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Filtering of Markov renewal queues, I: Feedback queues

Published online by Cambridge University Press:  01 July 2016

Jeffrey J. Hunter
Jeffrey J. Hunter*
Affiliation:
University of Auckland
*
∗Postal address: Department of Mathematics and Statistics, University of Auckland, Private Bag, Auckland, New Zealand.
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Abstract

Queueing systems which can be formulated as Markov renewal processes with basic transitions of three types, ‘arrivals', ‘departures' and ‘feedbacks' are examined. The filtering procedure developed for Markov renewal processes by Çinlar (1969) is applied to such queueing models to show that the queue-length processes embedded at any of the ‘arrival', ‘departure', ‘feedback', ‘input', ‘output' or ‘external' transition epochs are also Markov renewal. In this part we focus attention on the derivation of stationary and limiting distributions (when they exist) for each of the embedded discrete-time processes, the embedded Markov chains. These results are applied to birth–death queues with instantaneous state-dependent feedback including the special cases of M/M/1/N and M/M/1 queues with instantaneous Bernoulli feedback.

Keywords

MARKOV RENEWAL PROCESSESMARKOV CHAINSQUEUE LENGTHSLIMITING DISTRIBUTIONSSTATIONARY DISTRIBUTIONSBIRTH-DEATH QUEUES
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 2 , June 1983 , pp. 349 - 375
Copyright
Copyright © Applied Probability Trust 1983 

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References

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