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On the Time-Dependent Delta-Shock Model Governed by the Generalized PóLya Process

Author

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  • Dheeraj Goyal

    (Indian Institute of Technology Jodhpur)

  • Nil Kamal Hazra

    (Indian Institute of Technology Jodhpur
    Indian Institute of Technology Jodhpur)

  • Maxim Finkelstein

    (University of the Free State)

Abstract

One of the widely discussed in the literature and relevant in practice shock models is the delta-shock model that is described by the constant time of a system’s recovery after a shock. However, in practice, as time progresses and due to deterioration of a system, this recovery time is gradually increasing. This important phenomenon was not discussed in the literature so far. Therefore, in this paper, we are considering a time-dependent delta-shock model, i.e., the recovery time becomes an increasing function of time. Moreover, we assume that shocks occur according to the generalized Pólya process that contains the homogeneous Poisson process, the non-homogeneous Poisson process and the Pólya process as particular cases. For the defined survival model, we derive the corresponding survival function and the mean lifetime and study the related optimal replacement policy along with some relevant stochastic properties.

Suggested Citation

  • Dheeraj Goyal & Nil Kamal Hazra & Maxim Finkelstein, 2022. "On the Time-Dependent Delta-Shock Model Governed by the Generalized PóLya Process," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1627-1650, September.
    • Handle: RePEc:spr:metcap:v:24:y:2022:i:3:d:10.1007_s11009-021-09880-8
    DOI: 10.1007/s11009-021-09880-8
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    References listed on IDEAS

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    1. Yi Jiang, 2020. "A new δ-shock model for systems subject to multiple failure types and its optimal order-replacement policy," Journal of Risk and Reliability, , vol. 234(1), pages 138-150, February.
    2. Hidetoshi Konno, 2010. "On the Exact Solution of a Generalized Polya Process," Advances in Mathematical Physics, Hindawi, vol. 2010, pages 1-12, November.
    3. Parvardeh, A. & Balakrishnan, N., 2015. "On mixed δ-shock models," Statistics & Probability Letters, Elsevier, vol. 102(C), pages 51-60.
    4. Maxim Finkelstein & Ji Hwan Cha, 2013. "Burn-in for Heterogeneous Populations," Springer Series in Reliability Engineering, in: Stochastic Modeling for Reliability, edition 127, chapter 0, pages 261-312, Springer.
    5. Ji Hwan Cha & Maxim Finkelstein, 2018. "Point Processes for Reliability Analysis," Springer Series in Reliability Engineering, Springer, number 978-3-319-73540-5, September.
    6. Cha, Ji Hwan & Finkelstein, Maxim, 2016. "New shock models based on the generalized Polya process," European Journal of Operational Research, Elsevier, vol. 251(1), pages 135-141.
    7. Gut, Allan & Hüsler, Jürg, 2005. "Realistic variation of shock models," Statistics & Probability Letters, Elsevier, vol. 74(2), pages 187-204, September.
    8. Ming Ma & Zehui Li, 2010. "Life behavior of censored δ-shock model," Indian Journal of Pure and Applied Mathematics, Springer, vol. 41(2), pages 401-420, April.
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    10. A-Hameed, M. S. & Proschan, F., 1973. "Nonstationary shock models," Stochastic Processes and their Applications, Elsevier, vol. 1(4), pages 383-404, October.
    11. 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.
    12. Maxim Finkelstein & Ji Hwan Cha, 2013. "Shocks as Burn-in," Springer Series in Reliability Engineering, in: Stochastic Modeling for Reliability, edition 127, chapter 0, pages 313-361, Springer.
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    Cited by:

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    3. Stathis Chadjiconstantinidis & Altan Tuncel & Serkan Eryilmaz, 2023. "Α new mixed δ-shock model with a change in shock distribution," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(3), pages 491-509, October.
    4. Zhao, Xian & Dong, Bingbing & Wang, Xiaoyue, 2023. "Reliability analysis of a two-dimensional voting system equipped with protective devices considering triggering failures," Reliability Engineering and System Safety, Elsevier, vol. 232(C).
    5. 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).
    6. Chadjiconstantinidis, Stathis & Eryilmaz, Serkan, 2023. "Reliability of a mixed δ-shock model with a random change point in shock magnitude distribution and an optimal replacement policy," Reliability Engineering and System Safety, Elsevier, vol. 232(C).

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