Ann Oper Res (2006) 141: 85–107

DOI 10.1007/s10479-006-5295-7

A Discrete-Time Geo/G/1 retrial queue with the serversubject to starting failures∗

I. Atencia · P. Moreno

C© Springer Science + Business Media, Inc. 2006

Abstract This paper studies a discrete-time Geo/G/1 retrial queue where the server is

subject to starting failures. We analyse the Markov chain underlying the regarded queueing

system and present some performance measures of the system in steady-state. Then, we

give two stochastic decomposition laws and find a measure of the proximity between the

system size distributions of our model and the corresponding model without retrials. We

also develop a procedure for calculating the distributions of the orbit and system size as

well as the marginal distributions of the orbit size when the server is idle, busy or down.

Besides, we prove that the M/G/1 retrial queue with starting failures can be approximated

by its discrete-time counterpart. Finally, some numerical examples show the influence of the

parameters on several performance characteristics.

Keywords Discrete-time retrial queues . Recursive formulae . Stochastic decomposition .

Unreliable server

1. Introduction

During the last two decades considerable attention has been paid to the analysis of queueing

systems with repeated attempts (or retrial queues, queues with returning customers, . . . ),

see for example the bibliographies on retrial queues (Artalejo, 1999a, b), the surveys (Falin,

1990; Yang and Templeton, 1987) and the book (Falin and Templeton, 1997). Retrial queueing

∗This work is supported by the DGINV through the project BFM2002-02189.

I. AtenciaDepartamento de Matematica Aplicada, E.T.S.I. de Telecomunicacion, Universidad de Malaga,Malaga 29071, Spaine-mail: iatencia@ctima.uma.es

P. MorenoDepartamento de Economıa, Metodos Cuantitativos, e Historia Economica, Facultad de CienciasEmpresariales, Universidad Pablo de Olavide, Sevilla 41013, Spaine-mail: mpmornav@upo.es

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86 Ann Oper Res (2006) 141: 85–107

systems are characterized by the peculiarity that arriving customers who find the server

occupied join the retrial group (called orbit) to try again for their requests in random order

and at random intervals. These queueing models have been used to model many problems in

telephone switching systems, telecommunication networks, computer networks, computer

and communication systems. Some details on applications research in the area can be found

in the proceedings of successive International Teletraffic Congresses and in the review papers

or book mentioned previously.

On the other hand, discrete-time queueing models have been widely used over the past

years in view of their applicability in the study of many computer and communication sys-

tems in which time is slotted. Their importance has been further increased due to recent

developments in telecommunication technology caused by the emergence of broadband inte-

grated services digital network (BISDN). BISDN can provide the transfer of video, voice and

data communication signals through high speed local area network (LAN), on-demand video

distribution, and video telephony communications. The asynchronous transfer mode (ATM)

is a key technology for accommodating such a wide area of services. A detailed discussion

and applications of discrete-time queues can be found in the books (Bruneel and Kim, 1993;

Woodward, 1994) and the references therein.

The first work on discrete-time queues is due to Meisling (1958). Since 1958, many

researchers have dedicated time to the study of such systems, see for example (Artalejo and

Hernandez-Lerma, 2003; Atencia and Moreno, 2004c; Chaudhry, Templeton, and Gupta,

1996; Gupta and Goswami, 2002; Takagi, 1993). One of the most outstanding works of

the queueing theory has been recently carried out by Yang and Li (1995), who extended

the queues with repeated attempts to the discrete-time systems. However, contrary to what

happens in continuous-time retrial queues, until the moment little attention has been given

to the analysis of their discrete-time counterparts. In fact, only few people have analysed

discrete-time retrial queues (Atencia and Moreno, 2004a, b; Choi and Kim, 1997; Li and

Yang, 1998, 1999; Takahashi, Osawa, and Fujisawa, 1999; Yang and Li, 1995). That is why

efforts should be taken to fill up this gap.

A notable and inevitable phenomenon in the service facility of a queueing system is its

breakdown and consequent repair. Nevertheless, in most of the queueing literature the server is

always available, although this assumption is evidently unrealistic. Indeed, queueing systems

with server breakdowns are very common in communication systems. Developing analytical

models to be used for analysing their performance is a very important issue which has been

dealt with by several researchers. Most of the existing models focus on continuous-time

models, see for instance (Aissani and Artalejo, 1998; Artalejo, 1994; Krishna Kumar, Pavai

Madheswari, and Vijayakumar, 2002; Kulkarni and Choi, 1990; Wang, Cao, and Li, 2001;

Yang and Li, 1994); however, works related to discrete-time systems with server interruptions

and vacations can be found in (Fiems and Bruneel, 2002; Fiems, Steyaert, and Bruneel,

2002; Fiems, Steyaert, and Bruneel, 2003; Tian and Zhang, 2002; Zhang and Tian, 2001).

The current contribution is motivated by the recent interest in the discrete-time retrial queues

and the recognition of the importance of incorporating unreliable servers to these systems.

In fact, this aspect has not yet received attention in the context of the discrete-time retrial

queues.

The importance of the present paper lies in the effort to extend the queueing theory

concerning unreliable servers to the discrete-time retrial queues. Thus this work analyses a

discrete-time retrial queue with the service station subject to starting failures. It is assumed

the breakdowns take place in accordance with the preemptive non-resume strategy, i.e., the

customers whose services are unsuccessful enter the orbit and when the service resumes for

the preempted customers it starts again from the beginning. The above-mentioned papers

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Ann Oper Res (2006) 141: 85–107 87

together with their references reveal that no work has been done in discrete-time retrial

queues taking into account server failures and repairs.

