An Elementary Derivation of Mean Wait Time in Polling Systems

TL;DR

This paper presents a simple, algebraic derivation of the average wait time in polling systems, making the results more accessible for practical use and education.

Contribution

It introduces an elementary derivation method for mean wait time in polling systems, avoiding complex mathematical tools used in traditional approaches.

Findings

Derivation based solely on algebra and Poisson process properties

Simplifies calculation of average wait time in polling systems

Facilitates practical applications and teaching of queueing theory

Abstract

Polling systems are a well-established subject in queueing theory. However, their formal treatments generally rely heavily on relatively sophisticated theoretical tools, such as moment generating functions and Laplace transforms, and solutions often require the solution of large systems of equations. We show that, if you are willing to only have the average waiting of a system time rather than higher moments, it can found through an elementary derivation based only on algebra and some well-known properties of Poisson processes. Our result is simple enough to be easily used in real-world applications, and the simplicity of our derivation makes it ideal for pedagogical purposes.