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Continue, quit, restart probability model

Author

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  • Isaac M. Sonin

    (University of North Carolina at Charlotte)

  • Constantine Steinberg

    (University of North Carolina at Charlotte)

Abstract

We discuss a new applied probability model: there is a system whose evolution is described by a Markov chain (MC) with known transition matrix on a discrete state space and at each moment of a discrete time a decision maker can apply one of three possible actions: continue, quit, and restart MC in one of a finite number of fixed “restarting” points. Such a model is a generalization of a model due to Katehakis and Veinott (Math. Oper. Res. 12:262, 1987), where a restart to a unique point was allowed without any fee and quit action was absent. Both models are related to Gittins index and to another index defined in a Whittle family of stopping retirement problems. We propose a transparent recursive finite algorithm to solve our model by performing O(n 3) operations.

Suggested Citation

  • Isaac M. Sonin & Constantine Steinberg, 2016. "Continue, quit, restart probability model," Annals of Operations Research, Springer, vol. 241(1), pages 295-318, June.
    • Handle: RePEc:spr:annopr:v:241:y:2016:i:1:d:10.1007_s10479-012-1089-2
    DOI: 10.1007/s10479-012-1089-2
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    References listed on IDEAS

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    1. Yuri Kabanov & Robert Liptser, 2006. "From Stochastic Calculus to Mathematical Finance. The Shiryaev Festschrift," Post-Print hal-00488295, HAL.
    2. Isaac Sonin, 1999. "The Elimination algorithm for the problem of optimal stopping," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 49(1), pages 111-123, March.
    3. Sonin, Isaac M., 2008. "A generalized Gittins index for a Markov chain and its recursive calculation," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1526-1533, September.
    4. Yinyu Ye, 2011. "The Simplex and Policy-Iteration Methods Are Strongly Polynomial for the Markov Decision Problem with a Fixed Discount Rate," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 593-603, November.
    5. Eric V. Denardo & Uriel G. Rothblum & Ludo Van der Heyden, 2004. "Index Policies for Stochastic Search in a Forest with an Application to R&D Project Management," Mathematics of Operations Research, INFORMS, vol. 29(1), pages 162-181, February.
    6. Michael N. Katehakis & Arthur F. Veinott, 1987. "The Multi-Armed Bandit Problem: Decomposition and Computation," Mathematics of Operations Research, INFORMS, vol. 12(2), pages 262-268, May.
    7. Winfried K. Grassmann & Michael I. Taksar & Daniel P. Heyman, 1985. "Regenerative Analysis and Steady State Distributions for Markov Chains," Operations Research, INFORMS, vol. 33(5), pages 1107-1116, October.
    8. 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.
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    Cited by:

    1. 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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