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An iterative approximation scheme for repetitive Markov processes

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Tolga Tüfekçi  and
Refik Güllü
Tolga Tüfekçi*
Affiliation:
Middle East Technical University
Refik Güllü*
Affiliation:
Middle East Technical University
*
∗Postal address: Industrial Engineering Department, Information Technologies and Electronics Research Institute, (BILTEN), Middle East Technical University, 06531, Ankara, Turkey.
∗Postal address: Industrial Engineering Department, Information Technologies and Electronics Research Institute, (BILTEN), Middle East Technical University, 06531, Ankara, Turkey.
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Abstract

Repetitive Markov processes form a class of processes where the generator matrix has a particular repeating form. Many queueing models fall in this category such as M/M/1 queues, quasi-birth-and-death processes, and processes with M/G/1 or GI/M/1 generator matrices. In this paper, a new iterative scheme is proposed for computing the stationary probabilities of such processes. An infinite state process is approximated by a finite state process by lumping an infinite number of states into a super-state. What we call the feedback rate, the conditional expected rate of flow from the super-state to the remaining states, given the process is in the super-state, is approximated simultaneously with the steady state probabilities. The method is theoretically developed and numerically tested for quasi-birth-and-death processes. It turns out that the new concept of the feedback rate can be effectively used in computing the stationary probabilities.

Keywords

Quasi-birth-and-death processesapproximate solutionsnumerical algorithms

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J05: Discrete-time Markov processes on general state spaces 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 654 - 667
Copyright
Copyright © Applied Probability Trust 1999 

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