The workload in the<i>MAP/G/</i>1 queue with state-dependent services:its application to a queue with preemptive resume priority

Abstract

In this paper, we consider the workload process in the MAP/G/1 queue with state-dependent service time distributions. The arrival process of customers is assumed to be a MAP(Markovian arrival process), which is a class of semi-Markovian arrival processes. The amount of work brought into the system upon arrival is distributed according to a general distribution which may depend on the states of the underlying Markov chain immediately before and after arrivals. We first consider the first passage time to the idle state with an arbitrary initial condition. Based on this, we derive the time-dependent LST (Laplace-Stieltjes transform) for the amount of work in the system as well as the LST of its limiting distribution. As an application of the results, we consider a preemptive resume priority queue with independent MAP arrival streams. In the preemptive resume priority queue, the waiting time of customers in a particular priority class is considered as a first passage time (governed by higher priority customers) to an idle state starting from work in the system upon arrival. We derive the LST for the waiting time of customers in each class, as well as the mean waiting time