málaga 29071, Spain

c Departamento de Economı́a y Empresa, Facultad de Ciencias Empresariales, Universidad Pablo de Olavide,

Sevilla 41013, Spain

Received 1 April 2004; received in revised form 1 January 2005; accepted 8 February 2005Available online 25 March 2005

Abstract

This paper analyses a discrete-time Geo/G/1 retrial queue with batch arrivals in which individual arrivingcustomers have a control of admission. We study the underlying Markov chain at the epochs immediatelyafter the slot boundaries making emphasis on the computation of its steady-state distribution. To this endwe employ numerical inversion and maximum entropy techniques. We also establish a stochastic decompo-sition property and prove that the continuous-time M/G/1 retrial queue with batch arrivals and control ofadmission can be approximated by our discrete-time system. The outcomes agree with known results forspecial cases.� 2005 Elsevier Inc. All rights reserved.

Keywords: Control of admission; Discrete-time model; Markov chain; Maximum entropy; Numerical inversion;Stochastic decomposition

0307-904X/$ - see front matter � 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.apm.2005.02.005

* Corresponding author. Fax: +34 91 3944607.E-mail addresses: jesus_artalejo@mat.ucm.es (J.R. Artalejo), iatencia@ctima.uma.es (I. Atencia), mpmornav@

upo.es (P. Moreno).

mailto:jesus_artalejo@mat.ucm.esmailto:iatencia@ctima.uma.esmailto:mpmornav@ upo.esmailto:mpmornav@ upo.es

J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120 1101

1. Introduction

Queueing systems with repeated attempts have been widely used to model many problems intelecommunication and computer systems. The essential feature of a retrial queue is that arrivingcustomers who find all servers busy are obliged to abandon the service area and join a retrialgroup (called orbit) in order to try their luck again after some random time. For a detailed reviewof the main results and the literature on this topic the reader is referred to [1–5]. In addition, acomprehensive discussion of similarities and differences between retrial queues and their standardcounterparts was carried out by Artalejo and Falin in [6].

Retrial queues were initially investigated in continuous time, but some years ago Yang and Li[7] extended the study to the discrete-time systems. In spite of the importance of the discrete-timeretrial queues, little is still known about them; in fact, the only published works on this topic are[7–13]. However, since the publication of [14], discrete-time queues have been discussed exten-sively by different researchers, see for example [15–18]. Besides, the interest in discrete systemshas suffered a spectacular growth with the advent of the new technologies. A fundamental incen-tive to study discrete-time queues is that these systems are more appropriate than their continu-ous-time counterparts for modelling computer and telecommunication systems, since the basicunits in these systems are digital such as a machine cycle time, bits and packets, etc. Indeed, muchof the usefulness of discrete-time queues derives from the fact that they can be used in the perfor-mance analysis of Broadband Integrated Services Digital Network (BISDN), AsynchronousTransfer Mode (ATM) and related computer communication technologies wherein the continu-ous-time models do not adapt [15,18].

In this paper we deal with a discrete-time Geo[X]/G/1 retrial queue where individual arrivingcustomers have a control of admission. The applicability of a batch retrial queue relates to theperformance evaluation of Local Area Networks operating under transmission protocols likethe CSMA/CD (Carrier Sense Multiple Access with Collision Detection) [19]. In this context,messages which are transmitted could consist of a random number of packets. If, upon arri-val, the channel is free, one packet is randomly chosen to be transmitted and the rest of themare stored in the orbit; otherwise, if the channel is occupied, all the packets are stored in theorbit and will try the retransmission after a random period of time. On the other hand, withrespect to the control of admission, this work is an attempt to extend the existing controlmechanisms for the admission of customers in the continuous-time queues [20] to discrete-timesystems.

It should be noted that the analysis of most of the discrete-time retrial models is based on themethodology developed by Yang and Li [7]. In [7], solutions are given in terms of partial gener-ating functions. A recursive scheme for the computation of the stationary distribution is also pro-posed but it is given in terms of multiple infinite series. Consequently, the authors show theapplicability of their scheme only in the special case where the service times are geometric distrib-uted. The investigation of the queue length moments was reduced to the first order case, i.e., theexpected number of customers in orbit. In this paper, we also will use the methodology of Yangand Li but our goal is to make progress on the numerical calculation of the stationary probabil-ities. To reach this objective we propose to employ two computational techniques: (i) Numericalinversion based on the Fast Fourier Transform [21], and (ii) Maximum entropy approximationsbased on the knowledge of some mean value constraints [22–24]. We will obtain a numerically

1102 J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120

tractable expression for the second order moment as a prerequisite for the use of the maximumentropy approach.

The remainder of the paper is structured as follows. In the next section, the mathematicaldescription of the model is introduced. In Section 3, we analyse the Markov chain associated withthe system and obtain the distributions of the orbit and system sizes. The analysis includes a gen-erating function solution, expressions for the first two moments and a first recursive approach.The numerical discussion of the computation of the stationary probabilities based on the numer-ical inversion and the maximum entropy approaches is postponed to Section 6. Previously, in Sec-tion 4, we find a stochastic decomposition law and as consequence we give upper and lowerestimates of the distance between the steady-state distributions for our queueing system and itscorresponding standard system. Section 5 examines the relationship between our discrete-timesystem to its continuous-time counterpart. Some details concerning the computation of the max-imum entropy probabilities are indicated in Appendix A.

2. The mathematical description

We consider a single-server discrete-time queue where the time axis is divided into equal inter-vals (called slots). It is assumed that all queueing activities (arrivals, departures and retrials) occurat the slot boundaries and therefore they may happen at the same time. For mathematical conve-nience, we will suppose that the departures occur at the moment immediately before the slotboundaries, whereas arrivals and retrials occur at the moment immediately after the slot bound-aries. Thus, we will discuss the model for the early arrival system (EAS) policy. Details on theEAS discipline and related concepts can be found in [16,25].

