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Duality results for block-structured transition matrices

Part of: Operations research and management science Markov processes

Published online by Cambridge University Press:  14 July 2016

Yiqiang Q. Zhao ,
Wei Li  and
Attahiru Sule Alfa
Yiqiang Q. Zhao*
Affiliation:
University of Winnipeg
Wei Li*
Affiliation:
University of Louisiana at Lafayette
Attahiru Sule Alfa*
Affiliation:
University of Manitoba
*
∗Postal address: Department of Mathematics and Statistics, University of Winnipeg, Winnipeg, Manitoba, Canada R3B 2E9. Email address: [email protected]
∗∗Postal address: Department of Electrical and Computing Engineering, University of Louisiana at Lafayette, PO Box 43890, Lafayette, LA 70504–3890, USA.
∗∗∗Postal address: Department of Mechanical and Industrial Engineering, University of Manitoba, Winnipeg, Manitoba, Canada R3T 5V6.
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Abstract

In this paper, we consider a certain class of Markov renewal processes where the matrix of the transition kernel governing the Markov renewal process possesses some block-structured property, including repeating rows. Duality conditions and properties are obtained on two probabilistic measures which often play a key role in the analysis and computations of such a block-structured process. The method used here unifies two different concepts of duality. Applications of duality are also provided, including a characteristic theorem concerning recurrence and transience of a transition matrix with repeating rows and a batch arrival queueing model.

Keywords

Markov renewal processesdualityblock structures

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 90B22: Queues and service 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 1045 - 1057
Copyright
Copyright © Applied Probability Trust 1999 

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