The organization of the remaining paper is as follows. The next section gives a des-

cription of the queueing model. In Section 3, we study the Markov chain and the stability

condition of the system. The orbit and system size distributions are also obtained together

with several performance measures of the system. In Section 4, we derive two different

stochastic decomposition laws and as consequence we provide upper and lower estimates

for the distance between the steady-state distributions of our system and the corresponding

system without retrials. In Section 5, we investigate the recursive computation of the steady-

state distributions of the orbit and system size along with the marginal distributions of the

orbit size when the server is idle, busy or down. The relation between our discrete-time

system and its continuous-time counterpart is analysed in Section 6. Finally, some numerical

results are given and discussed in Section 7.

2. Description of the queueing system

We consider a discrete-time queueing system where the time axis is divided into intervals

of equal length, called slots. It should be pointed out that the probability of an arrival and a

departure occurring simultaneously is zero in continuous-time queues, whereas it is not so in

discrete-time queues. Since more than one different event may occur concurrently, to resolve

conflicts it is necessary to establish the order in which the arrivals and departures take place

in case of simultaneity. Essentially, there are two rules: (i) if an arrival takes precedence over

a departure, it is known as late arrival system (LAS), and (ii) if a departure takes precedence

over an arrival, then it is referred to early arrival system (EAS). They are also known as arrival

first (AF) and departure first (DF) policies, respectively. Moreover, if the server is idle and

a customer arrives, then either his service starts immediately or in the following slot. In the

former case it is known as immediate access (IA), whereas in the latter case it is known as

delayed access (DA). If these concepts are combined with LAS, then the discipline is known

as late-arrival system with immediate access (LAS-IA) and late-arrival system with delayed

access (LAS-DA), respectively. LAS-IA corresponds to EAS. For more details on these and

related concepts, see (Gravey and Hebuterne, 1992; Hunter, 1983). In the present paper we

discuss the model only for early arrival system.

The above disciplines are simply rules to solve the conflict of the simultaneity. Nev-

ertheless, when a discrete-time queueing system has more activities than just arrivals and

departures, such as having vacations and other interruptions both the definitions of Gravey

and Hebuterne (1992) and Hunter (1983) will have to be expanded. Therefore, let the time axis

be marked by 0, 1, 2, . . . , m, . . . Regard the epoch m and suppose that the departures and

the end of the repairs occur in (m−, m), and the arrivals, the retrials and the beginning of the

repairs in (m, m+). The model under consideration can be viewed through a self-explanatory

figure (see figure 1).

Customers arrive according to a Bernoulli arrival process with rate p, that is, p is the

probability that a customer arrives at a slot. If, upon arrival, the server is busy or down,

the customer is obliged to leave the service area and to come back to the system after a

random amount of time (in slots). Those customers who are waiting to retry services are

considered to be in orbit. An arriving (external or repeated) customer who finds the server

idle must turn on the service station. If the server is activated successfully (with a probability

θ ), the customer begins his service immediately and abandons the system forever after ser-

vice completion; otherwise, if the server is started unsuccessfully (with a complementary

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88 Ann Oper Res (2006) 141: 85–107

Fig. 1 Various time epochs inearly arrival system (EAS)

probability θ ), the server is sent to repair directly and the customer must join the

orbit.

Each customer in the orbit forms an independent retrial source with rate 1 − r ; that is,

at the end of each time slot, a repeated customer attempts service with probability 1 − r .

Therefore, the retrial time (the time between two successive attempts by the same customer)

follows a geometrical law with probability 1 − r . The retrial process of a repeated customer

concludes only if, upon one particular attempt, the server is idle, the repeated customer is

chosen for the service among all other repeated customers who are attempting the service at

that time and the service station is activated successfully.

The service times are independent and identically distributed with general distribution

{s1,i }∞i=1, generating function S1(x) = ∑∞i=1 s1,i x i and n-th factorial moments β1,n . Certainly,

after service completion, the served customer leaves the system forever and will have no

further effect on the system.

The repair times are independent and identically distributed with arbitrary distribution

{s2,i }∞i=1, generating function S2(x) = ∑∞i=1 s2,i x i and n-th factorial moments β2,n . Naturally,

after repair the service station is as good as new.

The load of the system is given by ρ1 + ρ2 where ρ1 = p β1,1 and ρ2 = θθ

p β2,1.

It is assumed that the interarrival times, the retrial times, the service times and the repair

times are mutually independent. In order to avoid trivial cases, it is also supposed 0 < p < 1,

0 ≤ r < 1 and 0 < θ ≤ 1.

3. The Markov chain

At time m+, the system can be described by the process

(Cm, ξ1,m, ξ2,m, Nm)

where Cm denotes the state of the server, 0, 1 or 2 according whether the server is free, busy or

down and Nm the number of repeated customers. If Cm = 1, then ξ1,m represents the remaining

service time of the customer currently being served and if Cm = 2, ξ2,m corresponds to the

remaining repair time.

It can be shown that {(Cm, ξ1,m, ξ2,m, Nm), m ∈ N} provides a Markovian description of

our queueing system, whose state space is

{(0, k) : k ≥ 0; (1, i, k) : i ≥ 1, k ≥ 0; (2, i, k) : i ≥ 1, k ≥ 1}.Springer

Ann Oper Res (2006) 141: 85–107 89

Our objective is to find the stationary distribution

π0,k = limm→∞ P[Cm = 0, Nm = k]; k ≥ 0

π1,i,k = limm→∞ P[Cm = 1, ξ1,m = i, Nm = k]; i ≥ 1, k ≥ 0

π2,i,k = limm→∞ P[Cm = 2, ξ2,m = i, Nm = k]; i ≥ 1, k ≥ 1

of the Markov chain {(Cm, ξ1,m, ξ2,m, Nm), m ∈ N}.The one-step transition probabilities are given by the formulae:

if k ≥ 0

p(0,k)(0,k) = p r k

p(1,1,k)(0,k) = p r k

p(2,1,k)(0,k) = p r k (k ≥ 1)

