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On the Dual Relationship Between Markov Chains of GI/M/1 and M/G/1 Type

Part of: Markov processes Operations research and management science

Published online by Cambridge University Press:  01 July 2016

P. G. Taylor  and
B. Van Houdt
P. G. Taylor*
Affiliation:
University of Melbourne
B. Van Houdt*
Affiliation:
University of Antwerp
*
∗ Postal address: Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia. Email address: p.taylor@ms.unimelb.edu.au
∗∗ Postal address: Performance Analysis of Telecommunications Systems Research Group, Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, Antwerp, B 2020, Belgium. Email address: benny.vanhoudt@ua.ac.be
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Abstract

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In 1990, Ramaswami proved that, given a Markov renewal process of M/G/1 type, it is possible to construct a Markov renewal process of GI/M/1 type such that the matrix transforms G(z, s) for the M/G/1-type process and R(z, s) for the GI/M/1-type process satisfy a duality relationship. In his 1996 PhD thesis, Bright used time reversal arguments to show that it is possible to define a different dual for positive-recurrent and transient processes of M/G/1 type and GI/M/1 type. Here we compare the properties of the Ramaswami and Bright dual processes and show that the Bright dual has desirable properties that can be exploited in the design of algorithms for the analysis of Markov chains of GI/M/1 type and M/G/1 type.

Keywords

Matrix-analytic modelsquasi-birth-and-death processprocesses of GI/M/1 typeprocesses of M/G/1 type

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 1 , March 2010 , pp. 210 - 225
Copyright
Copyright © Applied Probability Trust 2010 

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