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On Numerical Solution of the Markov Renewal Equation: Tight Upper and Lower Kernel Bounds

Author

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  • D. A. Elkins

    (General Motors R&D Center)

  • M. A. Wortman

    (Texas A&M University, College Station)

Abstract

We develop tight bounds and a fast parallel algorithm to compute the Markov renewal kernel. Knowledge of the kernel allows us to solve Markov renewal equations numerically to study non-steady state behavior in a finite state Markov renewal process. Computational error and numerical stability for computing the bounds in parallel are discussed using well-known results from numerical analysis. We use our algorithm and computed bounds to study the expected number of departures as a function of time for a two node overflow queueing network.

Suggested Citation

  • D. A. Elkins & M. A. Wortman, 2001. "On Numerical Solution of the Markov Renewal Equation: Tight Upper and Lower Kernel Bounds," Methodology and Computing in Applied Probability, Springer, vol. 3(3), pages 239-253, September.
    • Handle: RePEc:spr:metcap:v:3:y:2001:i:3:d:10.1023_a:1013767704349
    DOI: 10.1023/A:1013767704349
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    References listed on IDEAS

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    1. Erhan Çinlar, 1975. "Exceptional Paper--Markov Renewal Theory: A Survey," Management Science, INFORMS, vol. 21(7), pages 727-752, March.
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    Cited by:

    1. Gianfranco Corradi & Jacques Janssen & Raimondo Manca, 2004. "Numerical Treatment of Homogeneous Semi-Markov Processes in Transient Case–a Straightforward Approach," Methodology and Computing in Applied Probability, Springer, vol. 6(2), pages 233-246, June.
    2. Yunhui Hou & Nikolaos Limnios & Walter Schön, 2017. "On the Existence and Uniqueness of Solution of MRE and Applications," Methodology and Computing in Applied Probability, Springer, vol. 19(4), pages 1241-1250, December.
    3. Sophie Mercier, 2008. "Numerical Bounds for Semi-Markovian Quantities and Application to Reliability," Methodology and Computing in Applied Probability, Springer, vol. 10(2), pages 179-198, June.
    4. Inma T Castro & Sophie Mercier, 2016. "Performance measures for a deteriorating system subject to imperfect maintenance and delayed repairs," Journal of Risk and Reliability, , vol. 230(4), pages 364-377, August.

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