A Discrete-Time Geo/G/1 Retrial Queue with J Vacations and Two Types of Breakdowns

Abstract

This paper is concerned with a discrete-time $Geo/G/1$ retrial queueing model with $J$ vacations and two types of breakdowns. If the orbit is empty, the server takes at most $J$ vacations repeatedly until at least one customer appears in the orbit upon returning from a vacation. It is assumed that the server is subject to two types of different breakdowns and is sent immediately for repair. We analyze the Markov chain underlying the considered queueing system and derive the system state distribution as well as the orbit size and the system size distributions in terms of their generating functions. Then, we obtain some performance measures through the generating functions. Moreover, the stochastic decomposition property and the corresponding continuous-time queueing system are investigated. Finally, some numerical examples are provided to illustrate the effect of vacations and breakdowns on several performance measures of the system.