An M/G/1 retrial queue with recurrent customers and general retrial times

https://doi.org/10.1016/j.amc.2003.09.019Get rights and content

Abstract

This paper relates to an M/G/1 retrial queue with two types of customers: transit customers, who arrive according to a Poisson process, and a fixed number of recurrent customers, who immediately return to the orbit after having received a service. We first present the ergodicity condition for the system to be stable and derive analytical results for the stationary distribution as well as some performance measures of the system. A stochastic decomposition law for this retrial queuing system is established too. Finally, some numerical examples are studied.

Introduction

Retrial queues are characterized by the phenomenon that arriving customers who find the server busy join the retrial group (called orbit) to repeat their request for service after some random time. Retrial queuing systems have been widely used to model many practical problems in telephone switching systems, telecommunication networks, and computers competing to gain service from a central processing unit. For recent bibliographies on retrial queues, see [3], [4], [11], [12], [26].

Many of the queuing systems with repeated attempts operate under the classical retrial policy. However, there is a second discipline, called constant retrial policy, which arises naturally in problems where the server is required to search for customers [23] and in communication protocols of type carrier sense multiple access (CSMA). The latter discipline was introduced by Fayolle [15], who investigated an M/M/1 retrial queue in which the repeated customers form a queue and only the customer at the head of the orbit queue can request a service after an exponentially distributed retrial time. Farahmand [13] calls this discipline a retrial queue with FCFS orbit. Since Fayolle [15], there has been a fast development in the literature about retrial queues with constant repeated attempts [1], [2], [9], [13], [14], [22]. Choi et al. [8] generalized the constant retrial policy by considering an M/M/1 retrial queue with general retrial times where only the customer at the head of the orbit may attempt retrials from orbit. Later, Gómez-Corral [17] discussed extensively a retrial queuing system with FCFS discipline and general retrial times. Since 1999, several authors have spent their time studying retrial queues with general service times and non-exponential retrial time distributions [6], [18], [19], [20], [21].

On the other hand, Boxma and Cohen [7] studied an M/G/1 queue in which there is a fixed number of permanent customers present who rejoin the queue on their completion of service. This system with permanent customers in the retrial context was analyzed by Farahmand [14] from two points of view: classical and constant retrial policy. Our main objective is to generalize the second case in [14] considering a more general and natural situation: different service distribution for both types of customers and general retrial times for the transit customers. The fundamental reason for analyzing this queuing system is that its structure appears in many representations of computer and communication networks. Another reason is the connection with a vacation queue, since the service times of the recurrent customers can be viewed as server vacations. Thus the service times of the recurrent customers can be introduced in order to exploit the idle time of the server for other subsidiary tasks, such as machine repair, preventive maintenance, scanning for new work,…

The rest of the paper is organized as follows. The queuing model under consideration is described in detail in the next section. Section 3 is devoted to the study of the basic stochastic process describing the dynamic of our queue and its stability. The steady-state distribution of the server state and the orbit length is studied in Section 4. In Section 5, we show that a stochastic decomposition law also holds for our model. Finally, some numerical examples are presented.

Section snippets

Description of the queuing system

We consider a single-server retrial queue with two classes of customers: transit (also called ordinary) customers and a fixed number K (K⩾1) of recurrent (also called permanent) customers. After service completion, recurrent customers always return to the retrial group and transit customers leave the system forever.

Transit customers arrive according to a Poisson process with rate λ. If a transit customer finds the server free on his arrival, he occupies the server; otherwise, he enters the

Basic stochastic process

The state of the system at time t can be described by the Markov processX(t)=(C(t),N(t),ξ0(t),ξ1(t)),where C(t) denotes the server state at time t (0 or 1 or 2, according to the server is idle or busy with a transit customer or busy with a recurrent customer, respectively) and N(t) is the number of repeated customers at time t. If C(t)=0 and N(t)>K, then ξ0(t) represents the elapsed retrial time of the transit repeated customer at the head of the orbit. If C(t)∈{1,2}, then ξ1(t) corresponds to

Analysis of the steady-state probabilities

It is well-known that the limiting probabilities for the stochastic process {X(t),t⩾0} exist and are positive if the embedded Markov chain is ergodic. If the stationary regime exists, we can define the limiting probabilityp0,K=limt→∞P[C(t)=0,N(t)=K]and the limiting probability densitiesp0,n(x)=limt→∞P[C(t)=0,N(t)=n,x<ξ0(t)⩽x+dx],n⩾K+1,p1,n(x)=limt→∞P[C(t)=1,N(t)=n,x<ξ1(t)⩽x+dx],n⩾K,p2,n(x)=limt→∞P[C(t)=2,N(t)=n,x<ξ1(t)⩽x+dx],n⩾K−1for x⩾0.

