Random Overlapping Intervals--A Generalization of Erlang's Loss Formula

Abstract

Consider a queueing system with infinitely many servers, a general distribution of service times and an instantaneous rate $\alpha_k$ of new arrivals, where $\alpha_k$ depends only on the number of busy servers. This is called a generalized Erlang model (GEM) since if $\alpha_k = \alpha (k < N), \alpha_k = 0 (k \geqq N)$, then Erlang's model for a telephone exchange with $N$ lines is recovered. The synchronous and asynchronous stationary distributions of the GEM are determined and several interesting properties of the process are discussed. In particular the stationary GEM is shown to be reversible.