A BMAP/G/1 Retrial Queue with a Server Subject to Breakdowns and Repairs

Abstract

In this paper, we consider a BMAP/G/1 retrial queue with a server subject to breakdowns and repairs, where the life time of the server is exponential and the repair time is general. We use the supplementary variable method, which combines with the matrix-analytic method and the censoring technique, to study the system. We apply the RG-factorization of a level-dependent continuous-time Markov chain of M/G/1 type to provide the stationary performance measures of the system, for example, the stationary availability, failure frequency and queue length. Furthermore, we use the RG-factorization of a level-dependent Markov renewal process of M/G/1 type to express the Laplace transform of the distribution of a first passage time such as the reliability function and the busy period. KeywordsRetrial queues . Batch Markov arrival processes (BMAP) . Markov chains of M/G/1 type . Markov renewal processes of M/G/1 type . Supplementary variable method . Matrix-analytic method . Censoring technique . RG-factorization . Reliability

On a BMAP/G/1 G-queue with setup times and multiple vacations

“…In particular, Lee [15] used the supplementary variable method, which is combined with the embedded Markov chain, to analyze the BMAP/G/1 queues with finite or infinite waiting room. Li et al [18] used the supplementary variable method, which is combined with the matrix-analytic method and the censoring technique, to study the BMAP/G/1 retrial queue with a server subject to breakdowns and repairs.…”

Multi-server retrial queue with negative customers and disasters

“…see [14,15,18,21] but the case of sample path approach is quite a few, e.g. He et al [18] where the sample path method is used only for necessary condition and the mean drift method is used for the sufficient condition.…”

“…see [9,14,21,24]. Neuts and Rao [24] proposed an algorithm called generalized truncation method to compute the stationary distribution of level-dependent quasi-birth-and-death (QBD) process.…”

“…Bright and Taylor [9] improved the direct truncation method for the general class of level-dependent QBD process by presenting an elaborate method for determining the truncation level and computing the boundary probability vectors. For computing the stationary distribution of the level-dependent M/G/1 type Markov chain, Dudin and Klimenok [14] used a finite state Markov chain that is constructed by the censoring technique and Li et al [21] used the RG-factorization method after modifying the level-dependent Markov chain to the level independent one except some boundary levels and the boundary vectors are obtained by solving the finite system of linear equations that comes from the balance equation of a censored Markov chain. In this paper, we combine the methods in [9,24] to compute the stationary queue length distribution.…”

Tail asymptotics for the queue length in an M/G/1 retrial queue

“…In addition to a large number of research papers published on retrial queues, the survey papers by Yang and Templeton [23], Falin [8], Kulkarni and Liang [14], and Li et al [16], a book by Falin and Templeton [9], and a bibliography by Artalejo [1] testify their significance.…”