Services within a Busy Period of an M/M/1 Queue and Dyck Paths

Authors:
Moez Draief;Jean Mairesse
Affiliations:
LIAFA, CNRS, Université Paris 7, Paris Cedex 05, France 75251;LIAFA, CNRS, Université Paris 7, Paris Cedex 05, France 75251
Venue:
Queueing Systems: Theory and Applications
Year:
2005

Citing 3
Cited 1

Enumerative combinatorics

Enumerative combinatorics
Concrete Mathematics: A Foundation for Computer Science

Concrete Mathematics: A Foundation for Computer Science
On the area swept under the occupation process of an M/M/1 queue in a busy period

Queueing Systems: Theory and Applications

A direct approach to sojourn times in a busy period of an M/M/1 queue

Queueing Systems: Theory and Applications

Quantified Score

Hi-index	0.02

Visualization

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.