M/M/$c$ Queues and the Poisson Clumping Heuristic

TL;DR

This paper revisits approximations of maximum queue length in M/M/c queues using a discrete Gumbel model, supported by simulations and graphical analysis, to improve understanding of queue behavior.

Contribution

It introduces a discrete Gumbel-based approximation for maximum queue length in M/M/c queues, enhancing previous models with detailed simulation comparisons.

Findings

Gumbel approximation closely matches simulation results

Graphical analysis clarifies queue length distribution

Improved understanding of maximum queue length behavior

Abstract

In continuous time, customers arrive at random. Each waits until one of $c$ servers is available; each thereafter departs at random. The distribution of maximum line length of idle customers was studied over 25 years ago. We revisit two good approximations of this, employing a discrete Gumbel formulation and detailed graphics to describe simulation outcomes.