Batches of customers arrive at the system according to a geometrical process with probability p.The number of individual external customers arriving in each batch is k P 1 with probability ckand a 2 (0,1] is the individual admission probability for each customer. Hence, the probabilitythat a group of size n P 0 joins the system is given by

a0 ¼X1k¼1

ckð1� aÞk;

an ¼X1k¼n

ckk

n

� �anð1� aÞk�n; n P 1:

We will denote by c(z) and a(z) the probability generating functions of the sequences fckg1k¼1 andfang1n¼0 respectively. From the above equalities, we can easily obtain the following relation:

aðzÞ ¼ cða zþ 1� aÞ:

Moreover, we will denote by �ak and �ck the corresponding kth factorial moments. The ratep� = p(1 � a0) will designate the probability that in a slot a batch of customers arrives and at leasta customer of the batch is admitted to the system, and �p� ¼ 1� p�.

It should be pointed out that this Bernoulli admission mechanism can be regarded as a schemeto model situations where a proportion of the arriving packets are corrupted and accordingly theymust be deleted.

J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120 1103

The policy for those admitted customers is that if the server is idle then one customer starts hisservice immediately and the rest enter the retrial group, whereas all the admitted customers go tothe retrial group when the server is busy. The discipline to access a repeated customer to the serveris governed by a geometrical distribution with probability 1 � r, where r is the probability that arepeated customer does not make a retrial in a slot. If more than one customer retries at the sameslot when the server is unoccupied, any of them is randomly chosen to be served and the othersmust return to the orbit. In case of simultaneous coalescence of primary admitted customers andretrials, one of them occupies the server and the rest join the orbit. Obviously, the identity of theselected customer does not modify the system state distribution. The service times follow a generaldistribution fsig1i¼1 with probability generating function S(x) and nth factorial moments bn.

It is assumed that the input flow, the batch sizes, the intervals between successive repeated at-tempts and the service times are mutually independent.

In order to avoid trivial cases, we will suppose 0 < p < 1 and 0 6 r < 1. The load of the system isgiven by q ¼ p�a1b1.

3. The Markov chain

At time m+ (the instant immediately after the m-th slot), the system can be described by theprocess

Xm ¼ ðCm; nm;NmÞ;

where Cm symbolizes the server state (0 if the server is idle or 1 if the server is busy) and Nm is thenumber of repeated customers. If Cm = 1, nm represents the remaining service time of the cus-tomer currently being served.

It can be shown that fXm; m 2 Ng is a Markov chain of our queueing system and its state spaceis

S ¼ fð0; kÞ : k P 0; ð1; i; kÞ : i P 1; k P 0g:

It is not difficult to see that fXm; m 2 Ng is irreducible and aperiodic.

The first question to be studied is the condition for the positive recurrence of the Markov chain.For this purpose, we will use Foster�s criterion where the choice of the test function is essential.

Theorem 1. The Markov chain fXm; m 2 Ng is positive recurrent if and only if q < 1.

Proof. Foster�s criterion establishes that an irreducible and aperiodic Markov chain Xm with statespaceS is ergodic if there exists a non-negative function f(s), s 2 S, called test function, and e > 0such that the mean drift

xs ¼ E½f ðXmþ1Þ � f ðXmÞ j Xm ¼ s�

is finite for all s 2 S and xs 6 �e for all s 2 S except perhaps a finite number.

In our case, we consider the following test function on the state space

f ðsÞ ¼b1k if s 2 fð0; kÞ : k P 0g;i� 1þ b1k if s 2 fð1; i; kÞ : i P 1; k P 0g:

�

1104 J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120

In order to obtain the mean drifts, it is necessary to specify the relations:

• if Xm 2 {(0,k) : k P 0; (1,1,k) : kP 0}:

Xmþ1 ¼ð1; i; k þ lÞ with probability palþ1si; i P 1; l P 0;ð1; i; k � 1Þ with probability �p�ð1� rkÞsi; i P 1;ð0; kÞ with probability �p�rk;

8>:

• if Xm 2 {(1, i,k) : iP 2, k P 0}:

Xmþ1 ¼ð1; i� 1; kÞ with probability �p�;ð1; i� 1; k þ lÞ with probability pal; l P 1:

�

Therefore, the mean drift is given by

xs ¼q� 1þ �p�rk if s 2 fð0; kÞ : k P 0; ð1; 1; kÞ : k P 0g;q� 1 if s 2 fð1; i; kÞ : i P 2; k P 0g:

�

Let q < 1. Then e = (1 � q)/2 is positive and for all states s 2 S we have

• if s 2 {(0,k) : k P 0; (1,1,k) : k P 0}:

limk!1

xs ¼ limk!1

q� 1þ �p�rk� �

¼ q� 1 ¼ �2e < �e;

• if s 2 {(1, i,k) : i P 2, k P 0}:

xs ¼ q� 1 ¼ �2e < �e:

Thus, xs < �e for all states except for a finite number. Therefore, q < 1 is a sufficient condition forthe positive recurrence of the Markov chain. At the end of the proof of the Theorem 2, we willshow that this condition is also necessary. h

Our next objective is to study the stationary distribution of the Markov chain fXm; m 2 Ng,which will be denoted by

ps ¼ limm!1

P ½Xm ¼ s�; s 2 S:

The evolution of the chain is governed by the one-step transition probabilities given by

• if k P 0:

pð0;kÞð0;kÞ ¼ �p�rk;pð1;1;kÞð0;kÞ ¼ �p�rk;

• if iP 1, k P 0:

pð0;lÞð1;i;kÞ ¼ pak�lþ1si; l ¼ 0; . . . ; k;pð0;kþ1Þð1;i;kÞ ¼ �p�ð1� rkþ1Þsi;

J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120 1105

pð1;1;lÞð1;i;kÞ ¼ pak�lþ1si; l ¼ 0; . . . ; k;pð1;1;kþ1Þð1;i;kÞ ¼ �p�ð1� rkþ1Þsi;pð1;iþ1;lÞð1;i;kÞ ¼ pak�l; l ¼ 0; . . . ; k � 1 ðk P 1Þ;pð1;iþ1;kÞð1;i;kÞ ¼ �p�:

The Kolmogorov equations together with the normalization condition for the stationary distri-bution are given by

p0;k ¼ �p�rkp0;k þ �p�rkp1;1;k; k P 0; ð1Þ

p1;i;k ¼ psiXkl¼0

ak�lþ1p0;l þ �p� 1� rkþ1� �

sip0;kþ1 þ psiXkl¼0

ak�lþ1p1;1;l þ �p�ð1� rkþ1Þsip1;1;kþ1

þ ð1� d0;kÞpXk�1l¼0

ak�lp1;iþ1;l þ �p�p1;iþ1;k; i P 1; k P 0; ð2Þ

1 ¼X1k¼0

p0;k þX1i¼1

X1k¼0

p1;i;k; ð3Þ

where da,b denotes Kronecker�s delta.To resolve the system of Eqs. (1)–(3), we define the following generating functions:

u0ðzÞ ¼X1k¼0

p0;kzk;

u1ðx; zÞ ¼X1i¼1

X1k¼0

p1;i;kxizk

and we enunciate a lemma, whose proof is easily obtained and accordingly omitted.

