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Attainability for Markov and Semi-Markov Chains

Author

Listed:
  • Brecht Verbeken

    (Department of Business Technology and Operations, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
    Data Analytics Laboratory, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium)

  • Marie-Anne Guerry

    (Department of Business Technology and Operations, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
    Data Analytics Laboratory, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium)

Abstract

When studying Markov chain models and semi-Markov chain models, it is useful to know which state vectors n , where each component n i represents the number of entities in the state S i , can be maintained or attained. This question leads to the definitions of maintainability and attainability for (time-homogeneous) Markov chain models. Recently, the definition of maintainability was extended to the concept of state reunion maintainability ( S R -maintainability) for semi-Markov chains. Within the framework of semi-Markov chains, the states are subdivided further into seniority-based states. State reunion maintainability assesses the maintainability of the distribution across states. Following this idea, we introduce the concept of state reunion attainability, which encompasses the potential of a system to attain a specific distribution across the states after uniting the seniority-based states into the underlying states. In this paper, we start by extending the concept of attainability for constant-sized Markov chain models to systems that are subject to growth or contraction. Afterwards, we introduce the concepts of attainability and state reunion attainability for semi-Markov chain models, using S R -maintainability as a starting point. The attainable region, as well as the state reunion attainable region, are described as the convex hull of their respective vertices, and properties of these regions are investigated.

Suggested Citation

  • Brecht Verbeken & Marie-Anne Guerry, 2024. "Attainability for Markov and Semi-Markov Chains," Mathematics, MDPI, vol. 12(8), pages 1-14, April.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:8:p:1227-:d:1378816
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    References listed on IDEAS

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    1. Vincent A. Amenaghawon & Virtue U. Ekhosuehi & Augustine A. Osagiede, 2023. "Markov manpower planning models: a review," International Journal of Operational Research, Inderscience Enterprises Ltd, vol. 46(2), pages 227-250.
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    4. Pablo Azcue & Esther Frostig & Nora Muler, 2023. "Optimal Strategies in a Production Inventory Control Model," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-43, March.
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