On an M/G/1 Queue with Vacation in Random Environment

Abstract

We study an M / G / 1 queue with multiple vacation and vacation interruption. Both normal vacation (type I) and working vacation (type II) are considered. The exhaustive service discipline is assumed in this paper. At the end of a busy period, depending on the environment, the server either opts for normal vacation or working vacation. On completion of type I vacation if the server finds the system empty he goes for type II vacation. On completion of type II vacation if the server finds the system empty goes for another type II vacation. On completion of service in type II vacation, if the server finds one or more customers in queue he returns to normal service, interrupting the vacation. An arriving customer, during type I vacation, joins the queue with probability q or leaves the system with probability $$\displaystyle 1-q$$ and during type II vacation all the arriving customers join the queue. Using supplementary variable technique we derive the distributions for the queue length and service status under steady state condition. Laplace-Stieltjes transform of the stationary waiting time is also developed. Some numerical illustrations are also given.