Lemma 1

(1) The inequality S(x) 6 x holds for 0 6 x 6 1.(2) If q < 1, the inequality Sð�p þ paðzÞÞ > z holds for 0 6 z < 1, where �p ¼ 1� p.

The following result provides us the solution of the Kolmogorov equations in terms of the pre-ceding generating functions. The proof follows the methodology given by Yang and Li [7].

Theorem 2. If q < 1, the stationary distribution of the Markov chain fXm; m 2 Ng has thefollowing generating functions:

u0ðzÞ ¼ ð1� qÞ

Q1k¼1

GðrkzÞQ1k¼1

GðrkÞ;

1106 J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120

u1ðx; zÞ ¼SðxÞ � Sð�p þ paðzÞÞx� ð�p þ paðzÞÞ

px½1� aðzÞ�Sð�p þ paðzÞÞ � zu0ðzÞ;

where

GðzÞ ¼ �p� Sð�p þ paðzÞÞ � z½�p þ paðzÞ�½Sð�p þ paðzÞÞ � z�½�p þ paðzÞ� :

Proof. To solve the Kolmogorov equations (1)–(3), we need to define the next auxiliary generat-ing functions

u1;iðzÞ ¼X1k¼0

p1;i;kzk; i P 1:

Multiplying Eqs. (1) and (2) by zk and summing over k yields

u0ðzÞ ¼ �p�u0ðrzÞ þ �p�u1;1ðrzÞ; ð4Þ

u1;iðzÞ ¼ ½�p þ paðzÞ�u1;iþ1ðzÞ þaðzÞ � 1

zpsiu0ðzÞ þ

�p þ paðzÞz

siu1;1ðzÞ; i P 1: ð5Þ

Multiplying both sides of Eq. (5) by xi and summing over i leads to

x� ½�p þ paðzÞ�x

u1ðx; zÞ ¼aðzÞ � 1

zpSðxÞu0ðzÞ þ

�p þ paðzÞz

½SðxÞ � z�u1;1ðzÞ: ð6Þ

Putting x ¼ �p þ paðzÞ in the previous equation gives

u1;1ðzÞ ¼p½1� aðzÞ�Sð�p þ paðzÞÞ

½Sð�p þ paðzÞÞ � z�½�p þ paðzÞ�u0ðzÞ: ð7Þ

Using the Lemma 1 and taking into account the limit

limz!1

1� aðzÞSð�p þ paðzÞÞ � z ¼

�a11� q ;

we observe that if q < 1 then the generating function u1,1(z) is well-defined in [0,1) and can beextended by continuity in 1.

Substituting Eq. (7) into Eq. (6), we get

u1ðx; zÞ ¼SðxÞ � Sð�p þ paðzÞÞx� ð�p þ paðzÞÞ

px½1� aðzÞ�Sð�p þ paðzÞÞ � zu0ðzÞ:

In order to find out u0(z), we substitute u1,1(rz) into Eq. (4) obtaining

u0ðzÞ ¼ GðrzÞu0ðrzÞ: ð8Þ

Applying recursively Eq. (8) yields

u0ðzÞ ¼ u0ð0ÞY1k¼1

GðrkzÞ: ð9Þ

J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120 1107

Letting z = 1 in the preceding equation, we achieve

u0ð0Þ ¼ u0ð1ÞY1k¼1

GðrkÞ( )�1

;

where u0(1) = 1 � q is calculated from the normalization condition u0(1) + u1(1,1) = 1.To conclude this proof it will be enough to prove that the infinite product in Eq. (9) converges if

q < 1. That is why we will rewrite the auxiliary function G(z) as

GðzÞ ¼ 1þ F ðzÞ; ð10Þ

where

F ðzÞ ¼ p�z½�p þ paðzÞ� � p½aðzÞ � a0�Sð�p þ paðzÞÞ

½Sð�p þ paðzÞÞ � z�½�p þ paðzÞ� :

Using Lemma 1, it is easy to show that if q < 1 then F(z) is a non-negative function in [0,1].Regarding Eq. (10), the infinite product in Eq. (9) can be written as

Y1

k¼1GðrkzÞ ¼

Y1k¼1

½1þ F ðrkzÞ�:

It is well-known that the infinite product in previous equation is convergent if the seriesP1k¼1F r

kzð Þ converges, which is clear seeing that limk!1F rkþ1zð ÞF rkzð Þ ¼ r < 1.

Lastly, since u0(1) > 0, we have that q < 1 is a necessary condition for the ergodicity of theMarkov chain. h

Corollary 1(1) The marginal generating function of the orbit size when the server is idle is given by

u0ðzÞ:

(2) The marginal generating function of the orbit size when the server is busy is given by

u1ð1; zÞ ¼1� Sð�p þ paðzÞÞSð�p þ paðzÞÞ � z u0ðzÞ:

(3) The probability generating function of the orbit size (i.e., of the variable N) is given by

WðzÞ ¼ u0ðzÞ þ u1ð1; zÞ ¼1� z

Sð�p þ paðzÞÞ � zu0ðzÞ:

(4) The probability generating function of the system size (i.e., of the variable L) is given by

UðzÞ ¼ u0ðzÞ þ zu1ð1; zÞ ¼ð1� zÞSð�p þ paðzÞÞSð�p þ paðzÞÞ � z u0ðzÞ:

Remark 1. It is held the relation UðzÞ ¼ WðzÞSð�p þ paðzÞÞ.

