The exact transient distribution of the queue length of the Mt/Mt/1 single server queue with time-dependent Poisson arrival rate and time-dependent exponential service rate was recently obtained by Zhang and Coyle in terms of a solution to a Volterra equation. Their method involved the use of generating functions and complex analysis. In this paper, we present an approach that ties the computation of these transient distributions directly to the random sample path behaviour of the Mt/Mt/1 queue. We show the versatility of this method by applying it to the Mt/Mt/c multiserver queue, and indicate how it can be applied to queues with time-dependent phase arrivals or time-dependent phase service.
