The transient solution to a class of Markovian queues

Abstract

In this paper, we study the transient behavior of Markovian queues such as M/ M/ s queues and bulk-arrival M/ M/1 queues. It will be shown that the transition probabilities of a birth-death process on the non-negative integers, governed by parameters { λ n , μ n } n=0 ∞ such that λ n−1 = λ and μ n = μ for all n ≥ N and for some N ≥ 1 with μ 0 = 0, can be represented in terms of the busy period density of an M/ M/1 queue having the arrival rate λ and the service rate μ and some exponential functions. The transition probabilities of a bulk-arrival M/ M/1 queue can also be expressed in terms of its busy period density.