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On the Existence and Uniqueness of Solution of MRE and Applications

Author

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  • Yunhui Hou

    (Sorbonne Universités)

  • Nikolaos Limnios

    (Sorbonne Universités)

  • Walter Schön

    (Sorbonne Universités)

Abstract

In this paper, we study the existence and uniqueness of the solution for Markov renewal equation (MRE) of a semi-Markov process with countable state space. This method and its proof are based on an iterative scheme. A numerical solution is also given as well as a case study on system reliability assessment.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:19:y:2017:i:4:d:10.1007_s11009-017-9570-7
    DOI: 10.1007/s11009-017-9570-7
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    References listed on IDEAS

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    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. 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.
    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. Nikolaos Limnios, 2014. "Interval Reliability, Corrections and Developments of “Reliability Measures of Semi-Markov Systems with General State Space”," Methodology and Computing in Applied Probability, Springer, vol. 16(3), pages 765-770, September.
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    Cited by:

    1. Vlad Stefan Barbu & Nicolas Vergne, 2019. "Reliability and Survival Analysis for Drifting Markov Models: Modeling and Estimation," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1407-1429, December.
    2. Bei Wu & Brenda Ivette Garcia Maya & Nikolaos Limnios, 2021. "Using Semi-Markov Chains to Solve Semi-Markov Processes," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1419-1431, December.
    3. Guglielmo D’Amico & Raimondo Manca & Filippo Petroni & Dharmaraja Selvamuthu, 2021. "On the Computation of Some Interval Reliability Indicators for Semi-Markov Systems," Mathematics, MDPI, vol. 9(5), pages 1-23, March.

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