An unreliable retrial queue with delaying repair and general retrial times under Bernoulli vacation schedule

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Abstract

This paper deals with the steady state behavior of an M/G/1 queue with general retrial time and Bernoulli vacation schedule for an unreliable server, which consist of a breakdown period and delay period. While the server is working with primary customers, it may breakdown at any instant and server will be down for short interval of time. Further concept of the delay time is also introduced. The primary customer finding the server busy, down or vacation enters a group of unsatisfied customer called orbit. After the completion of a service, the server either goes for a vacation of random length with probability p or may continue serve for the next unit, if any with probability (1  p). For this model, we first derive the system size distribution at a departure epoch. Secondly, we derive the probability generating function of joint distributions of state of the server and queue size, which is one of chief objectives of the paper. Finally, we obtain some important performance measures and reliability indices of this model.

Introduction

Retrial queues (or queues with repeated attempts) are characterized by the feature that a customer who finds the server busy upon arrival is obliged to leave the service area and repeat its demand after some time called “retrial time”. Between trials, the blocked customer joins a group of unsatisfied customers called “orbit” or ‘retrial group’. For a review of main results and methods, the reader is referred to the survey papers by Yang and Templeton [51], Falin [21], Kulkarni and Liang [37] and the book by Falin and Templeton [23]. For more recent references see the bibliographical overviews in [8], [9], [28]. Further, a comprehensive comparison between retrial queue and their standard counterpart with classical waiting line can be found in Artalejo and Falin [10].

Many of the queueing systems with repeated attempts operate under the classical retrial policy, where each block of customer generates a stream of repeated attempts independently of the rest of the customers in the orbit. However, there is a second kind of policy, called constant retrial policy, which arises naturally in problems where the server is required to search for customers (e.g. see [42]) and in communication protocols of type carrier sense multiple access (CSMA). The latter discipline was introduced by Fayolle [25], who investigated an M/M/1 retrial queue in which the repeated customers from a queue and only the customers head of the orbit queue can requested a service after an exponentially distributed retrial time with some parameter ɛ (say) i.e. retrial rate is (1  δ0,n)ɛ (where δi,j denotes Kronecker’s delta), when number of units in the orbit is ‘n’. Farahmand [24] called this discipline a retrial queue with FCFS orbit. Choi et al. [15] generalized the constant retrial policy by considering an M/M/1 retrial queue with general retrial times. Since Fayolle [25], there has been rapid growth in literature e.g. see [16], [20]. This retrial policy is a useful device for modeling the retrial phenomenon in communication and computer networks where repeated attempts are made by processor units independently of the numbers of messages stored in each node of the network. Beside, this kind of retrial control policy was used for stability of the ALOHA protocol and unslotted CSMA/CD (carrier sense multiple access with collision detection) protocol in communication systems.

Retrial queueing system with general service time and non exponential retrial times have also received considerable attention during last decade. The first investigation with general retrial time done by Kapyrin [30], who assumed that each customer in the orbit generates a stream of repeated attempts that are independent of the customer in the orbit and state of the server. However, this methodology was found to be incorrect by Falin [22]. Subsequently, Yang et al. [52] have developed an approximation method to obtain the steady state performance for the model of Kapyrin. Later, Gomez-Corral [29] discussed extensively an M/G/1 retrial queue with FCFS discipline and general retrial times. In recent years, several retrial models have been analyzed with general retrial times, details of which may be found in [5], [12], [35], [41], [50].

A wide class of retrial policies for governing the vacation mechanism have also been discussed in the literature. Most of the analysis for retrial queues concerns the exhaustive service schedule [7], gated service policy [38] and recently modified vacation policy [32]. Keilson and Servi [33] introduced Bernoulli vacation schedule: if the queue is empty after service completion then the server become inactive, i.e. begins a vacation period. If the queue is not empty then service begins with specific probability q or a vacation period begins with probability p = 1  q. At the end of a vacation period service begins if a customer is present in the queue. Otherwise, the server waits for the first customer to arrive. A number of papers [31], [34], [49] have recently appeared in the queueing literature in which concept of general retrial times have been introduced along with Bernoulli vacation schedule under the constant retrial policy. Such type of queueing situations occur in many real life situations where the server may be used for other secondary jobs, for instance to serve customers in other systems. Allowing server to take vacations makes the queueing model more realistic and flexible in studying real word queueing situations. Applications arise naturally in call centers with multi task employees, customized manufacturing, telecommunication and computer networks, maintenance activities, production and quality control problems, etc.

