Probabilistic Decomposition Method on the ServerIndices of an /G/1 Vacation Queue

This paper develops a probabilistic decomposition method for an /G/1 repairable queueing system with multiple vacations, in which the customers who arrive during server vacations enter the system with probability p. Such a novel method is used to analyze the main performance indices of the server, such as the unavailability and the mean failure number during . It is derived that the structures of server indices are two convolution equations. Further, comparisons with existing methods indicate that our method is effective and applicable for studying server performances in single-server /G/1 vacation queues and their complex variants. Finally, a stochastic order and production system with a multipurpose production facility is numerically presented for illustrative purpose. 1. Introduction There are some effective and convenient analytic methods for single-server queues with a repairable server or service station. For example, the Markov renewal process method is used to study an M/G/1 queueing system with repairable service station in [1], the geometric process method introduced by Lam is applied to analyze the lifetime behaviors and repair times of deteriorating service station in [2, 3], and the matrix-geometric method is available for GI/M/1 and /1 repairable queues in [4, 5]. It is well known that the supplementary variable method posed by Cox [6] is very important in dealing with some Poisson input queues with a repairable server. Many researchers, such as Wang [7], Ke et al. [8], Liu et al. [9], and Cao [10], have utilized this method for lots of repairable single-server queueing systems. The above approaches were applied to analyze some queueing indices, such as queue size, waiting time, and their stochastic decompositions, and the performance measures of the server, such as the mean times to the first failure, unavailability and failure frequency. However, the common methods mentioned above usually become too complicated to be solved especially when dealing with some Poisson input bulk arrival queues with a repairable server and their complex vacation variants. In this paper, based on the renewal process theory and Laplace and Laplace-Stieltjes transforms we develop a probabilistic decomposition method to analyze the performance measures of the repairable server for a single-server /G/1 queue with variable input rate and multiple vacations. Our method is completely different from the methods used in [1–10] and reveals that the structures of the server indices in Poisson input single-server bulk arrival vacation queues are two convolution