Transient analysis of the Erlang A model

TL;DR

This paper derives explicit Laplace transform expressions for the transient distribution and first passage times in the Erlang A queue, a model with abandonment, using advanced mathematical techniques, which were previously considered intractable.

Contribution

It provides the first explicit formulas for the transient characteristics of the Erlang A model using Green's functions and contour integrals, extending understanding of this queueing system.

Findings

Explicit Laplace transforms for transient distribution and first passage times.

Specialized results for M/M/∞, M/M/m, and M/M/m/m queues.

Results applicable to diffusion approximations.

Abstract

We consider the Erlang A model, or $M/M/m+M$ queue, with Poisson arrivals, exponential service times, and $m$ parallel servers, and the property that waiting customers abandon the queue after an exponential time. The queue length process is in this case a birth-death process, for which we obtain explicit expressions for the Laplace transforms of the time-dependent distribution and the first passage time. These two transient characteristics were generally presumed to be intractable. Solving for the Laplace transforms involves using Green's functions and contour integrals related to hypergeometric functions. Our results are specialized to the $M/M/\infty$ queue, the $M/M/m$ queue, and the $M/M/m/m$ loss model. We also obtain some corresponding results for diffusion approximations to these models.