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Discrete and Continuous Operational Calculus in N-Critical Shocks Reliability Systems with Aging under Delayed Information

Author

Listed:
  • Jewgeni H. Dshalalow

    (Department of Mathematical Sciences, College of Engineering and Science, Florida Institute of Technology, Melbourne, FL 32901, USA)

  • Hend Aljahani

    (Department of Mathematical Sciences, College of Engineering and Science, Florida Institute of Technology, Melbourne, FL 32901, USA)

Abstract

We study a reliability system subject to occasional random shocks of random magnitudes W 0 , W 1 , W 2 , … occurring at times τ 0 , τ 1 , τ 2 , … . Any such shock is harmless or critical dependent on W k ≤ H or W k > H , given a fixed threshold H . It takes a total of N critical shocks to knock the system down. In addition, the system ages in accordance with a monotone increasing continuous function δ , so that when δ T crosses some sustainability threshold D at time T , the system becomes essentially inoperational. However, it can still function for a while undetected. The most common way to do the checking is at one of the moments τ 1 , τ 2 , … when the shocks are registered. Thus, if crossing of D by δ occurs at time T ∈ τ k , τ k + 1 , only at time τ k + 1 , can one identify the system’s failure. The age-related failure is detected with some random delay. The objective is to predict when the system fails, through the N th critical shock or by the observed aging moment, whichever of the two events comes first. We use and embellish tools of discrete and continuous operational calculus ( D -operator and Laplace–Carson transform), combined with first-passage time analysis of random walk processes, to arrive at fully explicit functionals of joint distributions for the observed lifetime of the system and cumulative damage to the system. We discuss various special cases and modifications including the assumption that D is random (and so is T ). A number of examples and numerically drawn figures demonstrate the analytic tractability of the results.

Suggested Citation

  • Jewgeni H. Dshalalow & Hend Aljahani, 2023. "Discrete and Continuous Operational Calculus in N-Critical Shocks Reliability Systems with Aging under Delayed Information," Mathematics, MDPI, vol. 11(16), pages 1-27, August.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:16:p:3568-:d:1219492
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    References listed on IDEAS

    as
    1. Jewgeni H. Dshalalow & Ryan T. White, 2022. "Fluctuation Analysis of a Soft-Extreme Shock Reliability Model," Mathematics, MDPI, vol. 10(18), pages 1-33, September.
    2. Jewgeni H. Dshalalow, 1997. "On the level crossing of multi-dimensional delayed renewal processes," International Journal of Stochastic Analysis, Hindawi, vol. 10, pages 1-7, January.
    3. Coskun Kus & Altan Tuncel & Serkan Eryilmaz, 2022. "Assessment of Shock Models for a Particular Class of Intershock Time Distributions," Methodology and Computing in Applied Probability, Springer, vol. 24(1), pages 213-231, March.
    4. Ali Doostmoradi & Mohammad Reza Akhoond & Mohammad Reza Zadkarami, 2023. "Reliability of a system under a new mixed shock model," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 52(1), pages 156-169, January.
    5. Fermín Mallor & Javier Santos, 2003. "Reliability of systems subject to shocks with a stochastic dependence for the damages," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 12(2), pages 427-444, December.
    6. Mohammad Hossein Poursaeed, 2021. "A run shock-erosion model," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 50(5), pages 1228-1239, March.
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