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The M/M/1 queue with mass exodus and mass arrivals when empty

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Anyue Chen  and
Eric Renshaw
Anyue Chen*
Affiliation:
University of Greenwich
Eric Renshaw*
Affiliation:
University of Strathclyde
*
∗Postal address: School of Computing and Mathematical Sciences, University of Greenwich, Wellington Street, Woolwich, London SE18 6PF, UK.
∗∗Postal address: Department of Statistics and Modelling Science, Livingstone Tower, University of Strathclyde, 26 Richmond Street, Glasgow Gl 1XH, UK.
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Abstract

An M/M/1 queue is subject to mass exodus at rate β and mass immigration at rate when idle. A general resolvent approach is used to derive occupation probabilities and high-order moments. This powerful technique is not only considerably easier to apply than a standard direct attack on the forward p.g.f. equation, but it also implicitly yields necessary and sufficient conditions for recurrence, positive recurrence and transience.

Keywords

M/M/1 QUEUECATASTROPHEMASS IMMIGRATIONQ-PROCESSRECURRENCETRANSIENCEIDLE PROBABILITYEQUILIBRIUM PROBABILITIESFACTORIAL MOMENTS

MSC classification

Primary: 60J35: Transition functions, generators and resolvents
Secondary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 34 , Issue 1 , March 1997 , pp. 192 - 207
Copyright
Copyright © Applied Probability Trust 1997 

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