On the transient behavior of the Erlang loss model: Heavy usage asymptotics

The Erlang loss model, which is the M/M/m/m queue is considered. Asymptotic expansions are constructed for systemswith a large number of servers (m>>1) and a large arrival rate that is O(m). Fromulas are given for the probability pn(t) that n servers are occupied at time t. Several cases are treated of initial conditions and several regions in the (n,t) plane. The approximations are constructed by using singular perturbation techniques such as the ray method, boundary layer theory, and the method of matched asymptotic expansions.