1108 J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120

Remark 2. The stationary distribution of the server state

u0ð1Þ ¼ 1� q; u1ð1; 1Þ ¼ q

hinges on the service times distribution only through its mean b1 and is independent of the inter-retrial times distribution.

Corollary 2(1) The probability that the server is idle is u0(1) = 1 � q.(2) The probability that the server is busy is u1(1,1) = q.(3) The mean orbit size is given by

E½N ; idle� ¼ ddz

u0ðzÞ�z¼1

¼ ð1� qÞX1k¼1

G0ðrkÞGðrkÞ r

k;

E½N ; busy� ¼ ddz

u1ð1; zÞ�z¼1

¼ p p�a21b2 þ �a2b12ð1� qÞ þ q

X1k¼1

G0ðrkÞGðrkÞ r

k;

E½N � ¼ ddz

WðzÞ�z¼1

¼ p p�a21b2 þ �a2b12ð1� qÞ þ

X1k¼1

G0ðrkÞGðrkÞ r

k:

(4) The mean system size is given by

E½L� ¼ ddz

UðzÞ�z¼1

¼ qþ p p�a21b2 þ �a2b12ð1� qÞ þ

X1k¼1

G0ðrkÞGðrkÞ r

k:

(5) The mean time a customer spends in the system (including the service time) is given byW = E[L]/p*.

(6) The second factorial moments of the orbit size are given by

E½NðN � 1Þ; idle� ¼ d2

dz2u0ðzÞ

�z¼1

¼ ð1� qÞX1k¼1

G0ðrkÞGðrkÞ r

k

!2þ ð1� qÞ

X1k¼1

G00ðrkÞGðrkÞ � ðG0ðrkÞÞ2

ðGðrkÞÞ2r2k;

E½NðN � 1Þ;busy� ¼ d2

dz2u1ð1; zÞ

�z¼1

¼ q1� qE½NðN � 1Þ; idle� þ p

�a3b1 þ 3p�a1�a2b2 þ p2�a31b33ð1� qÞ

þp�a2b1 þ p2�a21b2� �2

2ð1� qÞ2þ p �a2b1 þ p�a

21b2

1� qX1k¼1

G0 rkð ÞG rkð Þ r

k:

J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120 1109

The proof of both corollaries is cumbersome but standard from Theorem 2 and thus over-looked. The above formulas (6) will be helpful in Section 6 for constructing the maximum entropyapproximation of the stationary probabilities.

Remark 3. To obtain numerical approximations of the previous performance characteristics wehave to estimate the series

X1

k¼1

G0 rkð ÞG rkð Þ r

k �Xn0ðeÞk¼1

G0 rkð ÞG rkð Þ r

k þ 11� r

Z rn0ðeÞþ10

G0ðuÞGðuÞ du ¼

Xn0ðeÞk¼1

G0 rkð ÞG rkð Þ r

k þlnG rn0ðeÞþ1

� �1� r ;

X1k¼1

G00 rkð ÞG rkð Þ � G0 rkð Þð Þ2

G rkð Þð Þ2r2k �

Xn0ðeÞk¼1

G00 rkð ÞG rkð Þ � G0 rkð Þð Þ2

G rkð Þð Þ2r2k þ 1

1� r

Z rn0ðeÞþ10

� G00ðuÞGðuÞ � ðG0ðuÞÞ2

ðGðuÞÞ2udu;

where, for each e > 0, n0(e) is chosen such that rn0ðeÞþ1 < e.

The following corollary provides recursive formulas for the stationary probabilities of theMarkov chain fXm; m 2 Ng. Its proof is a generalization of the arguments given in [7].Corollary 3. If q < 1, then we have:(1) The stationary probabilities of the system state are given by

p0;0 ¼1� qQ1k¼1G r

kð Þ ;

p0;k ¼

Pk�1n¼0

dk�n � rnbk�nð Þrk�np0;n

1� rkð ÞSð�p�Þ ; k P 1;

p1;k ¼1� Sð�p�ÞSð�p�Þ p0;k þ

1� d0;kSð�p�Þ

Xk�1n¼0

dk�nðp0;n þ p1;nÞ; k P 0;

where p1;k ¼P1

i¼1p1;i;k and the coefficients bm and dm are given by

bm ¼ �p�Xmn¼0

X1j¼0

Xjl¼0

j

l

� �Am�nBjþ1�pj�lplþ1aðlÞn ; m P 1;

dm ¼ pAm þXmn¼0

X1j¼1

Xjl¼0

j

l

� �Am�nBj�pj�lplþ1aðlÞn ; m P 1;

Aj ¼X1i¼jþ1

ai; Bj ¼X1i¼jþ1

si; j P 0

and faðlÞn g1n¼0 denotes the lth fold convolution of the sequence fang

1n¼0.

1110 J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120

(2) Let wk = p0,k + p1,k, kP 0, be the stationary probability of the number of customers in orbit.Then, we have

wk ¼p0;k þ ð1� d0;kÞ

Pk�1n¼0

dk�nwn

Sð�p�Þ ; k P 0:

(3) Let /k = p0,k + (1 � d0,k)p1,k�1, kP 0, be the stationary probability of the total number ofcustomers in the system. Then, we have

/k ¼ p0;k þ1� d0;kSð�p�Þ

Xk�1n¼0

ðek�np0;n þ dk�n/nÞ; k P 0;

where

ek ¼X1i¼1

Xij¼0

i

j

� �si�pi�jpja

ðjÞk ; k P 1:

The recursive formulas in Corollary 3 are expressed in terms of multiple infinite series involv-ing the convolutions faðlÞn g

1n¼0. At the light of these expressions it seems very difficult, or indeed

impossible, to get simplifications leading to a tractable computational scheme even for the caseof simple service times particularizations. This drawback will be solved in Section 6 by employ-ing numerical inversion of the generating functions u0(z) and u1(1,z), and maximum entropymethodologies.

Remark 4 (Special cases). This observation judges three particular cases: the case of a batcharrival queue with control of admission and random service discipline, the case of a Geo[X]/G/1retrial queueing system and the case of a Geo/G/1 retrial queue.

