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

Abstract

This paper develops a probabilistic decomposition method for an ${\text{M}}^{\xi }$/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 $(0,t]$. 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 ${\text{M}}^{\xi }$/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.