THE TRANSIENT SOLUTION TO A CLASS OF MARKOVIAN QUEUES

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 {lambda(n), mu(n)}n = 0(infinity) such that lambda(n-1) = lambda and mu(n) = mu for all n greater-than-or-equal-to N and for some N greater-than-or-equal-to 1 with mu-0 = 0, can be represented in terms of the busy period density of an M/M/1 queue having arrival rate-lambda and the service rate-mu 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.