if i ≥ 1, k ≥ 0

p(0,k)(1,i,k) = p θ s1,i

p(0,k+1)(1,i,k) = p(1 − rk+1)θ s1,i

p(1,1,k)(1,i,k) = p θ s1,i

p(1,1,k+1)(1,i,k) = p(1 − rk+1)θ s1,i

p(1,i+1,k−1)(1,i,k) = p (k ≥ 1)

p(1,i+1,k)(1,i,k) = p

p(2,1,k)(1,i,k) = p θ s1,i (k ≥ 1)

p(2,1,k+1)(1,i,k) = p(1 − rk+1)θ s1,i

if i ≥ 1, k ≥ 1

p(0,k−1)(2,i,k) = p θ s2,i

p(0,k)(2,i,k) = p(1 − rk)θ s2,i

p(1,1,k−1)(2,i,k) = p θ s2,i

p(1,1,k)(2,i,k) = p(1 − rk)θ s2,i

p(2,1,k−1)(2,i,k) = p θ s2,i (k ≥ 2)

p(2,1,k)(2,i,k) = p(1 − rk)θ s2,i

p(2,i+1,k−1)(2,i,k) = p (k ≥ 2)

p(2,i+1,k)(2,i,k) = p

where p = 1 − p.

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90 Ann Oper Res (2006) 141: 85–107

The Kolmogorov equations for the stationary distribution are

π0,k = p r k π0,k + p r k π1,1,k + (1 − δ0,k) p r k π2,1,k ; k ≥ 0 (1)

π1,i,k = p θ s1,i π0,k + p(1 − rk+1)θ s1,i π0,k+1 + p θ s1,i π1,1,k ++ p(1 − rk+1)θ s1,i π1,1,k+1 + (1 − δ0,k) p π1,i+1,k−1 ++ p π1,i+1,k + (1 − δ0,k) p θ s1,i π2,1,k ++ p(1 − rk+1)θ s1,i π2,1,k+1; i ≥ 1, k ≥ 0 (2)

π2,i,k = p θ s2,i π0,k−1 + p(1 − rk)θ s2,i π0,k + p θ s2,i π1,1,k−1 ++ p(1 − rk)θ s2,i π1,1,k + (1 − δ1,k) p θ s2,i π2,1,k−1 ++ p(1 − rk)θ s2,i π2,1,k + (1 − δ1,k) p π2,i+1,k−1 + p π2,i+1,k ;

i ≥ 1, k ≥ 1 (3)

and the normalizing condition is

∞∑k=0

π0,k +∞∑

i=1

∞∑k=0

π1,i,k +∞∑

i=1

∞∑k=1

π2,i,k = 1.

To solve Eqs. (1)–(3), we define the generating functions:

ϕ0(z) =∞∑

k=0

π0,k zk

ϕ1(x, z) =∞∑

i=1

∞∑k=0

π1,i,k xi zk

ϕ2(x, z) =∞∑

i=1

∞∑k=1

π2,i,k xi zk .

The following theorem gives us the solution of the Kolmogorov equations in terms of the

preceding generating functions and its proof can be found in the Appendix.

Theorem 1. If ρ1 + ρ2 < 1, the stationary distribution of the Markov chain {(Cm, ξ1,m,

ξ2,m, Nm), m ∈ N} has the generating functions:

ϕ0(z) = (1 − ρ1 − ρ2)

∏∞k=1 G(rk z)∏∞k=1 G(rk)

ϕ1(x, z) = S1(x) − S1( p + p z)

x − ( p + p z)

p x (1 − z) θ

θ S1( p + p z) + θ z S2( p + p z) − zϕ0(z)

ϕ2(x, z) = S2(x) − S2( p + p z)

x − ( p + p z)

p x (1 − z) θ z

θ S1( p + p z) + θ z S2( p + p z) − zϕ0(z)

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Ann Oper Res (2006) 141: 85–107 91

where

G(z) = θ S1( p + p z) + θ z S2( p + p z) − z ( p + p z)

( p + p z) [θ S1( p + p z) + θ z S2( p + p z) − z]p.

Corollary 1.

(1) The marginal generating function of the number of customers in the orbit when the serveris idle is given by ϕ0(z).

(2) The marginal generating function of the number of customers in the orbit when the serveris busy is given by

ϕ1(1, z) = θ [1 − S1( p + p z)]

θ S1( p + p z) + θ z S2( p + p z) − zϕ0(z).

(3) The marginal generating function of the number of customers in the orbit when the serveris down is given by

ϕ2(1, z) = θ z [1 − S2( p + p z)]

θ S1( p + p z) + θ z S2( p + p z) − zϕ0(z).

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

(z) = ϕ0(z) + ϕ1(1, z) + ϕ2(1, z) = θ (1 − z)

θ S1( p + p z) + θ z S2( p + p z) − zϕ0(z).

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

(z) = ϕ0(z) + z ϕ1(1, z) + ϕ2(1, z) = θ (1 − z) S1( p + p z)

θ S1( p + p z) + θ z S2( p + p z) − zϕ0(z).

Corollary 2.

(1) The probabilities that the server is idle, busy and down are, respectively,

ϕ0(1) = 1 − ρ1 − ρ2, ϕ1(1, 1) = ρ1 and ϕ2(1, 1) = ρ2.

(2) The mean orbit size is given by

E[N ] = θ p β1,2 + θ (2 β2,1 + p β2,2)

2 θ (1 − ρ1 − ρ2)p +

∞∑k=1

G ′(rk)

G(rk)rk .

(3) The mean system size is given by E[L] = E[N ] + ρ1.(4) The mean time a customer spends in the system (including the service time) is given by

W = E[L]/p.

