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Filtering of Markov renewal queues, III: semi-Markov processes embedded in 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, The University of Auckland, Private Bag, Auckland, New Zealand.
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Abstract

In Part I (Hunter) a study of feedback queueing models was initiated. For such models the queue-length process embedded at all transition points was formulated as a Markov renewal process (MRP). This led to the observation that the queue-length processes embedded at any of the ‘arrival', ‘departure', ‘feedback', ‘input', ‘output' or ‘external' transition epochs are also MRP. Part I concentrated on the properties of the embedded discrete-time Markov chains. In this part we examine the semi-Markov processes associated with each of these embedded MRP and derive expressions for the stationary distributions associated with their irreducible subspaces. The special cases of birth-death queues with instantaneous state-dependent feedback, M/M/1/N and M/M/1 queues with instantaneous Bernoulli feedback are considered in detail. The results obtained complement those derived in Part II (Hunter) for birth-death queues without feedback.

Keywords

MARKOV RENEWAL PROCESSESQUEUE LENGTHSLIMITING DISTRIBUTIONSSTATIONARY DISTRIBUTIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 2 , June 1984 , pp. 422 - 436
Copyright
Copyright © Applied Probability Trust 1984 

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