Abstract
This paper considers an M/G/1 retrial G-queue with general retrial times, in which the server is subject to working breakdowns and repairs. If the system is not empty during a normal service period, the arrival of a negative customer can cause the server breakdown, and the failed server still works at a lower service rate rather than stopping the service completely. Applying the embedded Markov chain, we obtain the necessary and sufficient condition for the stability of the system. Using the supplementary variable method, we deal with the generating functions of the number of customers in the orbit. Various system performance measures are also developed. Finally, some numerical examples and a cost optimization analysis are presented.
Highlights
Queues with negative arrivals, called G-queues, were first introduced by Gelenbe [1]
Case 1: Suppose that λ2 = 0, i.e., λ = λ1, and our model reduces to the M/G/1 retrial queue with general retrial times
If we assume that the normal service time and the lower service time are governed by the exponential distribution with parameters μ and η, respectively, our model reduces to the M/M/1 queue with working breakdowns
Summary
Queues with negative arrivals, called G-queues, were first introduced by Gelenbe [1]. In many practical cases, the failed server still can serve a customer at a lower service rate during the breakdown period. This type of breakdown is called a working breakdown, as introduced by Kalidass and Kasturi [11], and the M/M/1 queue they analyzed can be applied to studying the behavior of the communication system or the machine replacement problem. To the authors’ best knowledge, there is no research work investigating a retrial queue with working breakdowns This motivates us to deal with an M/G/1 retrial G-queue with working breakdowns in this paper, where a breakdown may occur due to the arrival of a negative customer.
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