Remark 1. It can be observed the relation (z) = (z) S1( p + p z), and as a consequence

we find the formula

(n(1) =n∑

m=0

(n

m

)pm β1,m (n−m(1), n ≥ 1

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92 Ann Oper Res (2006) 141: 85–107

where (n(1) and (n(1) are the n-th factorial moments for the distribution of the random

variables L and N respectively.

Remark 2. The stationary distribution of the server state

ϕ0(1) = 1 − ρ1 − ρ2, ϕ1(1, 1) = ρ1, ϕ2(1, 1) = ρ2

depends on the service and repair times distributions only through their means and is inde-

pendent of the interretrial times parameter r .

Remark 3. To approximate numerically the previous performance measures, it is necessary

to estimate the series∑∞

k=1G ′(rk )G(rk )

rk . It is easy to see the relation

∞∑k=1

G ′(rk)

G(rk)rk = 1

1 − r

∞∑k=1

G ′(rk)

G(rk)(rk − rk+1).

Consequently, the former series can be approximated by an integral as follows:

(i)

∞∑k=1

G ′(rk)

G(rk)rk ≈ r

1 − r

∫ 1

0

G ′(z)

G(z)dz = r ln G(1)

1 − r

= r

1 − rln

[p

p + θ (1 − ρ1 − ρ2)

θ (1 − ρ1 − ρ2)

]if r is close to 1.

(ii)

∞∑k=1

G ′(rk)

G(rk)rk ≈ 1

1 − r

n0(ε)∑k=1

G ′(rk)

G(rk)(rk − rk+1) + 1

1 − r

∫ rn0(ε)+1

0

G ′(z)

G(z)dz

=n0(ε)∑k=1

G ′(rk)

G(rk)rk + ln G(rn0(ε)+1)

1 − rwhere, for each ε > 0,

n0(ε) is chosen such that rn0(ε)+1 < ε.

The approximation (ii) is applicable for any r ∈ [0, 1), although it is complex to obtain

the functions G(z) and G ′(z).

Remark 4 (Special cases). In this remark we analyse two special models: the case of a queue

with random service discipline and the case of a system with reliable server. Moreover our

results agree with the one reported by Yang and Li (1995).

(a) When r = 0, (z) has the expression

(z) = (1 − ρ1 − ρ2) θ (1 − z) S1( p + p z)

θ S1( p + p z) + θ z S2( p + p z) − z,

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Ann Oper Res (2006) 141: 85–107 93

which is the probability generating function of the system size in the model

Geo/G/1/∞ with starting failures. This is not surprising because as r = 0, the blocked

customers try to get service at every slot boundary. This system is equivalent to the model

Geo/G/1/∞ with starting failures and random service discipline, since the system size

distribution does not depend on the service discipline.

(b) When θ = 1, our model becomes a discrete-time Geo/G/1 retrial queue with reliable

server and the generating functions of Theorem 1 are reduced to

ϕ0(z) = (1 − ρ1)

∏∞k=1 G(rk z)∏∞k=1 G(rk)

ϕ1(x, z) = S1(x) − S1( p + p z)

x − ( p + p z)

p x (1 − z)

S1( p + p z) − zϕ0(z)

where

G(z) = S1( p + p z) − z ( p + p z)

( p + p z) [S1( p + p z) − z]p,

which coincide with the generating functions of Theorem 1 in (Yang and Li, 1995).

4. Stochastic decomposition laws

In this section we investigate the stochastic decomposition property for the number of cus-

tomers in our model. The stochastic decomposition law has been analysed extensively for

the queueing systems with server vacations, see for example (Fuhrmann and Cooper, 1985).

This property allows to study the system by considering separately the distribution of the

system size without vacations and the additional system size due to vacations. Specifically,

the stochastic decomposition law establishes that the number of customers in the system

can be decomposed as sum of two independent random variables: one being the num-

ber of customers in the corresponding standard system and the other random variable can

have different probabilistic meanings depending on how the vacation discipline has been

defined.

The stochastic decomposition has been considered in some discrete-time queueing sys-

tems, see for example (Takagi, 1993; Tian and Zhang, 2002; Zhang and Tian, 2001). This

decomposition has also been studied for discrete-time retrial queues in (Atencia and Moreno,

2004a,b; Li and Yang, 1999; Takahashi, Osawa, and Fujisawa, 1999; Yang and Li, 1995).

In the context of our system, we present two different stochastic decompositions of the

system size distribution:

(z) = (1 − ρ1) (1 − z) S1( p + p z)

S1( p + p z) − z

ϕ0(z) + ϕ2(1, z)

ϕ0(1) + ϕ2(1, 1)(4)

(z) = (1 − ρ1 − ρ2) θ (1 − z) S1( p + p z)

θ S1( p + p z) + θ z S2( p + p z) − z

ϕ0(z)

ϕ0(1). (5)

In the formula (4) the first fraction is the probability generating function of the number

of customers in the standard Geo/G/1/∞ queue and the second fraction is the probability

generating function of the number of repeated customers given that the server is idle or

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94 Ann Oper Res (2006) 141: 85–107

down. In the same way, in the expression (5) the first fraction corresponds to the probability

generating function of the number of customers in the Geo/G/1/∞ queueing system with

starting failures and the second fraction is the probability generating function of the number

of blocked customers given that the server is idle.

These results can be summed up in the next theorem.

Theorem 2.

(a) The random variable “system size” (L) can be decomposed as the sum of two independentrandom variables, one of which is the number of customers in the standard Geo/G/1/∞queue (L ′) and the other is the number of repeated customers given that the server is idleor down (M ′). That is, L = L ′ + M ′.

(b) The random variable “system size” (L) can be decomposed as the sum of two independentrandom variables, one of which is the number of customers in the Geo/G/1/∞ queueingsystem with the server subject to starting failures (L ′′) and the other is the number ofrepeated customers given that the server is idle (M ′′). That is, L = L ′′ + M ′′.