The study of queueing models with service interruptions goes back to the 1950s. Among some early papers on service interruptions, we refer the readers to see the papers by Gaver [27], Avi-ltzhak and Naor [11], Thirurengadan [47] and Mitrany and Avi-ltzhak [40] for some fundamental works. While Li et al. [39], Sengupta [43], Takin and Sengupta [45] and Tang [46], among others have studied some queueing systems with interruptions wherein one of the underlying assumption is that, the service channel undergoes repair instantaneously, as soon as it fails. On the other hand retrial queues that take into account servers failures and repairs were introduced by Aissani [1] and Kulkarni and Choi [36]. As related literature, we should mention some papers studied in [2], [3], [4]. Wang et al. [48] studied a repairable M/G/1 retrial queueing model from the viewpoint of reliability for the first time, and both of the queueing indices and reliability characteristics are obtained. Recently, Choudhury and Deka [13] investigate such a repairable M/G/1 retrial queueing model with two phases of service. A model of similar nature with Bernoulli schedule has also been studied by Atencia et al. [6]. More recently, Ke and Chang [31], investigates similar type of M/G/1 retrial queueing model with two phases of service and Bernoulli vacation schedule starting with failure having general retrial times. Although some aspects have been discussed separately on queueing systems with service interruptions, Bernoulli schedule vacation, repeated attempts with general retrial time, however, no work have been found that combine these features together for unreliable server queueing systems, even in the most recent studies. Moreover, another important characteristic for considering the retrial model with general retrial times is that we always obtain analytical solution in term of closed form expression. Hence to fill up to this gap, in this article an attempt has been made to study an M/G/1 retrial queue with general retrial times and Bernoulli vacation schedule for an unreliable server, which is consist of a server’s breakdown period and delayed period. To this end, the methodology will be based on embedded Markov chain and inclusion of supplementary variables.

The rest of the paper is organized as follows. In Section 2, we give a brief description of the mathematical model. Section 3 deals with the derivations of the stability criteria for existence of stationary regime and study the embedded Markov chain describing the behavior of the system size distribution at a departure epoch. Section 4 deals with the steady state joint distribution of the server state and the number of customers in the orbit. Some important particular cases of this model are discussed briefly in Section 5. Further, existence of the Stochastic decomposition property is also demonstrated in Section 6. Some important performance measures and reliability indices of this model are derived in Section 7. Finally a simple numerical example is given to study the cost effectiveness maximization model in Section 8.

Section snippets

The mathematical model

We consider an M/G/1 queueing system where the primary customers arrive according to a Poisson process with mean arrival rate λ. The service times {Bn;n1} of the customers are identically and independently distributed (i.i.d.) random variables with probability distribution function (d.f) B(x), Laplace Stieltjes Transform (LST) β(θ) = E[eθB] and finite kth moment βk, for k1. While the server is serving the primary customers, it may breakdown at any time and the server will be down for a short

Embedded Markov chain

We first investigate the necessary and sufficient condition for the system to be stable of our model. Let {tn;nZ+} be the sequence epochs at which nth either service completed or vacation terminated i.e. we consider the epochs at which total service requested by a customer expires. Then the sequence Xm = N(tm+) forms a Markov chain, which is embedded in our queueing system. We observe that {Xn;nZ+} satisfies the following state equationXn=Xn-1-Wn+Un,where Un is the number of units arriving

Jonit distribution of number in the orbit and state of the server

In this section, attempts have been made to obtain the PGF of the joint distribution of the state of the server and number in the orbit by treating elapsed retrial time, elapsed service time, of the customer and elapsed vacation time, elapsed delay time and elapsed repair time of the server as supplementary variables. Assuming the system is in steady state conditions. Let N(t) be the orbit size (i.e. number of customers in the retrial group) at time t, R0(t), B0(t) be the elapsed retrial time

Some particular case

In this section we discuss briefly some particular cases of our model, which are consistent with the existing literature. Suppose we assume that retrial time distribution is exponential with d.f. C(x)=1-exp{-νx};x>0, then C(λ)=ν(λ+ν)-1 and therefore Eqs. (4.19), (3.4) yieldsΦ(z)=ν(1-z)(1-σ){q+pϖ(a(z))}β(A(z))-z(λ+ν),andπ(z)=ν(1-z)(1-σ)[q+pϖ(a(z))]β(A(z)){q+pϖ(a(z))}β(A(z))-z(λ+ν),where σ=ρH(λ+ν)ν.

These are the PGFs of the orbit size and system size distributions of the unreliable M/G/1

Stochastic decomposition

In this section we study the stochastic decomposition property of the system size distribution of our model. The literature on vacation models recognizes this property one of the most interesting features in this mater, e.g. see Doshi [19], and Fuhrmann and Cooper [26]. Stochastic decomposition for the retrial models has also been found in Yang et al. [52] and Yang and Tampleton [51]. The existence of the stochastic decomposition property for our model can be demonstrated easily by showing thatπ

System performance measures

Our next object is to provide explicit expression for the system state probabilities and some important performance measures of the system. First of all we derive the system state probabilities and results are summaries in the following theorem.

Theorem 7.1

If the system is in steady state conditions, then we have

  • (i)

    the probability that the server is idle and system is empty isPE=(C(λ)-ρH)C(λ)

  • (ii)

    the probability that the server is idle but system is nonempty isPNE=(C(λ)-ρH)ρHC(λ)

  • (iii)

    the probability that the server

A real world application with the cost effectiveness maximization model

In this section, we present a possible application and some numerical examples in some situations to explain that our model can represent the possible application reasonably well. In the transfer model of e-mail system, mail system uses the Simple Mail Transfer Protocol (SMTP) to deliver the messages between mail servers. When a mail transfer program contacts a server on a remote machine, it forms a TCP connection over which it communicates. Once the connection is in plane, the two programs

Conclusion

In this paper we have studied an M/G/1 retrial queue with following features: a customer who finds the server busy joins an orbit and retries for service after a random time generally distributed, the server is unreliable and may breakdown during the service, the server is subject to a Bernoulli vacation schedule. As soon as breakdown occurs it is sent for repair. In many real life situations it may not feasible to start the repairs immediately due to non-availability of the server and the

Acknowledgments

The author would like to thank the referee for their constructive comments on the earlier version of the paper.

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