(a) When r = 0, then G(rkz) = 1, k P 1 and u0(z) = 1 � q. Consequently, U(z) changes into

UðzÞ ¼ ð1� qÞð1� zÞSð�p þ paðzÞÞSð�p þ paðzÞÞ � z ;

which is the probability generating function of the number of customers in the Geo[X]/G/1/1queueing system with control of admission. This result is not surprising since when r = 0, the re-peated customers make retrials at every slot boundaries until they receive their services. Thismodel is equal to the Geo[X]/G/1/1 with control of admission and random service discipline, sincethe system size distribution is independent of the service discipline.

(b) When a = 1, our model becomes a discrete-time Geo[X]/G/1 retrial queue and the generatingfunctions of the Theorem 2 change into

u0ðzÞ ¼ ð1� qÞ

Q1k¼1

G rkzð ÞQ1k¼1

G rkð Þ;

u1ðx; zÞ ¼SðxÞ � Sð�p þ pcðzÞÞx� ð�p þ pcðzÞÞ

px½1� cðzÞ�Sð�p þ pcðzÞÞ � zu0ðzÞ;

J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120 1111

where

GðzÞ ¼ �p Sð�p þ pcðzÞÞ � z½�p þ pcðzÞ�½Sð�p þ pcðzÞÞ � z�½�p þ pcðzÞ� :

(c) When a(z) = z, our model becomes a discrete-time Geo/G/1 retrial queue and the generatingfunctions of the Theorem 2 change into

u0ðzÞ ¼ ð1� qÞ

Q1k¼1

G rkzð ÞQ1k¼1

G rkð Þ;

u1ðx; zÞ ¼SðxÞ � Sð�p þ pzÞx� ð�p þ pzÞ

pxð1� zÞSð�p þ pzÞ � zu0ðzÞ;

where

GðzÞ ¼ �p Sð�p þ pzÞ � zð�p þ pzÞ½Sð�p þ pzÞ � z�ð�p þ pzÞ ;

which agree with the generating functions of the Theorem 1 in [7].

4. Stochastic decomposition

This section investigates the stochastic decomposition property of the system size distribution,which permits the system to be analysed by considering severally the distribution of the systemsize without vacations and the additional system size due to vacations. This significant outcomewas first established by Fuhrmann and Cooper [26] for a versatile class of M/G/1 queueing sys-tems. Yang and Templeton [5] studied the stochastic decomposition applied to retrial queues,whose applications were discussed afterwards by Artalejo and Falin [27] and Yang et al. [28].

In general, the stochastic decomposition connects one performance measure for the system withvacations to the corresponding one for the equivalent model without vacations. Expressly, the sto-chastic decomposition property asserts that the system size distribution can be represented as thesum of two independent random variables: one of which is the corresponding random variable ofthe system size of the queueing model without vacations and the other is the system size due to thevacations effects.

In our case the probability generating function of system size can be written as

UðzÞ ¼ ð1� qÞð1� zÞSð�p þ paðzÞÞSð�p þ paðzÞÞ � z

u0ðzÞu0ð1Þ

;

where the first fraction is the probability generating function of the number of customers in the stan-dard Geo[X]/G/1/1 queue with control of admission and the second fraction is the probability gen-erating function of the number of customers in the model under study given that the server is idle.

This outcome is summed up in the following theorem.

1112 J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120

Theorem 3. The total number of customers in the system under consideration (L) can be representedas the sum of two independent random variables, one of which is the total number of customers in thestandard Geo[X]/G/1/1 queueing system with control of admission (L0) and the other one is thenumber of customers in the model under study given that the server is idle (M). That is, L = L0 + M.

It is not unexpected that our discrete system fulfils this property, since our model can be con-sidered as a classical queue with server vacations. In this vacation model, the server begins a vaca-tion when a service ends and there is neither arrival nor retrial. The vacations time depends on thearrival process, the orbit size and the interretrial times. Vacations stop whenever the server ran-domly chooses one of the repeated customers or an external customer arrives. Under these con-siderations, the stochastic decomposition law observed earlier for our system is consistent with theone reported by Fuhrmann and Cooper [26] for the M/G/1 vacation models.

On the other hand, excluding the term

X1k¼1

G0 rkð ÞG rkð Þ r

k ¼ 1u0ð1Þ

d

dzu0ðzÞ

�z¼1

;

the remaining expression in E[L], which is independent of r, is the mean system size in the Geo[X]/G/1/1 queueing system with control of admission. Consequently, the term

X1

k¼1

G0 rkð ÞG rkð Þ r

k

represents the effect of the retrial policy and is the mean orbit size given that the server is idle. Letus indicate that this remark was expected once we had proven the stochastic decomposition law.

Theorem 4. The following inequalities hold

2ð1� qÞ 1� p0;01� q

� �6X1j¼0

jP ½L ¼ j� � P ½L0 ¼ j�j 6 2 1�p0;01� q

� �:

The proof of the preceding theorem follows the steps given in the paper [27] and accordingly itis omitted.

Finally, let us comment that the distanceP1

j¼0 j P ½L ¼ j� � P ½L0 ¼ j� j between the distributionsof the variables L and L0 decreases as r goes to zero. Besides, the importance of the previous the-orem is to give lower and upper estimates for the distance between both distributions.

5. Relation to the continuous-time system

In this section, in order to unify the results of both the discrete-time and the corresponding con-tinuous-time models, we give below a succinct proof as to how to get the continuous-time resultsfrom those of the discrete-time ones. Using appropriate limits, we can get the results of the con-tinuous-time M[X]/G/1 retrial queue with control of admission derived in [29] from those derivedhere. To this objective, time axis is divided into small intervals of equal length, thus the approx-imation approaches the exact value when the length of the intervals tends to zero.

J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120 1113

We regard the continuous-time M[X]/G/1 retrial queue with control of admission in whichbatches of customers arrive according to a Poisson stream with rate k. The interretrial times ofa repeated customer are independent random variables exponentially distributed with parameterc. The service times follow a general law with distribution function B(x), Laplace–Stieltjes trans-form b(s) and finite mean l�1. It is assumed that interarrival times, batch sizes, interretrial timesand service times are mutually independent.

