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

The paper develops an asymptotic technique for analyzing the transient behavior of queues and other stochastic models. As an example it considers the Erlang loss model which is the M/M/m/m queue. Asymptotic expansions are constructed for systems with a large number of servers (m>>1) and a large arrival rate that is 𝒪(m). The paper gives formulas for the probability pn(t) that n servers are occupied at time t. It treats several cases of initial conditions and several regions in the (n,t) plane. The present approach is based on applying the ray method to a scaled form of the forward Kolmogorov equation which describes the time evolution of pn(t).