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Mathematical formulation and numerical treatment based on transition frequency densities and quadrature methods for non-homogeneous semi-Markov processes

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  • Moura, Márcio das Chagas
  • Droguett, Enrique López

Abstract

Non-homogeneous semi-Markov processes (NHSMP) are important stochastic tools for modeling reliability metrics over time for systems where the future behavior depends on the current and next states as well as on sojourn and process times. The classical method to solve the interval transition probabilities of NHSMPs consists of directly applying any general quadrature method to some non-convolution integral equations. However, this approach has a considerable computational effort. Namely, N2-coupled integral equations with two variables must be solved, where N is the number of states. Therefore, this article proposes a more efficient mathematical formulation and numerical treatment, which are based on transition frequency densities and general quadrature methods respectively, for NHSMPs. The approach consists of only solving N-coupled integral equations with one variable and N straightforward integrations. Two examples in the context of reliability are also presented. The first one addresses a case where a semi-analytical solution is available. Then an example of application concerning pressure–temperature optical monitoring systems for oil wells is discussed. In both cases, the proposed approach is validated via the comparison against the results obtained from the semi-analytical solution (for the first example) as well as from both the classic and the Monte Carlo methods.

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  • Moura, Márcio das Chagas & Droguett, Enrique López, 2009. "Mathematical formulation and numerical treatment based on transition frequency densities and quadrature methods for non-homogeneous semi-Markov processes," Reliability Engineering and System Safety, Elsevier, vol. 94(2), pages 342-349.
    • Handle: RePEc:eee:reensy:v:94:y:2009:i:2:p:342-349
    DOI: 10.1016/j.ress.2008.03.032
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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.
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    5. Ouhbi, Brahim & Limnios, Nikolaos, 2002. "The rate of occurrence of failures for semi-Markov processes and estimation," Statistics & Probability Letters, Elsevier, vol. 59(3), pages 245-255, October.
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    Cited by:

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    3. Vlad Stefan Barbu & Guglielmo D’Amico & Thomas Gkelsinis, 2021. "Sequential Interval Reliability for Discrete-Time Homogeneous Semi-Markov Repairable Systems," Mathematics, MDPI, vol. 9(16), pages 1-18, August.
    4. Moura, Márcio das Chagas & Zio, Enrico & Lins, Isis Didier & Droguett, Enrique, 2011. "Failure and reliability prediction by support vector machines regression of time series data," Reliability Engineering and System Safety, Elsevier, vol. 96(11), pages 1527-1534.
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    7. Moghaddass, Ramin & Zuo, Ming J., 2012. "A parameter estimation method for a condition-monitored device under multi-state deterioration," Reliability Engineering and System Safety, Elsevier, vol. 106(C), pages 94-103.
    8. D׳Amico, Guglielmo & Petroni, Filippo & Prattico, Flavio, 2015. "Reliability measures for indexed semi-Markov chains applied to wind energy production," Reliability Engineering and System Safety, Elsevier, vol. 144(C), pages 170-177.
    9. 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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