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Markov-Renewal Programming. I: Formulation, Finite Return Models

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  • William S. Jewell

    (Operations Research Center and Department of Industrial Engineering, University of California, Berkeley)

Abstract

A special structure in dynamic programming which has been studied by Bellman, Blackwell, D'Épenoux, Derman, Howard, Manne, Oliver, Wolfe and Dantzig, and others is the problem of programming over a Markov chain This paper extends their results and solution algorithms to programming over a Markov-renewal process---in which the intervals between transitions of the system from state i to state j are independent samples from a distribution that may depend on both i and j . For these processes, a general reward structure and a decision mechanism are postulated, the problem is to make decisions at each transition to maximize the total expected reward at the end of the planning horizon. The paper is divided into two parts. This part describes the properties of Markov-renewal processes, the reward structure, and the decision process. Algorithms for finite-horizon, and infinite-horizon models with discounting are presented. The second part will investigate the models that have infinite return.

Suggested Citation

  • William S. Jewell, 1963. "Markov-Renewal Programming. I: Formulation, Finite Return Models," Operations Research, INFORMS, vol. 11(6), pages 938-948, December.
    • Handle: RePEc:inm:oropre:v:11:y:1963:i:6:p:938-948
    DOI: 10.1287/opre.11.6.938
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    Cited by:

    1. Prasenjit Mondal, 2020. "Computing semi-stationary optimal policies for multichain semi-Markov decision processes," Annals of Operations Research, Springer, vol. 287(2), pages 843-865, April.
    2. Chris P. Lee & Glenn M. Chertow & Stefanos A. Zenios, 2008. "Optimal Initiation and Management of Dialysis Therapy," Operations Research, INFORMS, vol. 56(6), pages 1428-1449, December.
    3. Luyao Niu & Bhaskar Ramasubramanian & Andrew Clark & Radha Poovendran, 2023. "Robust Satisfaction of Metric Interval Temporal Logic Objectives in Adversarial Environments," Games, MDPI, vol. 14(2), pages 1-23, March.
    4. Zhang, Xueqing & Gao, Hui, 2012. "Road maintenance optimization through a discrete-time semi-Markov decision process," Reliability Engineering and System Safety, Elsevier, vol. 103(C), pages 110-119.
    5. Nooshin Salari & Viliam Makis, 2020. "Application of Markov renewal theory and semi‐Markov decision processes in maintenance modeling and optimization of multi‐unit systems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 67(7), pages 548-558, October.

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