The semi-markovian queue: theory and applications

Abstract

In this paper, we study a first-come-first-served single server semi-Markovian queue in which both the arrival and service mechanisms are semi-Markov processes. The interarrival time and service times may depend on one another and the marginal distribution of the service times is assumed to be phase-type. For this queue, we show that the distributions of waiting time, time in system and virtual waiting time are matrix-exponential. Further, these matrix-exponential distributions have phase-type representations. For the special case when the interarrival times are independent of the service times, we show that the queue length distribution is matrix-geometric. For this special case, we prove that the queue length distribution problem is the dual of the waiting time distribution problem, i.e., finding the solution of one problem immediately gives the solution of the other. We show that our methods are computationally feasible and report our numerical experience. We give Examples where such queues arise naturally. In particular, we discuss an application in manufacturing, a periodic queue and a queue with Markov modulated arrivals and services.