Services within a busy period of an M/M/1 queue and Dyck paths

TL;DR

This paper studies the service times of customers in an M/M/1 queue during busy periods, revealing their distribution and correlation structure, and connects these findings to Dyck paths and their generating functions.

Contribution

It provides the distribution of service times at different points in a busy period and links the process to Dyck paths, offering new insights into queue dynamics and combinatorial structures.

Findings

Service time distributions depend on customer position in the busy period.

The process of service start times is not Poisson.

Dyck paths encode the correlation functions of successive services.

Abstract

We analyze the service times of customers in a stable M/M/1 queue in equilibrium depending on their position in a busy period. We give the law of the service of a customer at the beginning, at the end, or in the middle of the busy period. It enables as a by-product to prove that the process of instants of beginning of services is not Poisson. We then proceed to a more precise analysis. We consider a family of polynomial generating series associated with Dyck paths of length 2n and we show that they provide the correlation function of the successive services in a busy period with (n+1) customers.