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Filtering of Markov renewal queues, II: Birth-death 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

In this part we extend and particularise results developed by the author in Part I (pp. 349–375) for a class of queueing systems which can be formulated as Markov renewal processes. We examine those models where the basic transition consists of only two types: ‘arrivals' and ‘departures'. The ‘arrival lobby' and ‘departure lobby' queue-length processes are shown, using the results of Part I to be Markov renewal. Whereas the initial study focused attention on the behaviour of the embedded discrete-time Markov chains, in this paper we examine, in detail, the embedded continuous-time semi-Markov processes. The limiting distributions of the queue-length processes in both continuous and discrete time are derived and interrelationships between them are examined in the case of continuous-time birth–death queues including the M/M/1/M and M/M/1 variants. Results for discrete-time birth–death queues are also derived.

Keywords

MARKOV RENEWAL PROCESSESMARKOV CHAINSSEMI-MARKOV PROCESSESQUEUE LENGTHSLIMITING DISTRIBUTIONSSTATIONARY DISTRIBUTIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 2 , June 1983 , pp. 376 - 391
Copyright
Copyright © Applied Probability Trust 1983 

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References

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