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Idle and busy periods in stable M/M/k queues

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

D. J. Daley  and
L. D. Servi
D. J. Daley*
Affiliation:
Australian National University
L. D. Servi*
Affiliation:
GTE Laboratories Incorporated
*
∗Postal address: Stochastic Analysis Group, Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, ACT 0200, Australia. Email address: Daryl.Daley@anu.edu.au
∗∗Postal address: GTE Laboratories Incorporated, 40 Sylvan Rd., Waltham MA 02254, USA. Email address: lds0@gte.com
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Abstract

This paper finds the first and second moments of the number of arrivals in a stable M/M/k queue during an idle period, i.e. in a period when at least one server is idle. These and similar results are used along with renewal theory asymptotics to find the first and second moments of the proportion of arrivals occurring while there are at least ξ customers waiting for service. The asymptotics of the second moment for large ξ are established.

Keywords

Many-server queuebusy periodidle period

MSC classification

Primary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 4 , December 1998 , pp. 950 - 962
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

Work done with support from GTE while visiting GTE Laboratories Incorporated.

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