Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T03:06:41.827Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asmussen, S. (1987). Applied Probability and Queues. John Wiley, New York.Google Scholar
Asmussen, S. (1995). In Advances in Queueing, ed. Dshalalow, J., CRC Press, Boca Raton, FL, Chap. 3.Google Scholar
Asmussen, S., and Ramaswami, V. (1990). Probabilistic interpretations of some duality results for the matrix paradigms in queueing theory. Stochastic Models 6, 715–733.CrossRefGoogle Scholar
Kemeny, J. G., Snell, J. L., and Knapp, A. W. (1976). Denumerable Markov Chains, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Li, H., and Zhao, Y. Q. (1999). Stochastic block-monotonicity in the approximation of the stationary distribution of infinite Markov chains. Accepted by Stochastic Models.Google Scholar
Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Johns Hopkins University Press, Baltimore, MD.Google Scholar
Neuts, M. F. (1985). Structured Stochastic Matrices of M/G/1 Type and Their Applications. Marcel Decker, New York.Google Scholar
Ramaswami, V. (1990). A duality theorem for the matrix paradigms in queueing theory. Stochastic Models 6, 151–161.CrossRefGoogle Scholar
Siegmund, D. (1976). The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes. Ann. Prob. 4, 914–924.CrossRefGoogle Scholar
Zhao, Y. Q., Li, W., and Braun, W. J. (1998). Infinite block-structured transition matrices and their properties. Adv. Appl. Prob. 30, 365–384.Google Scholar