As a consequence of the first stochastic decomposition law, we derive a measure of the

proximity between the steady-state distribution of the standard Geo/G/1/∞ queue and

the ours. In the same way, using the second decomposition law, we provide a measure of

the proximity between the steady-state distribution of the Geo/G/1/∞ queueing system in

which the server is subject to starting failures and the ours.

Theorem 3. The following inequalities hold

2 (1 − ρ1 − π0,0) ≤∞∑j=0

|P[L = j] − P[L ′ = j]| ≤ 21 − ρ1 − π0,0

1 − ρ1

2 (1 − ρ1 − ρ2 − π0,0) ≤∞∑j=0

|P[L = j] − P[L ′′ = j]| ≤ 21 − ρ1 − ρ2 − π0,0

1 − ρ1 − ρ2

.

The proof of this theorem is based on the stochastic decomposition laws and follows the

steps explained in (Artalejo and Falin, 1994).

Finally, let us observe that the distance∑∞

j=0 |P[L = j] − P[L ′ = j]| between the

distributions of the variables L and L ′ decreases as (θ, r ) approaches (1,0); similarly,

the distance∑∞

j=0 |P[L = j] − P[L ′′ = j]| between the distributions of the variables Land L ′′ diminishes when r goes towards 0. We also comment that the interest of the

former theorem is to provide lower and upper estimates for the distance between these

distributions.

5. Calculation of the steady-state probabilities

This section is devoted to develop recursive formulae for calculating the stationary distribu-

tions associated with our system.

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Ann Oper Res (2006) 141: 85–107 95

Theorem 4. The stationary distribution of the orbit size when the server is idle is given bythe following recursive formulae

π0,0 = 1 − ρ1 − ρ2∏∞k=1 G(rk)

π0,k =∑k−1

n=0[(θ ck−n + θ dk−n) − rn(θ ak−n + θ bk−n)]rk−n π0,n

θ (1 − rk)S1( p), k ≥ 1

where for n ≥ 1 we have defined the coefficients

an =∞∑

j=n

(j

n

)B1, j+1 p j−n+1 pn+1, bn =

∞∑j=n−1

(j

n − 1

)B2, j+1 p j−n+2 pn,

cn =∞∑

j=n

(j

n

)B1, j p j−n pn+1, dn =

∞∑j=n−1

(j

n − 1

)B2, j p j−n+1 pn,

and B1, j = ∑∞k= j+1 s1,k is the probability that a service lasts more than j slots and B2, j =∑∞

k= j+1 s2,k is the probability that a repair lasts more than j slots.

Proof: Firstly, we define the functions

ϒ(z) = p

1 − z

[θ

S1( p + p z)

p + p z+ θ z

S2( p + p z)

p + p z− z

]=

∞∑n=0

υn zn

�(z) = θ S1( p + p z) + θ z S2( p + p z) − z

1 − z=

∞∑n=0

ωn zn .

Our first task is to find the sequences {υn}∞n=0 and {ωn}∞n=0. Using properties of the gen-

erating functions and the Newton’s binomial (see Theorem 3 in (Yang and Li, 1995)), we

get

ϒ(z) = θ S1( p) −∞∑

n=1

(θ an + θ bn)zn

�(z) = θ S1( p) −∞∑

n=1

(θ cn + θ dn)zn .

Since G(z) = ϒ(z)

�(z), we obtain from (21) that ϕ0(z) �(r z) = ϕ0(r z) ϒ(r z). Now com-

paring the coefficients of zk on both sides of the previous equation yields

k∑n=0

π0,n ωk−n rk−n =k∑

n=0

π0,n rn υk−n rk−n, k ≥ 0.

Finally, the proof of the theorem is concluded taking into account the relations υ0 = ω0 =θ S1( p), −υn = θ an + θ bn and −ωn = θ cn + θ dn for n ≥ 1. �

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96 Ann Oper Res (2006) 141: 85–107

Throughout this section, we will denote by π1,·,k = ∑∞i=1 π1,i,k the stationary probability

that the server is busy and there are k repeated customers, and by π2,·,k = ∑∞i=1 π2,i,k the

stationary probability that the server is down and there are k repeated customers.

Theorem 5. The stationary distribution of the orbit size when the server is busy is given bythe following recursive formulae

π1,·,k = (1 − δ0,k)θ

∑k−1n=0 ck−n π0,n + ∑k−1

n=0(θ ck−n + θ dk−n)π1,·,nθ S1( p)

+

+ 1 − S1( p)

S1( p)π0,k, k ≥ 0.

Proof: Let us observe the equation

ϕ1(1, z) �(z) = θ ϕ0(z)1 − S1( p + p z)

1 − z(6)

where �(z) and its expression in power series are given in the proof of the preceding theorem.

Taking into account the relation

1 − S1( p + p z)

1 − z= 1 − S1( p) +

∞∑n=1

cn zn

and after equalling the coefficients of zk on both sides of Eq. (6), we obtain

k∑n=0

π1,·,n ωk−n = (1 − δ0,k)k−1∑n=0

θ π0,n ck−n + θ [1 − S1( p)] π0,k, k ≥ 0.

To end the proof it suffices to consider the expression of {ωn}∞n=0. �

Theorem 6. The stationary distribution of the orbit size when the server is down is given bythe following recursive formulae

π2,·,k = θ∑k−1

n=0 dk−n π0,n + (1 − δ1,k)∑k−1

n=1(θ ck−n + θ dk−n)π2,·,nθ S1( p)

, k ≥ 1.