If we suppose that time axis is divided into equal intervals of length D, the continuous-timemodel can be approximated by our discrete-time model consideringZ

p ¼ kD; r ¼ 1� cD and si ¼iD

ði�1ÞDdBðxÞ; i P 1;

where D is sufficiently small so that p and r are probabilities.We will show that limD!0UðzÞ is the probability generating function of the system size in the

M[X]/G/1 retrial queue with control of admission. It is easy to see the following equalities usingthe definition of the Lebesgue integral:

limD!0

pb1 ¼ limD!0

kDX1i¼1

i½BðiDÞ � Bðði� 1ÞDÞ� ¼ k limD!0

X1i¼1

iD½BðiDÞ � Bðði� 1ÞDÞ�

¼ kZ 10

xdBðxÞ ¼ kl;

limD!0

q ¼ limD!0

p�a1b1 ¼kl�a1;

limD!0

Sð�p þ paðzÞÞ ¼ limD!0

X1i¼1

½BðiDÞ � Bðði� 1ÞDÞ�½1� kDþ kDaðzÞ�i

¼ limD!0

X1i¼1

½BðiDÞ � Bðði� 1ÞDÞ�f1� k½1� aðzÞ�DgiDD

¼Z 10

e�k½1�aðzÞ�x dBðxÞ ¼ bðk� kaðzÞÞ:

From Eqs. (8) and (10), seeing that r ! 1 when D! 0, we get

d

dzu0ðzÞ ¼ lim

r!1

u0ðzÞ � u0ðrzÞz� rz ¼ limr!1

F ðrzÞð1� rÞzu0ðrzÞ�

;

but

limr!1

F ðrzÞð1� rÞz ¼

kcz

limr!1

ð1� a0Þrz½�p þ paðrzÞ� � ½aðrzÞ � a0�Sð�p þ paðrzÞÞ½Sð�p þ paðrzÞÞ � rz�½�p þ paðrzÞ�

¼ kcz

ð1� a0Þz� ½aðzÞ � a0�bðk� kaðzÞÞbðk� kaðzÞÞ � z :

And as a result we achieve this differential equation as D! 0

d

dzu0ðzÞ ¼

kcz

ð1� a0Þz� ½aðzÞ � a0�bðk� kaðzÞÞbðk� kaðzÞÞ � z u0ðzÞ;

1114 J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120

whose solution is given by

Fig

u0ðzÞ ¼ u0ð1Þ expkc

Z z1

ð1� a0Þu� ½aðuÞ � a0�bðk� kaðuÞÞu½bðk� kaðuÞÞ � u� du

� :

Taking into account the stochastic decomposition law together with the previous outcomes, wehave

limD!0

UðzÞ ¼1� kl �a1

�

ð1� zÞbðk� kaðzÞÞbðk� kaðzÞÞ � z

� exp kc

Z z1

ð1� a0Þu� ½aðuÞ � a0�bðk� kaðuÞÞu½bðk� kaðuÞÞ � u� du

� ;

which coincides with the formula (3.17) of the probability generating function of the system size inthe M[X]/G/1 retrial queue with control of admission [29].

6. Numerical examples

In this section, we present some numerical examples on the performance measures derived inSection 3 making emphasis on the computation of the stationary probabilities.

Firstly, we will consider the arrival rate p = 0.1 and service times geometric distributed withmean b1 = 2. For simplicity, it is also assumed that the batch size follows a geometrical distribu-tion. For the numerical evaluation, we will use the approximation explained in Remark 3, wheree = 10�8. Of course, the values of the parameters satisfy the stability condition q < 1.

Fig. 1(a) and (b) depict the behaviour of the mean number of repeated customers against theparameter a. As was expected, E[N] is increasing as function of a. In Fig. 1(a), we present threecurves corresponding to �c1 ¼ 2; 3; 4. As is to be expected, E[N] increases with increasing values ofthe parameter �c1. Fig. 1(b) studies how the retrial rate affects the mean orbit size. This graphiccorroborates that the expectation E[N] increases with increasing values of r. It should be pointed

12

10

8

6

4

2

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

12

10

8

6

4

2

(a) (b)

. 1. The mean orbit size versus a: (a) the effect of the mean batch size and (b) the effect of the retrial rate r.

J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120 1115

out that the mean orbit size in the standard Geo[X]/Geo/1/1 queue with control of admission pro-vides a lower bound for the corresponding mean value in our queueing system.

In Tables 1 and 2 we consider the Geo[X]/Geo/1 retrial queue with batch distributionck = (1 � h)hk�1, k P 1, h 2 [0,1). The system parameters are p = 1/6, q = 0.5, h = 0.4 anda = 0.9 so the traffic load is q = 0.5. Both tables defer on the retrial parameter which is r = 0.9(low retrial probability) and r = 0.1 (high retrial probability), respectively.

We carry out the numerical inversion of the partial generating functions u0(z) and u1(1,z) usingthe Fast Fourier Transform (FFT) method. The method is based on the computation of the gen-erating functions over 2m complex arguments z 0s, where 2m satisfies that the tail of the distributionis lesser than a prespecified value e 0. We have chosen e 0 = 10�2, then m is calculated with the helpof Chebychev�s inequality. At this point we notice that the resulting inverted probabilities epj;k are

Table 1p = 1/6, q = 0.5, r = 0.9, h = 0.4, a = 0.9ep0;k ep1;k bp10;k bp11;k bp20;k bp21;k bp2;10;k bp2;11;k0 0.1381 0.0511 0.1682 0.1070 0.1488 0.0702 0.1391 0.05231 0.1223 0.0775 0.1116 0.0841 0.1126 0.0711 0.1200 0.07512 0.0877 0.0813 0.0740 0.0661 0.0819 0.0687 0.0882 0.08113 0.0581 0.0727 0.0491 0.0519 0.0572 0.0633 0.0590 0.07374 0.0368 0.0594 0.0326 0.0408 0.0385 0.0556 0.0372 0.06035 0.0227 0.0458 0.0216 0.0320 0.0248 0.0466 0.0227 0.04616 0.0138 0.0339 0.0143 0.0252 0.0154 0.0372 0.0136 0.03387 0.0082 0.0244 0.0095 0.0198 0.0092 0.0284 0.0081 0.02428 0.0049 0.0172 0.0063 0.0155 0.0052 0.0206 0.0047 0.01699 0.0028 0.0119 0.0041 0.0122 0.0029 0.0143 0.0028 0.011710 0.0016 0.0081 0.0027 0.0096 0.0015 0.0094 0.0016 0.0080

