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On a Mixed Transient–Asymptotic Result for the Sequential Interval Reliability for Semi-Markov Chains

Author

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  • Guglielmo D’Amico

    (Department of Economics, University “G. d’Annunzio” of Chieti-Pescara, 65127 Pescara, Italy
    These authors contributed equally to this work.)

  • Thomas Gkelsinis

    (Laboratory of Mathematics Raphaël Salem, University of Rouen-Normandy, UMR 6085, 76801 Saint-Étienne-du-Rouvray, France
    These authors contributed equally to this work.)

Abstract

In this paper, we are concerned with the study of sequential interval reliability, a measure recently introduced in the literature. This measure represents the probability of the system working during a sequence of nonoverlapping time intervals. In the cited work, the authors proposed a recurrent-type formula for computing this indicator in the transient case and investigated the asymptotic behavior as all the time intervals go to infinity. The purpose of the present work is to further explore the asymptotic behavior when only some of the time intervals are allowed to go to infinity while the remaining ones are not. In this way, we provide a unique indicator that is able to describe the process evolution in the transient and asymptotic cases as well. It is important to mention that this is not a straightforward result since, in order to achieve it, we need to develop several mathematical ingredients that generalize the classical renewal and Markov renewal frameworks. A numerical example illustrates our theoretical results.

Suggested Citation

  • Guglielmo D’Amico & Thomas Gkelsinis, 2024. "On a Mixed Transient–Asymptotic Result for the Sequential Interval Reliability for Semi-Markov Chains," Mathematics, MDPI, vol. 12(12), pages 1-18, June.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:12:p:1842-:d:1414351
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    References listed on IDEAS

    as
    1. 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.
    2. 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.
    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.
    4. 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.
    5. P.-C.G. Vassiliou & Andreas C. Georgiou, 2021. "Markov and Semi-Markov Chains, Processes, Systems, and Emerging Related Fields," Mathematics, MDPI, vol. 9(19), pages 1-6, October.
    6. Samis Trevezas & Nikolaos Limnios, 2011. "Exact MLE and asymptotic properties for nonparametric semi-Markov models," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(3), pages 719-739.
    7. D’Amico, Guglielmo & Petroni, Filippo, 2023. "ROCOF of higher order for semi-Markov processes," Applied Mathematics and Computation, Elsevier, vol. 441(C).
    8. Guglielmo D’Amico & Jacques Janssen & Raimondo Manca, 2016. "Downward migration credit risk problem: a non-homogeneous backward semi-Markov reliability approach," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 67(3), pages 393-401, March.
    9. Guglielmo D’Amico & Jacques Janssen & Raimondo Manca, 2010. "Initial and Final Backward and Forward Discrete Time Non-homogeneous Semi-Markov Credit Risk Models," Methodology and Computing in Applied Probability, Springer, vol. 12(2), pages 215-225, June.
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