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Reduced System Algorithms for Markov Chains

Author

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  • Ram Lal

    (Department of Management Science, California State University, Fullerton, California 92634)

  • U. Narayan Bhat

    (Department of Statistics, Southern Methodist University, Dallas, Texas 75275)

Abstract

A reduced system is a smaller system derived in the process of analyzing a larger system. In solving for steady state probabilities of a Markov chain, generally the solution can be found by first solving a reduced system of equations which is obtained by appropriately partitioning the transition probability (or rate) matrix. Following Lal (Lal, R. 1981. A unified study of algorithms for steady state probabilities in Makov chains. Ph.D. Dissertation, Department of Operations Research and Engineering Management, School of Engineering and Applied Sciences, Southern Methodist University, Dallas, TX 75275.), a Markov chain can be categorized as standard or nonstandard depending on the location of an invertible submatrix necessary for an efficient solution in a transition probability (or rate) matrix. In this paper, algorithms for the determination of steady state probabilities are developed by using (i) a backward recursion which is efficient for standard systems and (ii) a forward recursion which is efficient for nonstandard systems. It is also shown that the backward recursion can be used for finding the first passage time distribution and its mean and variance.

Suggested Citation

  • Ram Lal & U. Narayan Bhat, 1988. "Reduced System Algorithms for Markov Chains," Management Science, INFORMS, vol. 34(10), pages 1202-1220, October.
    • Handle: RePEc:inm:ormnsc:v:34:y:1988:i:10:p:1202-1220
    DOI: 10.1287/mnsc.34.10.1202
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