An Analysis of the Modified Offered-Load Approximation for the Nonstationary Erlang Loss Model

Abstract

A fundamental problem that led to the development of queueing theory is the probabilistic modelling of the number of busy lines in telephone trunk groups. Based on the behavior of real telephone systems, a natural model to use would be the $M_t/G/s/0$ queue, which has $s$ servers, no extra waiting space and a nonhomogeneous Poisson arrival process $(M_t)$. Unfortunately, so far queueing theory has provided an exact analysis for only the $M/G/s/0$ queue in steady state, which yields the Erlang blocking formula, and the $M_t/G/\infty$ queue, which treats nonstationary arrivals at the expense of having infinitely many servers. However, these results can be synthesized to create a modified offered-load (MOL) approximation for the $M_t/G/s/0$ queue: the distribution of the number of busy servers in the $M_t/G/s/0$ queue at time $t$ is approximated by the steady-state distribution of the stationary $M/G/s/0$ queue with an offered load (arrival rate times mean service time) equal to the mean number of busy servers in the $M_t/G/\infty$ queue at time $t$. In addition to being a simple effective approximation scheme, the MOL approximation makes all of the exact results for infinite server queues relevant to the analysis of nonstationary loss systems. In this paper, we provide a rigorous mathematical basis for the MOL approximation. We find an expression for the difference between the $M_t/G/s/0$ queue length distribution and its MOL approximation. From this expression we extract bounds on the error and deduce when one distribution stochastically dominates the other.