On the transient behavior of the m/m/m/m loss model

Abstract

We develop an asymptotic technique for analyzing the transient behavior of queues and other stochastic models. As an example we consider the Erlang loss model which is the M/M/m/m queue. Asymptotic expansions are constructed for systems with a large number of servers and a large arrival rate that is O(m). We give formulas for the probability p n(t) that n servers are occupied at time t. We treat several cases of initial conditions and several regions in the (n, t) plane. Our approach is based on applying the ray method to a scaled form of the forward Kolmogorov equation which describes the time evolution of p n(t).