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Technical Note—A Markov Chain Partitioning Algorithm for Computing Steady State Probabilities

Author

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  • Theodore J. Sheskin

    (Cleveland State University, Cleveland, Ohio)

Abstract

This paper presents a partitioning algorithm for recursively computing the steady state probabilities for a finite, irreducible Markov chain or a Markov process. The algorithm contains a matrix reduction routine, followed by a vector enlargement routine. The matrix reduction routine repeatedly partitions the transition matrix for the Markov chain, creating a sequence of smaller, reduced transition matrices. The vector enlargement routine computes the components of the steady state probability vector by starting with the smallest reduced matrix and working sequentially toward the original transition matrix. This procedure produces an exact solution for the steady state probabilities. No special structure is required for the Markov chain. In theory, the procedure imposes no limit on the size of the largest Markov chain to which the partitioning algorithm can be applied. In practice, roundoff errors may require modifications to the method.

Suggested Citation

  • Theodore J. Sheskin, 1985. "Technical Note—A Markov Chain Partitioning Algorithm for Computing Steady State Probabilities," Operations Research, INFORMS, vol. 33(1), pages 228-235, February.
    • Handle: RePEc:inm:oropre:v:33:y:1985:i:1:p:228-235
    DOI: 10.1287/opre.33.1.228
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    Cited by:

    1. Karakoyun, Ece Cigdem & Avci, Harun & Kocaman, Ayse Selin & Nadar, Emre, 2023. "Deviations from commitments: Markov decision process formulations for the role of energy storage," International Journal of Production Economics, Elsevier, vol. 255(C).
    2. Isaac M. Sonin & Constantine Steinberg, 2016. "Continue, quit, restart probability model," Annals of Operations Research, Springer, vol. 241(1), pages 295-318, June.
    3. Yiqiang Q. Zhao & W. John Braun & Wei Li, 1999. "Northwest corner and banded matrix approximations to a Markov chain," Naval Research Logistics (NRL), John Wiley & Sons, vol. 46(2), pages 187-197, March.
    4. Zhenya Li & Zulfiqar Ali & Tong Cui & Sadia Qamar & Muhammad Ismail & Amna Nazeer & Muhammad Faisal, 2022. "A comparative analysis of pre- and post-industrial spatiotemporal drought trends and patterns of Tibet Plateau using Sen slope estimator and steady-state probabilities of Markov Chain," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 113(1), pages 547-576, August.
    5. Amod J. Basnet & Isaac M. Sonin, 2022. "Parallel computing for Markov chains with islands and ports," Annals of Operations Research, Springer, vol. 317(2), pages 335-352, October.

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