Proof: We now notice the equation

ϕ2(1, z) �(z) = θ ϕ0(z) z1 − S2( p + p z)

1 − z(7)

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Ann Oper Res (2006) 141: 85–107 97

where �(z) and its development in power series are given in the proof of Theorem 4. From

the expression

z1 − S2( p + p z)

1 − z=

∞∑n=1

dn zn

and equalizing the coefficients of zk on both sides of Eq. (7), we get

k∑n=1

π2,·,n ωk−n =k−1∑n=0

θ π0,n dk−n, k ≥ 1.

Now taking into account the expression of the sequence {ωn}∞n=0, we end the proof of the

theorem. �

Theorem 7. The stationary distribution of the orbit size is given by the following recursiveformulae

ψk = P[N = k] = θ π0,k + (1 − δ0,k)∑k−1

n=0

(θ ck−n + θ dk−n

)ψn

θ S1( p), k ≥ 0.

Proof: Let us observe the equation (z) �(z) = θ ϕ0(z) where �(z) and its development in

power series are given in the proof of Theorem 4. Comparing the coefficients of zk on both

sides of the above equation leads to

k∑n=0

ψn ωk−n = θ π0,k, k ≥ 0.

Then, by observing the expression of the sequence {ωn}∞n=0, we finish the proof of the theorem.�

Theorem 8. The stationary distribution of the system size is given by the following recursiveformulae

φk = P[L = k] = θ∑k

n=0 ek−n π0,n + (1 − δ0,k)∑k−1

n=0

(θ ck−n + θ dk−n

)φn

θ S1( p), k ≥ 0

where

e0 = S1( p) and en =∞∑

j=n

(j

n

)s1, j p j−n pn, n ≥ 1.

Proof: Firstly, we note the next relationship

(z) �(z) = θ ϕ0(z) S1( p + p z) (8)

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98 Ann Oper Res (2006) 141: 85–107

where �(z) and its development in power series are given in the proof of Theorem 4. Com-

paring the coefficients on both sides of Eq. (8) and taking into account the relation

S1( p + p z) =∞∑

n=0

en zn,

we attain

k∑n=0

φn ωk−n =k∑

n=0

θ π0,n ek−n, k ≥ 0.

In order to conclude this proof, we only have to regard the sequence {ωn}∞n=0. �

To finish this section, we observe that the discrete-time systems have the advantage of

finding recursive formulae to calculate the steady-state distributions. These recursive formu-

lae are not usually easy to find out in the continuous-time cases and they involve methods

such as partial fraction expansion and roots finding. Nevertheless, since a continuous-time

system can be approximated by its discrete-time counterpart, the formulae developed for

discrete-time models can be applied to continuous-time models for the approximation of the

steady-state distributions.

6. Relation to the continuous-time system

This section is dedicated to the analysis of the relationship between the discrete-time system

to its continuous-time counterpart. More specifically, we will show that the continuous-time

M/G/1 retrial queue with the server subject to starting failures can be approximated by the

corresponding discrete-time system; to this end, time is slotted into intervals of equal length,

so the approximation tends to the exact value when the length of the intervals goes to zero.

We consider the continuous-time M/G/1 retrial queue with starting failures in which

customers arrive according to a Poisson stream with rate λ. Upon arrival, the customer who

finds the server busy or down joins a group of unsatisfied customers in order to apply for his

service later on after a random time. Retrial times of a returning customer are independent

exponential random variables with parameter γ . If the server is started successfully (with a

probability θ ), the customer gets service directly. Otherwise, if the server is started without

success (with a complementary probability θ ), the server undergoes a repair immediately and

the customer must leave the service area in order to retry for service after some random time.

Service and repair times are independent and identically distributed with common distribution

functions B1(x) and B2(x), Laplace-Stieltjes transform β1(s) and β2(s), and finite means μ−11

and μ−12 , respectively.

If we assume that time is slotted into intervals of equal length �, the above continuous-time

system can be approximated by a discrete-time system for which

p = λ �; r = 1 − γ �; s j,i =∫ i �

(i−1) �

d B j (x), j = 1, 2, i ≥ 1

where � is sufficiently small so that p and r are probabilities.

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Ann Oper Res (2006) 141: 85–107 99

Our first objective is to prove that lim�→0

(z) is the probability generating function of the

number of customers in the M/G/1 retrial queue with starting failures.

It is not difficult to show the following equalities using the definition of the Lebesgue

integral:

lim�→0

ρ1 = λ

μ1

; lim�→0

ρ2 = θ

θ

λ

μ2

; lim�→0

Sj ( p + p z) = β j (λ (1 − z)), j = 1, 2.

The proof of these relationships is omitted here since the technique used can be found

Theorem 5 in (Yang and Li, 1995).

Since r → 1 as � → 0, from Eqs. (21)–(22), we obtain

d

dzϕ0(z) = lim

�→0

ϕ0(z) − ϕ0(r z)

z − r z= lim

�→0

{F(r z)

(1 − r ) zϕ0(r z)

}

but

lim�→0

F(r z)

(1 − r ) z

= λ

γlim�→0

{1 − λ (1 − r z) � − θ S1( p + p r z) − θ r z S2( p + p r z)

[1 − λ (1 − r z) �] [θ S1( p + p r z) + θ r z S2( p + p r z) − r z]

}= λ

γ

1 − θ β1(λ (1 − z)) − θ z β2(λ (1 − z))

θ β1(λ (1 − z)) + θ z β2(λ (1 − z)) − z

and consequently

d

dzϕ0(z) = λ

γ

1 − θ β1(λ (1 − z)) − θ z β2(λ (1 − z))

θ β1(λ (1 − z)) + θ z β2(λ (1 − z)) − zϕ0(z). (9)

Solving the differential equation (9), we get

ϕ0(z) → ϕ0(1) exp

{λ

γ

∫ z

1

1 − θ β1(λ (1 − u)) − θ u β2(λ (1 − u))

θ β1(λ (1 − u)) + θ u β2(λ (1 − u)) − udu

}as � → 0.

