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Numerical Bounds for Semi-Markovian Quantities and Application to Reliability

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  • Sophie Mercier

    (Université Paris-Est)

Abstract

We propose new easily computable bounds for different quantities which are solutions of Markov renewal equations linked to some continuous-time semi-Markov process (SMP). The idea is to construct two new discrete-time SMP which bound the initial SMP in some sense. The solution of a Markov renewal equation linked to the initial SMP is then shown to be bounded by solutions of Markov renewal equations linked to the two discrete time SMP. Also, the bounds are proved to converge. To illustrate the results, numerical bounds are provided for two quantities from the reliability field: mean sojourn times and probability transitions.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:10:y:2008:i:2:d:10.1007_s11009-007-9035-5
    DOI: 10.1007/s11009-007-9035-5
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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. Jacques Janssen & Raimondo Manca, 2001. "Numerical Solution of non-Homogeneous Semi-Markov Processes in Transient Case," Methodology and Computing in Applied Probability, Springer, vol. 3(3), pages 271-293, September.
    3. 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.
    4. Mercier, Sophie, 2007. "Discrete random bounds for general random variables and applications to reliability," European Journal of Operational Research, Elsevier, vol. 177(1), pages 378-405, February.
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    Cited by:

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    3. Giuseppe Di Biase & Jacques Janssen & Raimondo Manca, 2010. "A Non-Homogeneous Continuous Time Semi-Markov Model for the Study of Accumulated Claim Process," Methodology and Computing in Applied Probability, Springer, vol. 12(2), pages 227-235, June.

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