An unreliable retrial queue with delaying repair and general retrial times under Bernoulli vacation schedule
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 . The service times 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[e−θB] and finite kth moment βk, for . 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 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 satisfies the following state equationwhere 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. , then and therefore Eqs. (4.19), (3.4) yieldswhere .
These are the PGFs of the orbit size and system size distributions of the unreliable
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 that
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 is
- (ii)
the probability that the server is idle but system is nonempty is
- (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 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.
References (52)
Unreliable queueing with repeated orders
Microelectron. Reliab.
(1993)- et al.
A single server retrial queue with general retrial time and Bernoulli schedule
Appl. Math. Comput.
(2005) Analysis of an M/G/1 queue with constant repeated attempts and server vacations
Comput. Oper. Res.
(1997)Accessible bibliography on retrial queues
Math. Comput. Model.
(1999)- et al.
An retrial queueing system with two phases of service subject to the server breakdown and repair
Perform. Eval.
(2008) - et al.
Approximation method for retrial queues with phase type inter-retrial times
Eur. J. Oper. Res.
(1999) - et al.
M[x]/(G1, G2)/1 retrial queue under Bernoulli vacation schedules with general repeated attempts and starting failures
Appl. Math. Model.
(2009) - et al.
Modified vacation policy for M/G/1 retrial queue with balking and feedback
Comput. Ind. Eng.
(2009) - et al.
The M/G/1 retrial queue with Bernoulli schedule and general retrial time
Comput. Math. Appl.
(2002) An M/G/1 retrial queue with recurrent customers and general retrial times
Appl. Math. Comput.
(2004)
Analysis of a single server retrial queue with FCFS orbit and Bernoulli vacation
Appl. Math. Comput.
An approximation method for the M/G/1 retrial queue with general retrial times
Eur. J. Oper. Res.
On the queueing system with repeated orders and unreliable
J. Technol.
A retrial queue with redundancy and unreliable server
Queue. Syst.
On the single server retrial queue subject to breakdowns
Queue. Syst.
An retrial queue with active breakdowns and Bernoulli schedule in the server
Int. J. Inform. Manage. Sci.
A classical bibliography of research on retrial queues: progress in 1990–1999
TOP
Standard and retrial queueing systems: a comparative analysis
Rev. Math. Comput.
Some queueing problems with the service station subject to breakdowns
Ann. Oper. Res.
An retrial queue with an additional phase of second service and general retrial time
Int. J. Inform. Manage. Sci.
A batch arrival queue with Bernoulli vacation schedule under multiple vacation policy
Int. J. Manage. Sci.
An retrial queue with control policy and general retrial time
Queue. Syst.
The retrial queue with retrial rate control policy
Probab. Eng. Inform. Sci.
The analysis of non Markovian Stochastic Process by the inclusion of supplementary variables
Proc. Cambridge Philos. Soc.
Introduction to Queueing Theory
Queueing systems with vacations – a survey
Queue. Syst.
Cited by (44)
Performance and sensitivity analysis of an M/G/1 queue with retrial customers due to server vacation
2020, Ain Shams Engineering JournalCitation Excerpt :Therefore, it is very meaningful to study retrial queues and vacation queues. Recently, retrial queues and vacation queues have caught extensive attention as very useful tools in analyzing and evaluating the performance of various important and practical systems, see the works by Artalejo and Gómez-Corral [1], Ayyappan and Karpagam [2], Choudhury and Ke [4], Dimitriou [6], Gao and Wang [10], Ke and Chang [12], Rajadurai et al. [15], Rajadurai et al. [16], Tian and Zhang [17], Wang et al. [19], Ye and Liu [20], and references therein. However, the majority of works on retrial queues and vacation queues assume that only one waiting queue exists, i.e., customers waiting in the orbit form an orbit queue or customers waiting in front of the server form an original queue.
Cost optimization and ANFIS computing for admission control of M/M/1/K queue with general retrial times and discouragement
2019, Applied Mathematics and ComputationCitation Excerpt :To develop the non-Markovian queueing model, Cox [5] first introduced the supplementary variable technique (SVT) to study M/G/1 queueing model. Choudhury and Ke [6] dealt with the unreliable server non-Markovian queueing model with general retrial times by considering Bernoulli vacation and two successive phases of service. The non-Markovian queueing model with general retrial attempts and Bernoulli schedule was studied by Gao et al. [7].
A study on M/G/1 feedback retrial queue with subject to server breakdown and repair under multiple working vacation policy
2018, Alexandria Engineering JournalCitation Excerpt :Thus, it is important to study retrial queue with breakdowns to implement retrial queueing models practically. Choudhury and Ke [10] have studied the batch arrival retrial queue with Bernoulli vacations and delaying repair. Recently authors like Choudhury and Deka [11], Yang et al. [12], Rajadurai et al. [13–15] and Dimitriou [16,17] discussed about the retrial queueing systems with the concept of breakdown and repair.
On an unreliable-server retrial queue with customer feedback and impatience
2018, Applied Mathematical ModellingCitation Excerpt :Choudhury et al. [14] extended the model of Choudhury and Deka [15] by taking the Bernoulli admission mechanism, delay times and batch arrivals into consideration. For some other articles on unreliable retrial queues, interested researchers can refer to Atencia et al. [16], Choudhury and Deka [15, 17], Efrosinin and Winkler [18], Gharbi and Dutheillet [19], Choudhury and Ke [20, 21], Rajadurai et al. [22], Singh et al. [23] and Yang et al. [24]. Motivated by the above situations, this paper analyzes a feedback retrial queueing system with reneging, balking and multiple unreliable servers.
Optimal joining strategies in a repairable retrial queue with reserved time and N-policy
2024, Operational ResearchA LITERATURE REVIEW ON RETRIAL QUEUEING SYSTEM WITH BERNOULLI VACATION
2024, Yugoslav Journal of Operations Research