Combining the previous result with the second stochastic decomposition law yields

lim�→0

(z) =(

1 − λ

μ1

− θ

θ

λ

μ2

)θ (1 − z) β1(λ (1 − z))

θ β1(λ (1 − z)) + θ z β2(λ (1 − z)) − z×

× exp

{λ

γ

∫ z

1

1 − θ β1(λ (1 − u)) − θ u β2(λ (1 − u))

θ β1(λ (1 − u)) + θ u β2(λ (1 − u)) − udu

}

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100 Ann Oper Res (2006) 141: 85–107

Fig. 2 The prob. that the server is idle vs. θ . (a) The effect of the arrival rate. (b) The effect of the service time

which agrees with the probability generating function of the number of customers in the

M/G/1 retrial queue with starting failures (see in (Yang and Li, 1994) the Eq. (26) for

δ = α).

7. Numerical examples

In this section, we present some numerical examples to study the effect of the parameters on

the main performance characteristics. To this end, it is assumed that service and repair times

follow a Negative Binomial with generating functions

S1(x) =(

x

2 − x

)n1

and S2(x) =(

2 x

3 − x

)n2

respectively. Let us note that service and repair times are the sum of n1 and n2 independent

random variables geometrically distributed with means 2 and 1.5, respectively.

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Ann Oper Res (2006) 141: 85–107 101

Fig. 3 The effect of the repair time. (a) The prob. that the server is idle vs. θ . (b) The prob. that the server isdown vs. θ

We will concentrate our attention on three important performance descriptors: the proba-

bility that the server is idle or down, and the mean number of customers in the retrial group.

We will also remark that all the considered performance characteristics are plotted versus θ ,

because this is the most specific parameter of our model. From the stability condition, one

finds the value

θ∗(p, β1,1, β2,1) = p β2,1

1 − p β1,1 + p β2,1

such that the system is stable if and only if θ > θ∗. Hence, the domain of the functions, whose

graphics are represented, will be (θ∗, 1]. And obviously, when θ = 1 the server becomes

reliable and therefore the performance measures agree with the corresponding in the standard

discrete-time retrial queue (Yang and Li, 1995).

In figures 2(a), (b) and 3(a), the probability that the server is idle is shown against the

parameter θ . On the other hand, the probability that the server is down is depicted versus

θ in figure 3(b). As expected, ϕ0(1) and ϕ2(1, 1) are respectively increasing and decreasing

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102 Ann Oper Res (2006) 141: 85–107

Fig. 4 The mean orbit size vs. θ . (a) The effect of the arrival rate. (b) The effect of the service time

functions of the success probability. If the system size is considerable, what happens when

θ approaches θ∗, the probability that there is any attempt at every slot is close to 1. In this

particular case, the retrial group is transformed into a queue with random service discipline

and consequently ϕ0(1) ≈ π0,0 ≈ 0. This situation is known in continuous-time as high rate

of retrials.

In figure 2(a), we study the influence of the arrival rate on the probability that the server is

idle. Specifically, we present three curves which correspond to p = 0.1, 0.2, 0.3. As intuition

tells us, ϕ0(1) increases with decreasing values of p. And the behaviour of ϕ0(1) with respect

to the service time is displayed in figure 2(b). The curves correspond to n1 = 1, 2, 3. As is to

be expected, when the service times increase the probability that the server is idle decreases.

The influence of the repair times on ϕ0(1) and ϕ2(1, 1) is illustrated in figure 3 for the same

parameters n2 = 1, 3, 9, 30. It is observed that ϕ0(1) and ϕ2(1, 1) are respectively decreasing

and increasing functions of the repair time, which also agrees with the intuitive expectations.

It should be pointed out that the graphics in figure 3 are consistent when n2 increases with

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Ann Oper Res (2006) 141: 85–107 103

Fig. 5 The mean orbit size vs. θ . (a) The effect of the repair time. (b) The effect of the retrial rate

the intuition and the limiting result

limβ2,1→∞

θ∗(p, β1,1, β2,1) = 1.

Besides, it can be analytically and numerically checked that the probability that the server

is down does not depend on the repair time distribution when θ tends to θ∗, which is really

surprising.

In figures 4–5, for the numerical evaluation, we will use the approximation (ii), explained

in Remark 3, where ε = 0.001. Figures 4–5 describe the effect of the arrival and retrial rate

and the service and repair times on the mean orbit size. For different choices of the parameters,

the curves show that E[N ] is decreasing as function of θ , as we expected. In addition, the

mean number of repeated customers diverges when the parameter θ approximates to θ∗.

These graphics corroborate that the expectation E[N ] increases with increasing values of p,

n1, n2 and r . Finally, we would like to remark that the mean orbit size in the Geo/G/1/∞queue with starting failures provides a lower bound for the corresponding mean value in our

queueing system.

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104 Ann Oper Res (2006) 141: 85–107

Appendix

To resolve the Kolmogorov equations (1)–(3), we introduce the auxiliary generating func-

tions:

ϕ1,i (z) =∞∑

k=0

π1,i,k zk, i ≥ 1; ϕ2,i (z) =∞∑

k=1

π2,i,k zk, i ≥ 1.

