Services within a busy period of an M/M/1 queue and Dyck paths
Discrete Mathematics
2007-07-30 v1
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.
Cite
@article{arxiv.0707.4124,
title = {Services within a busy period of an M/M/1 queue and Dyck paths},
author = {Moez Draief and Jean Mairesse},
journal= {arXiv preprint arXiv:0707.4124},
year = {2007}
}