A discrete time retrial queue with general retrial times and balking customers

Abstract

In this work, we consider a discrete time Geo/G/1 retrial queue with general retrial times and balking customers. If a new arriving customer finds the server busy, he may join the orbit to retry getting the required service again or depart completely from the system. Using the supplementary variable technique, this queueing system is modelled using a Markov chain. We derive the generating functions of the steady state distribution of this Markov chain. Hence, we establish the generating functions of the orbit size and the system size distributions. This set of generating functions is used to derive various performance measures. We prove a stochastic decomposition law and use it to a derive a measure of the proximity between the distributions of the system size in the present model and the corresponding one without retrials. A set of recursive formulae is built up to facilitate computing the orbit size and the system size distributions. Numerical results are presented with a focus on the effect of balking on the system performance.

A Discrete-Time UnreliableGeo/G/1Retrial Queue with Balking Customers, Second Optional Service, and General Retrial Times

“…In addition, Figures 3 and 4 show how the mean repair time affects the mean customer of customers in the orbit. In general, compared with the previous model of Aboul-Hassan et al [11] and Atencia and Moreno [24], our simulation results show the impact of various practical factors on the mean number of customers in the orbit ( ) simultaneously.…”

“…From the curves in Figures 1 and 2, we can also obtain the comparative results between our model and the corresponding discrete-time queueing model without second optional service and server breakdowns. Figure 2 also shows that the mean number of customers in the orbit ( ) increases significantly compared with the model of Aboul-Hassan et al [11]. That is, our simulation results show that the breakdown of the server has a heavy impact on the performance measures of the discrete-time retrial queueing model with balking customer.…”

“…The main objective in this paper is to extend the analysis of continuous-time M/G/1 retrial queueing system with second optional service [13], exponential server lifetime [17], and general retrial times to discrete-time case. Moreover, our model generalized the discrete-time retrial queueing system studied by Atencia and Moreno [7] and Aboul-Hassan et al [11] directly. Our queueing system can be used to model many practical situations.…”

“…Queueing models with balking or impatience customers arise in many diverse applications. Although the incorporation of balking or impatience customers in retrial queueing systems is very important, so far there is only little work on retrial queueing systems with balking or impatience customers in the literature [10][11][12]. On the other hand, queueing models in which the server may provide a second phase service have been proved very useful to analyze many practical situations, arising in manufacturing systems, call center, and computer systems.…”

Retrial queueing models in discrete time: a short survey of some late arrival models

“…the complicated mixed arrival pattern of primary and secondary customers, is absent in models with servers who upon a service completion start to search for a customer. In these models the choice of the next customer to be served lies in the hands of the server and is not determined by the decision of a customer to approach the server from the orbit [see Atencia and Moreno (2004a), Wang and Zhao (2007) and Aboul-Hassan et al (2008) for three examples in discrete time]. We think that the term retrial queue for these latter models is a misnomer, and we propose to rename these models as queueing models with an actively searching server (everybody who has a better name is welcome of course!)…”