Erlang Redux: An Ansatz Method for Solving the M/M/m Queue

TL;DR

This paper introduces a new algebraic method using an ansatz transformation based on the Erlang B function to solve for waiting and residence times in M/M/m queues, simplifying previous approaches.

Contribution

It presents a novel algebraic approach for solving M/M/m queues that avoids complex probability derivations, relying only on elementary Poisson distribution knowledge.

Findings

Provides exact solutions for waiting and residence times in M/M/m queues.

Simplifies queue analysis by replacing complex derivations with an ansatz transformation.

Builds on and improves previous approximate methods.

Abstract

This exposition presents a novel approach to solving an M/M/m queue for the waiting time and the residence time. The motivation comes from an algebraic solution for the residence time of the M/M/1 queue. The key idea is the introduction of an ansatz transformation, defined in terms of the Erlang B function, that avoids the more opaque derivation based on applied probability theory. The only prerequisite is an elementary knowledge of the Poisson distribution, which is already necessary for understanding the M/M/1 queue. The approach described here supersedes our earlier approximate morphing transformation.