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The Randomization Technique as a Modeling Tool and Solution Procedure for Transient Markov Processes

Author

Listed:
  • Donald Gross

    (The George Washington University, Washington, D.C.)

  • Douglas R. Miller

    (The George Washington University, Washington, D.C.)

Abstract

We present a randomization procedure for computing transient solutions to discrete state space, continuous time Markov processes. This procedure computes transient state probabilities. It is based on a construction relating a continuous time Markov process to a discrete time Markov chain. Modifications and extensions of the randomization method allow for computation of distributions of first passage times and sojourn times in Markov processes, and also the computation of expected cumulative occupancy times and expected number of events occurring during a time interval. Several implementations of the randomization procedure are discussed. In particular we present an implementation for a general class of Markov processes that can be described in terms of state space ( S ), event set ( E ), rate vectors ( R ), and target vectors ( T )—abbreviated as SERT . This general approach can handle systems whose state spaces are quite large, if they have sparse generators.

Suggested Citation

  • Donald Gross & Douglas R. Miller, 1984. "The Randomization Technique as a Modeling Tool and Solution Procedure for Transient Markov Processes," Operations Research, INFORMS, vol. 32(2), pages 343-361, April.
    • Handle: RePEc:inm:oropre:v:32:y:1984:i:2:p:343-361
    DOI: 10.1287/opre.32.2.343
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    Cited by:

    1. Michel Mandjes & Birgit Sollie, 2022. "A Numerical Approach for Evaluating the Time-Dependent Distribution of a Quasi Birth-Death Process," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1693-1715, September.
    2. Chen, Weiwei & Kumcu, Gül Çulhan & Melamed, Benjamin & Baveja, Alok, 2023. "Managing resource allocation for the recruitment stocking problem," Omega, Elsevier, vol. 120(C).
    3. Juan A. Carrasco, 2013. "A New General-Purpose Method for the Computation of the Interval Availability Distribution," INFORMS Journal on Computing, INFORMS, vol. 25(4), pages 774-791, November.
    4. Suñé, Víctor & Carrasco, Juan Antonio, 2017. "Implicit ODE solvers with good local error control for the transient analysis of Markov models," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 96-111.
    5. Congjin Zhou & Guojing Wang & Yinghui Dong & Pin Wang, 2024. "The Valuation at Origination of Mortgages with Full Prepayment and Default Risks," Methodology and Computing in Applied Probability, Springer, vol. 26(2), pages 1-26, June.
    6. Juan A. Carrasco, 2016. "Numerically Stable Methods for the Computation of Exit Rates in Markov Chains," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 307-334, June.
    7. Víctor Suñé, 2017. "Computing the Expected Markov Reward Rates with Stationarity Detection and Relative Error Control," Methodology and Computing in Applied Probability, Springer, vol. 19(2), pages 445-485, June.
    8. Donald Gross & Leonidas C. Kioussis & Douglas R. Miller, 1987. "Transient behavior of large Markovian multiechelon repairable item inventory systems using a truncated state space approach," Naval Research Logistics (NRL), John Wiley & Sons, vol. 34(2), pages 173-198, April.
    9. Ozen, Merve & Krishnamurthy, Ananth, 2020. "Resource allocation models for material convergence," International Journal of Production Economics, Elsevier, vol. 228(C).
    10. Jamal Temsamani & Juan A. Carrasco, 2006. "Transient analysis of Markov models of fault‐tolerant systems with deferred repair using split regenerative randomization," Naval Research Logistics (NRL), John Wiley & Sons, vol. 53(4), pages 318-353, June.
    11. Boquan Cheng & Rogemar Mamon, 2023. "A uniformisation-driven algorithm for inference-related estimation of a phase-type ageing model," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 29(1), pages 142-187, January.
    12. M. Arns & P. Buchholz & A. Panchenko, 2010. "On the Numerical Analysis of Inhomogeneous Continuous-Time Markov Chains," INFORMS Journal on Computing, INFORMS, vol. 22(3), pages 416-432, August.

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