On the Transient Behavior of the Erlang Loss Model: Heavy Usage asymptotics

Abstract

The Erlang loss model, which is the $M/M/m/m$ queue is considered. Asymptotic expansions are constructed for systems with a large number of servers $( m \gg 1 )$ and a large arrival rate that is $O( m )$. Formulas are given for the probability $p_n ( t )$ 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.