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A Method to Calculate Steady-State Distributions of Large Markov Chains by Aggregating States

Author

Listed:
  • Brion N. Feinberg

    (Stanford University, Stanford, California)

  • Samuel S. Chiu

    (Stanford University, Stanford, California)

Abstract

This paper develops an efficient iterative algorithm to calculate the steady-state distribution of nearly all irreducible discrete-time Markov chains. Computational experiences suggest that, for large Markovian systems (more than 130 states), the proposed algorithm can be ten times faster than standard Gaussian elimination in finding solutions to an accuracy of 0.1%. The proposed algorithm is developed in three stages. First, we develop a very efficient algorithm for determining steady-state distributions of a restricted class of Markovian systems. A second result establishes a relationship between a general irreducible Markovian system and a system in the restricted class of Markovian systems. Finally, we combine the two results to produce an efficient, iterative algorithm to solve Markov systems. The paper concludes with a discussion of the observed performance of the algorithm.

Suggested Citation

  • Brion N. Feinberg & Samuel S. Chiu, 1987. "A Method to Calculate Steady-State Distributions of Large Markov Chains by Aggregating States," Operations Research, INFORMS, vol. 35(2), pages 282-290, April.
    • Handle: RePEc:inm:oropre:v:35:y:1987:i:2:p:282-290
    DOI: 10.1287/opre.35.2.282
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    Cited by:

    1. Papadopoulos, H. T. & Heavey, C., 1996. "Queueing theory in manufacturing systems analysis and design: A classification of models for production and transfer lines," European Journal of Operational Research, Elsevier, vol. 92(1), pages 1-27, July.
    2. Chenxi Liu & Nan Chen & Jianing Yang, 2015. "New method for multi-state system reliability analysis based on linear algebraic representation," Journal of Risk and Reliability, , vol. 229(5), pages 469-482, October.
    3. Niek Baer & Ahmad Al Hanbali & Richard J. Boucherie & Jan-Kees van Ommeren, 2022. "A successive censoring algorithm for a system of connected LDQBD-processes," Annals of Operations Research, Springer, vol. 310(2), pages 389-410, March.

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