SE 2.8066 2.8525 2.8163 2.8067

Table 2p = 1/6, q = 0.5, r = 0.1, h = 0.4, a = 0.9ep0;k ep1;k bp10;k bp11;k bp20;k bp21;k bp2;10;k bp2;11;k0 0.4942 0.1830 0.4939 0.1843 0.4940 0.1838 0.4940 0.18321 0.0054 0.1172 0.0059 0.1163 0.0058 0.1164 0.0057 0.11682 0.0003 0.0739 7.1 · 10�5 0.0734 9.1 · 10�5 0.0736 0.0001 0.07403 1.9 · 10�5 0.0465 8.6 · 10�7 0.0463 1.8 · 10�6 0.0465 4.5 · 10�6 0.04664 1.2 · 10�6 0.0293 1.0 · 10�8 0.0292 4.9 · 10�8 0.0294 2.2 · 10�7 0.02935 7.7 · 10�8 0.0184 1.2 · 10�10 0.0184 1.6 · 10�9 0.0185 1.5 · 10�8 0.01846 4.9 · 10�9 0.0116 1.5 · 10�12 0.0116 7.6 · 10�11 0.0116 1.3 · 10�9 0.01167 3.0 · 10�10 0.0073 1.8 · 10�14 0.0073 4.5 · 10�12 0.0073 1.4 · 10�10 0.00728 1.9 · 10�11 0.0046 2.1 · 10�16 0.0046 3.4 · 10�13 0.0046 1.9 · 10�11 0.00459 1.2 · 10�12 0.0029 2.6 · 10�18 0.0029 3.4 · 10�14 0.0029 3.3 · 10�12 0.002810 7.6 · 10�14 0.0018 3.1 · 10�20 0.0018 4.5 · 10�15 0.0018 6.9 · 10�13 0.0018

SE 1.6185 1.6188 1.6187 1.6186

1116 J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120

based on some complex numbers which are calculated by replacing the generating functions u0(z)and u1(1,z) by the corresponding power series representations over the finite domain{0, . . . , 2m � 1}. From this point of view, the probabilities epj;k estimate pj,k. However, the FFTmethod is presented in the literature as a very powerful method to recover numerically the valuesof the unknown probabilities [21,30]. In fact, the accuracy can always be improved by decreasingthe value of e 0. In this sense, we may consider epj;k as the ‘‘true’’ probabilities.

To compute u0(z) we observe on the light of Remark 3 that it can be reexpressed asfollows:

u0ðzÞ � u0ð0Þ expXn0ðeÞk¼1

lnG rk z� �

þ 11� r

Z rn0ðeÞþ10

lnGðuzÞu

du

( );

where

u0ð0Þ ¼ p0;0 � ð1� qÞ exp �Xn0ðeÞk¼1

lnG rk� �

þ 11� r

Z rn0ðeÞþ10

lnGðuÞu

du

!( ):

Then, u1(1,z) is computed from formula (2) in Corollary 1. The above formula for p0,0 can becomputed and compared with ep0;0. In the case of the numerical examples presented along this sec-tion, we have observed that ep0;0 fits at least the first seven decimal digits of p0,0. This fact illus-trates that the probabilities epj;k are remarkably accurate.

The inverted probabilities epj;k have been summarized in the first two columns of each table. Itshould be noted that to increase r has the effect of distributing sparsely the probability mass. Con-sequently, the sequence fep1;kg1k¼0 in Table 1 has a mode at the point k�1 ¼ 2 whereas both se-quences fep0;kg1k¼0 and fep1;kg1k¼0 are decreasing in Table 2 corresponding to r = 0.1.

By using information theoretic techniques based on the principle of maximum entropy is pos-sible to find an estimation for the stationary probabilities subject to a few mean value constraints[24]. The maximum entropy distribution is the one which adds the less extraneous informationbeyond those supplied by the given constraints. A simple approach to the principle of maximumentropy presents this method as a possible approximation of any structurally complex system butit can also be viewed as a philosophical alternative to the classical queueing solutions. For theapplications of the principle of maximum entropy to the M/G/1 retrial queue we refer to the Refs.[22,23].

In case that we consider only the expected orbit size as constraint, the maximization of theShannon�s entropy functional

H fpj;kg� �

¼ �X1k¼0

pj;k ln pj;k; j ¼ 0; 1

leads to the explicit solution [23]

bp10;k ¼ ð1� qÞ21� qþ E½N ; idle� E½N ; idle�1� qþ E½N ; idle�� �k

; k P 0;

Tablep = 0.

012345678910

SE

J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120 1117

bp11;k ¼ q2qþ E½N ;busy� E½N ;busy�qþ E½N ; busy�� �k

; k P 0:

This first order solution is decreasing so it cannot exhibit a mode at the point k�1 ¼ 2. The esti-mation can be improved by adding the knowledge of the second moment as a constraint. The con-sideration of more than a single constraint drives to an optimization problem which can be carriedout with the help of the method of Lagrange�s multipliers [22,23]. In Table 1, we observe how thesecond order solution fbp21;kg1k¼0 has a mode but at the point k�1 ¼ 1. We can still improve the max-imum entropy solution by incorporating as a new constraint the value of the partial generatingfunctions at any given point z0. In this way we combine information about the generating functionat the point z = 1 (given by the first two moments) and information at another different point z0.In our numerical experiments we have chosen z0 = 0.6. The maximum entropy probabilities basedon the first two moments and the value of the generating function at z0 = 0.6 are denoted as bp2;1j;k .Some concrete steps for the computation of bp2;1j;k are detailed in an Appendix A. The entries in thelast row of each table give the Shannon�s entropy for each estimation. Of course, as far as weintroduce new constraints the entropy converges to the entropy of the theoretical probabilitiesepj;k.

