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A single-server queue with server vacations and a class of non-renewal arrival processes

Published online by Cambridge University Press:  01 July 2016

David M. Lucantoni ,
Kathleen S. Meier-Hellstern  and
Marcel F. Neuts
David M. Lucantoni*
Affiliation:
AT&T Bell Laboratories
Kathleen S. Meier-Hellstern*
Affiliation:
AT&T Bell Laboratories
Marcel F. Neuts*
Affiliation:
University of Arizona
*
∗Postal address: Room 3K-601, AT&T Bell Laboratories, Holmdel, NJ 07733, USA.
∗∗Postal address: Room 3J-612, AT&T Bell Laboratories, Holmdel, NJ 07733, USA.
∗∗∗Postal address: Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ 85721, USA.
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Abstract

We study a single-server queue in which the server takes a vacation whenever the system becomes empty. The service and vacation times and the arrival process are all assumed to be mutually independent. The successive service times and the vacation times each form independent, identically distributed sequences with general distributions. A new class of non-renewal arrival processes is introduced. As special cases, it includes the Markov-modulated Poisson process and the superposition of phase-type renewal processes.

Algorithmically tractable equations for the distributions of the waiting times at an arbitrary time and at arrivals, as well as for the queue length at an arbitrary time, at arrivals, and at departures are established. Some factorizations, which are known for the case of renewal input, are generalized to this new framework and new factorizations are obtained. The algorithmic implementation of these results is discussed.

Keywords

MARKOVIAN ARRIVAL PROCESSMATRIX ANALYTIC METHODSFACTORIZATIONSMARKOV MODULATED POISSON PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 3 , September 1990 , pp. 676 - 705
Copyright
Copyright © Applied Probability Trust 1990 

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Footnotes

The research of this author was supported in part by Grants ECS-8601203 from the National Science Foundation and AFOSR-88-0076 from the Air Force Office of Scientific Research.

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