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On Cold Standby Repairable Systems with a Random Change Point in Failure and/or Repair Times

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  • Stathis Chadjiconstantinidis

    (University of Piraeus)

  • Apostolos Bozikas

    (University of Piraeus)

Abstract

This paper is devoted to studying two repairable systems consisting of one active and one standby component for both cases, when the failure times of the units have discrete and continuous distributions. In Model I the failure times of the units do not have the same distribution, but it is assumed that a change occurs in the distribution of the failure times due to an environmental effect and hence this distribution changes after a random number of failures. Similarly, in Model II, it is assumed that the repair times of the units do not have the same distribution, but this changes after a random number of repairs. The two systems under consideration fail if either a damage size upon the failure of the active component is larger than a positive threshold (the repair limit) or the repair time of the failed unit exceeds the lifetime of the active unit, whichever happens first. The lifetimes of these systems are represented as compound random variables. For the continuous (discrete) time case, the Laplace transform (the probability generating function) of the system’s lifetime is obtained as well as its Mean Time to Failure. By assuming that the random change point has a discrete phase-type distribution, several explicit results for reliability characteristics of the systems are obtained. Under particular cases for the distributions of damages, of failure times and of repair times, it is shown that the Laplace transform (probability generating function) of system’s lifetime is rational, and the reliability evaluation of the systems are performed via well-known distributional properties of the matrix-exponential (matrix-geometric) distributions. To illustrate our results, some numerical examples, both for the discrete and the continuous case, are given. Finally, using the proposed model, an actuarial extension for the classical risk model is also discussed.

Suggested Citation

  • Stathis Chadjiconstantinidis & Apostolos Bozikas, 2025. "On Cold Standby Repairable Systems with a Random Change Point in Failure and/or Repair Times," Methodology and Computing in Applied Probability, Springer, vol. 27(2), pages 1-51, June.
    • Handle: RePEc:spr:metcap:v:27:y:2025:i:2:d:10.1007_s11009-025-10177-3
    DOI: 10.1007/s11009-025-10177-3
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    References listed on IDEAS

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    1. Eryilmaz, Serkan & Kan, Cihangir, 2019. "Reliability and optimal replacement policy for an extreme shock model with a change point," Reliability Engineering and System Safety, Elsevier, vol. 190(C), pages 1-1.
    2. Cui, Lirong & Chen, Jianhui & Wu, Bei, 2017. "New interval availability indexes for Markov repairable systems," Reliability Engineering and System Safety, Elsevier, vol. 168(C), pages 12-17.
    3. Kit Nam Francis Leung & Yuan Lin Zhang & Kin Keung Lai, 2010. "A bivariate optimal replacement policy for a cold standby repairable system with repair priority," Naval Research Logistics (NRL), John Wiley & Sons, vol. 57(2), pages 149-158, March.
    4. Yuan Lin Zhang & Guan Jun Wang, 2017. "A geometric process repair model for a cold standby repairable system with imperfect delay repair and priority in use," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(16), pages 8046-8058, August.
    5. Héléne Cossette & Etienne Marceau & Fouad Marri, 2010. "Analysis of ruin measures for the classical compound Poisson risk model with dependence," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2010(3), pages 221-245.
    6. Cihangir Kan & Serkan Eryilmaz, 2021. "Reliability assessment of a discrete time cold standby repairable system," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(3), pages 613-628, October.
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