In Tables 3 and 4 we turn our attention to the system Geo[X]/D/1 with retrials. We assume thateach service time demands exactly one slot so D = 1. Now the retrial probability is r = 0.8 in bothtables. In contrast, we choose the system parameters to get q = 0.75 in Table 3 and q = 0.25 inTable 4. A higher value of q implies a sparse distribution. In this sense, we note in Table 3 thatthe mode of the sequences fepj;kg1k¼0, j = 0, 1, are k�0 ¼ 1 and k�1 ¼ 3, respectively. The case q = 0.25yields decreasing stationary probabilities.

As a final remark we observe that our numerical results have been illustrated for two specificchoices of the service time distribution. However, it is not restrictive. The maximum entropy ap-proach only needs the knowledge of constraints independently of the underlying service time and/or batch size distributions. On the other hand, the numerical inversion can be performed for anyservice time and batch size distributions. We just need to specify the generating functions S(x) andc(z) in our FORTRAN codes.

35, D = 1, r = 0.8, h = 0.4, a = 0.9ep0;j ep1;j bp10;k bp11;k bp20;k bp21;k bp2;10;k bp2;11;k

0.0406 0.0359 0.0693 0.1248 0.0496 0.0674 0.0413 0.03670.0538 0.0728 0.0500 0.1040 0.0475 0.0741 0.0523 0.07060.0485 0.0942 0.0362 0.0867 0.0422 0.0783 0.0487 0.09420.0370 0.0992 0.0261 0.0723 0.0349 0.0795 0.0377 0.10070.0258 0.0928 0.0189 0.0602 0.0268 0.0775 0.0261 0.09400.0169 0.0805 0.0136 0.0502 0.0191 0.0727 0.0170 0.08080.0107 0.0663 0.0098 0.0418 0.0127 0.0655 0.0106 0.06580.0065 0.0526 0.0071 0.0349 0.0078 0.0567 0.0064 0.05190.0039 0.0406 0.0051 0.0290 0.0045 0.0472 0.0038 0.04000.0023 0.0307 0.0037 0.0242 0.0024 0.0377 0.0023 0.03030.0013 0.0229 0.0026 0.0202 0.0011 0.0290 0.0013 0.0227

3.0057 3.1228 3.0317 3.0058

Table 4p = 1/6, D = 1, r = 0.8, h = 0.4, a = 0.9ep0;k ep1;k bp10;k bp11;k bp20;k bp21;k bp2;10;k bp2;11;k0 0.5309 0.0983 0.5220 0.1117 0.5284 0.1019 0.5308 0.09861 0.1474 0.0710 0.1586 0.0618 0.1521 0.0658 0.1480 0.07022 0.0473 0.0403 0.0482 0.0341 0.0463 0.0394 0.0468 0.04073 0.0158 0.0208 0.0146 0.0188 0.0149 0.0219 0.0158 0.02114 0.0054 0.0102 0.0044 0.0104 0.0050 0.0113 0.0055 0.01025 0.0018 0.0049 0.0013 0.0057 0.0018 0.0054 0.0019 0.00486 0.0006 0.0023 0.0004 0.0031 0.0007 0.0023 0.0006 0.00227 0.0002 0.0010 0.0001 0.0017 0.0002 0.0009 0.0002 0.00108 8.0 · 10�5 0.0004 3.8 · 10�5 0.0009 0.0001 0.0003 7.5 · 10�5 0.00049 2.8 · 10�5 0.0002 1.1 · 10�5 0.0005 5.4 · 10�5 0.0001 2.4 · 10�5 0.000210 1.0 · 10�5 0.0001 3.5 · 10�6 0.0002 2.6 · 10�5 4.3 · 10�5 7.7 · 10�6 0.0001

SE 1.6046 1.6082 1.6053 1.6046

1118 J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120

Acknowledgments

The authors thank the referee for his comments on an earlier version of this paper. This re-search is supported by the DGINV through the project BFM2002-02189.

Appendix A. The computation of probabilities bp2;1j;kSince the probabilities bp2;1j;k give a good estimation of pj,k when they are compared with the

numerical inverted probabilities epj;k, we next summarize how to compute bp2;10;k . Of course, the com-putation of bp2;11;k follows parallel steps. Previously, we introduce some notation

F 1ðkÞ ¼ k; F 2ðkÞ ¼ k2; F 3ðkÞ ¼ zk0;

F 1 ¼ E½N ; idle�; F 2 ¼ E½N 2; idle�; F 3 ¼ u0ðz0Þ;

where z0 is any arbitrary fixed point in the interval (0,1).

Step 1. Compute the constraints Fi, 1 6 i 6 3 (see Corollary 2 and Remark 3).Step 2. For some prespecified e 0 (say e 0 = 10�2), employ Chebychev�s inequality in order to find an

integer 2m such thatP1

k¼2mp0;k 6 e0. (We construct the maximum entropy solution on thesame domain that the inverted probabilities ep0;j.)

Step 3. Solve the following system on the unknowns ki:

F iðk1; k2; k3Þ ¼X2m�1k¼0

ðF iðkÞ � F iÞ exp �X3n¼1

ðF nðkÞ � F nÞkn

( )¼ 0; 1 6 i 6 3:

(We employ the method of Lagrange�s multipliers to maximize the Shannon�s entropy functional.The interested reader is referred to papers [22–24] and the references therein.)

J.R. Artalejo et al. / Applied Mathematical Modelling 29 (2005) 1100–1120 1119

Step 4. Compute the multiplier k0 from the normalization condition u0(1) = 1 � q:

expfk0g ¼ ð1� qÞ�1X2m�1k¼0

exp �X3n¼1

F nðkÞkn

( ):

Step 5. Compute the maximum entropy probabilities bp2;10;k :

bp2;10;k ¼ expf�ðk0 þ k1k þ k2k2 þ k3zk0Þg; 0 6 k 6 2m � 1:

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A discrete-time Geo[X]/G/1 retrial queue with control of admissionIntroductionThe mathematical descriptionThe Markov chainStochastic decompositionRelation to the continuous-time systemNumerical examplesAcknowledgmentsThe computation of probabilities {\widehat{\pi}}_{j , k}^{2 , 1}References