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On a run-based δ-shock model with two critical levels

Author

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  • Finkelstein, Maxim
  • Lorvand, Hamed
  • Farhadian, Reza

Abstract

In reliability engineering, the δ-shock model is used to study shock-exposed systems that are sensitive to the length of the time distance between consecutive shocks. When the system failure depends on a certain number of consecutive shocks with an inter-arrival time within a critical range, we are dealing with a run-based δ-shock model. In this paper, a new run-based δ-shock model is introduced, under which the system fails when an inter-arrival time is less than a critical threshold δ1 for the first time or k consecutive inter-arrival times fall in the interval (δ1,δ2), for 0≤δ1<δ2. We study the probability behavior of the system’s stopping time as well as the survival of the system under the proposed model. As an illustrative example, we examine the survival of the system when the arrival of shocks follows a Poisson process. Furthermore, an example of applications is provided to illustrate possible application aspects.

Suggested Citation

  • Finkelstein, Maxim & Lorvand, Hamed & Farhadian, Reza, 2026. "On a run-based δ-shock model with two critical levels," Statistics & Probability Letters, Elsevier, vol. 232(C).
    • Handle: RePEc:eee:stapro:v:232:y:2026:i:c:s0167715225002779
    DOI: 10.1016/j.spl.2025.110632
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    References listed on IDEAS

    as
    1. Chadjiconstantinidis, Stathis & Eryilmaz, Serkan, 2024. "On δ-shock model with a change point in intershock time distribution," Statistics & Probability Letters, Elsevier, vol. 208(C).
    2. Goyal, Dheeraj & Finkelstein, Maxim & Hazra, Nil Kamal, 2025. "On repairable systems with time redundancy and operational constraints," Reliability Engineering and System Safety, Elsevier, vol. 255(C).
    3. Mohsen Bohlooli-Zefreh & Majid Asadi & Afshin Parvardeh, 2021. "On the reliability and optimal maintenance of systems under a generalized mixed δ -shock model," Journal of Risk and Reliability, , vol. 235(5), pages 909-922, October.
    4. Maxim Finkelstein & Ji Hwan Cha, 2024. "Discussing some approaches to delta-shock modeling," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(2), pages 245-262, July.
    5. Altan Tuncel & Serkan Eryilmaz, 2018. "System reliability under δ-shock model," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 47(19), pages 4872-4880, October.
    6. Mohammad Hossein Poursaeed, 2021. "Reliability analysis of an extended shock model," Journal of Risk and Reliability, , vol. 235(5), pages 845-852, October.
    7. Hamed Lorvand & Alireza Nematollahi & Mohammad Hossien Poursaeed, 2020. "Life distribution properties of a new δ - shock model," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 49(12), pages 3010-3025, June.
    8. Parvardeh, A. & Balakrishnan, N., 2015. "On mixed δ-shock models," Statistics & Probability Letters, Elsevier, vol. 102(C), pages 51-60.
    9. Eryilmaz, Serkan & Unlu, Kamil Demirberk, 2023. "A new generalized δ-shock model and its application to 1-out-of-(m+1):G cold standby system," Reliability Engineering and System Safety, Elsevier, vol. 234(C).
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