The M^X/M/1 queue with working breakdown

School of Mathematics and Statistics, Central South University, Changsha 410075, P.R. China
yy0605xx@126.com

Received: 15 May 2013
Accepted: 24 February 2014

Abstract

In this paper, we consider a batch arrival M^X/M/1 queue model with working breakdown. The server may be subject to a service breakdown when it is busy, rather than completely stoping service, it will decrease its service rate. For this model, we analyze a two-dimensional Markov chain and give its quasi upper triangle transition probability matrix. Under the system stability condition, we derive the probability generating function (PGF) of the stationary queue length, and then obtain its stochastic decomposition, which shows the relationship with that of the classical M^X/M/ 1 queue without vacation or breakdown. Besides, we also give the Laplace-Stieltjes transform (LST) of the stationary waiting time distribution of an arbitrary customer in a batch. Finally, some numerical examples are given to illustrate the effect of the parameters on the system performance measures.