Following the routine procedure of the method of

Stochastic decomposition

In this section we give a stochastic decomposition law for the system size, which is closely related to the vacation models. During the last three decades considerable attention has been paid to analyze queuing systems with server vacation. One of the most remarkable results that concerns with such type of queues is the stochastic decomposition result. This significant outcome was first established by Fuhrmann and Cooper [16] for M/G/1 type queues with generalized vacation; they proved that the

Numerical examples

To illustrate the effect of the parameters on the main performance measures, this section considers the model with the arrival rate λ=0.1. The service times for the transit and recurrent customers are assumed to be exponentially distributed with means β1,1=1 and β2,1=2 respectively. We also suppose that the interretrial times of any transit customer are governed by an Erlang distribution with parameters ν and l, i.e., α(s)=(ν/(ν+s))l. Let us remember that the Erlang distributed variable can be

References (26)

  • B.D. Choi et al.

    An M/M/1 retrial queue with control policy and general retrial times

    Queueing Systems

    (1993)
  • B.D. Choi et al.

    The M/G/1 retrial queue with retrial rate control policy

    Probability in the Engineering and Informational Sciences

    (1993)
  • R.B. Cooper

    Introduction to Queueing Theory

    (1981)

    • Cited by (30)

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

      2014, Applied Mathematics and Computation
      Citation Excerpt :

      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.

    • A batch arrival priority queue with recurrent repeated demands, admission control and hybrid failure recovery discipline

      2013, Applied Mathematics and Computation
      Citation Excerpt :

      Recently Kobza and Nachlas [19] studied the transient analysis of permanent customers in single server queue. In retrial context, a system with retrial permanent customers was investigated by Farahmand [15] in continues time, while in discrete time significant contribution has been made by Moreno [23], Atencia et al. [5], Gao et al. [17]. Recently, Kalyanaraman and Srinivasan [18] studied a non preemptive priority retrial queue with retrial permanent customers.

    • A batch arrival retrial queue with general retrial times under Bernoulli vacation schedule for unreliable server and delaying repair

      2012, Applied Mathematical Modelling
      Citation Excerpt :

      Many of the queueing systems with repeated attempts operate under the classical retrial policy, where each block of customers generate 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 [9]) and in communication protocols of type carrier sense multiple access (CSMA). The latter discipline was introduced by Fayolle [10], who investigated an M/M/1 retrial queue in which the repeat customers from a queue and only the head customers of the orbit queue can request a service after an exponentially distributed retrial time with some parameter ε (say), i.e., retrial rate is (1 − δ0,n)ε (where δi,j denotes Kronecker’s delta), when number of units in the orbit is ‘n’.

    • Analysis of an M / { D<inf>n</inf> } / 1 retrial queue

      2007, Journal of Computational and Applied Mathematics
      Citation Excerpt :

      Retrial queues with continuous service times are very interesting and an active research field over the last two decades. Recent work include the following [25,2,4,10,34,20,24,21,22,26,5,1,35]. The classical queueing systems with discrete service times have been considered by many authors such as Erlang [14], Crommelin [13], Prabhu [28,29], Riordan [30], Bertsekas and Gallager [6], Tijms [33], Brahimi and Worthington [7], Iversen and Staalhagen [18], Brun and Garcia [9], Franx [17], Brill [8], Shortle and Brill [31], and Shortle et al. [32].

    • A single-server retrial queue with general retrial times and Bernoulli schedule

      2005, Applied Mathematics and Computation

    • A note on infinite-server markov modulated and single-server retrial queues

      2014, Asia-Pacific Journal of Operational Research
    View all citing articles on Scopus
    View full text
    Copyright © 2003 Elsevier Inc. All rights reserved.