(M, MAP)/(PH, PH)/1 Queue with Non-preemptive Priority and Working Vacation Under N-Policy

Abstract

In this paper we consider two single server queueing models with non-preemptive priority and working vacation under two distinct N-policies. High priority (type I) customers are served even in vacation mode whereas low priority (type II) customers are served only when the server comes to normal mode of service. Type I customers have only a limited waiting space L whereas type II customers have unlimited capacity. The two distinct N-policies are as described below: In model I, while service of type I customers are in progress in vacation mode (working vacation), if the number of such customers present in the system hits N ($$\le L$$) or the vacation timer (clock) expires, whichever occurs first, the server is switched on to normal mode. In model II, switching the server to normal mode from vacation mode occurs as soon as the accumulated number (those served out plus those present in the system) of type I customers during that working vacation hits N or the vacation timer expires, whichever occurs first. Type I customers arrive according to a Poisson process whereas type II customer’s arrival is governed by Markovian Arrival Process. Service time of type I and type II customers follow distinct phase type distributions. At a service completion epoch, finding the system empty, server takes an exponentially distributed working vacation. During working vacation, type I customers are served at a reduced rate. On vacation expiration, the service of the type I customer already in service, will start from the beginning in the normal mode of service. We analyze these models in steady state to compute the distribution of duration of service time continuously in slow mode, expected number of returns to 0 type I customer state, starting from 0 type I customer state during vacation mode of service before the arrival of a type II customer, the distribution of a p-cycle in normal mode, LSTs of busy cycle, busy period of type I customers generated during the service time of a type II customer and LSTs of waiting time distributions of type I and type II customers. We compare these models in steady state by numerical experiments to identify the superior model.