Multiplying Eqs. (1)–(3) by zk and summing over k, these equations become

ϕ0(z) = p [ϕ0(r z) + ϕ1,1(r z) + ϕ2,1(r z)] (10)

ϕ1,i (z) = ( p + p z) ϕ1,i+1(z) + p + p z

zθ s1,i [ϕ0(z) + ϕ1,1(z) + ϕ2,1(z)] −

− p

zθ s1,i [ϕ0(r z) + ϕ1,1(r z) + ϕ2,1(r z)], i ≥ 1 (11)

ϕ2,i (z) = ( p + p z) ϕ2,i+1(z) + ( p + p z) θ s2,i [ϕ0(z) + ϕ1,1(z) + ϕ2,1(z)] −− p θ s2,i [ϕ0(r z) + ϕ1,1(r z) + ϕ2,1(r z)], i ≥ 1. (12)

By substituting Eq. (10) into Eqs. (11)–(12), we get

ϕ1,i (z) = ( p + p z) ϕ1,i+1(z) − 1 − z

zp θ s1,i ϕ0(z) +

+ p + p z

zθ s1,i [ϕ1,1(z) + ϕ2,1(z)], i ≥ 1 (13)

ϕ2,i (z) = ( p + p z) ϕ2,i+1(z) − (1 − z) p θ s2,i ϕ0(z) ++ ( p + p z) θ s2,i [ϕ1,1(z) + ϕ2,1(z)], i ≥ 1. (14)

Multiplying Eqs. (13)–(14) by xi and summing over i yields

x − ( p + p z)

xϕ1(x, z) = p + p z

zθ S1(x) ϕ2,1(z) + p + p z

z[θ S1(x) − z] ϕ1,1(z)

− 1 − z

zp θ S1(x) ϕ0(z) (15)

x − ( p + p z)

xϕ2(x, z) = ( p + p z)

[θ S2(x) − 1

]ϕ2,1(z) + ( p + p z) θ S2(x) ϕ1,1(z)

− (1 − z) p θ S2(x) ϕ0(z). (16)

Setting x = p + p z in (15)–(16), we have

(1 − z) p θ S1( p + p z) ϕ0(z) = ( p + p z) [θ S1( p + p z) − z] ϕ1,1(z)

+ ( p + p z) θ S1( p + p z) ϕ2,1(z) (17)

(1 − z) p θ S2( p + p z) ϕ0(z) = ( p + p z) θ S2( p + p z) ϕ1,1(z)

+ ( p + p z)[θ S2( p + p z) − 1

]ϕ2,1(z). (18)

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Ann Oper Res (2006) 141: 85–107 105

The following two lemmas, whose proofs can be readily obtained, will be used later on.

Lemma 1. The inequalities S1(x) ≤ x, S2(x) ≤ x hold for 0 ≤ x ≤ 1.

Lemma 2.

(1) The inequality θ S1( p + p z) + θ z S2( p + p z) − z > 0 holds for 0 ≤ z < 1 if and onlyif ρ1 + ρ2 < 1.

(2) The following limit is positive if and only if ρ1 + ρ2 < 1:

limz→1

1 − z

θ S1( p + p z) + θ z S2( p + p z) − z= 1

θ (1 − ρ1 − ρ2).

From (17)–(18) we obtain the auxiliary generating functions:

ϕ1,1(z) = S1( p + p z)

p + p z

p θ (1 − z)

θ S1( p + p z) + θ z S2( p + p z) − zϕ0(z) (19)

ϕ2,1(z) = S2( p + p z)

p + p z

p θ z (1 − z)

θ S1( p + p z) + θ z S2( p + p z) − zϕ0(z). (20)

Note that, by using Lemma 2, the above functions are defined for 0 ≤ z < 1 and in z = 1

can be extended by continuity if ρ1 + ρ2 < 1.

Inserting ϕ1,1(r z) and ϕ2,1(r z) into Eq. (10) leads to

ϕ0(z) = θ S1( p + p r z) + θ r z S2( p + p r z) − r z ( p + p r z)

( p + p r z) [θ S1( p + p r z) + θ r z S2( p + p r z) − r z]p ϕ0(r z)

= G(r z) ϕ0(r z). (21)

It follows, by using Eq. (21) recursively, that ϕ0(z) = ϕ0(0)∏∞

k=1 G(rk z). The conver-

gence of this infinite product is established in the following lemma.

Lemma 3. If ρ1 + ρ2 < 1, the infinite product∏∞

k=1 G(rk z) converges.

Proof: Firstly, we will express G(z) as

G(z) = 1 + F(z) (22)

where

F(z) = p + p z − θ S1( p + p z) − θ z S2( p + p z)

( p + p z) [θ S1( p + p z) + θ z S2( p + p z) − z]p z.

Applying Lemma 2 and the clear inequalities

p + p z − θ S1( p + p z) − θ z S2( p + p z) ≥ θ [ p + p z − S1( p + p z)]

+ θ [ p + p z − S2( p + p z)] ≥ 0

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106 Ann Oper Res (2006) 141: 85–107

valid for 0 ≤ z ≤ 1, it can be easily shown that

F(z) ≥ 0 for 0 ≤ z ≤ 1 if ρ1 + ρ2 < 1.

Considering Eq. (22), the infinite product can be rewritten as

∞∏k=1

G(rk z) =∞∏

k=1

[1 + F(rk z)]. (23)

It is well-known that infinite product in (23) converges if and only if the series∑∞

k=1 F(rk z) is convergent (Corollary 5.6 in (Conway, 1973)) which is obvious since limz→1F(rk+1 z)

F(rk z)= r < 1. �

Now putting the generating functions (19)–(20) into Eqs. (15)–(16), we obtain

ϕ1(x, z) = S1(x) − S1( p + p z)

x − ( p + p z)

p x (1 − z) θ ϕ0(z)

θ S1( p + p z) + θ z S2( p + p z) − z

ϕ2(x, z) = S2(x) − S2( p + p z)

x − ( p + p z)

p x (1 − z) θ z ϕ0(z)

θ S1( p + p z) + θ z S2( p + p z) − z.

The normalizing condition ϕ0(1) + ϕ1(1, 1) + ϕ2(1, 1) = 1 allows us to find out the con-

stant ϕ0(1) = 1 − ρ1 − ρ2 and therefore ϕ0(0).

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