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On terminating Poisson processes in some shock models

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  • Finkelstein, Maxim
  • Marais, Francois

Abstract

A system subject to a point process of shocks is considered. Shocks occur in accordance with the homogeneous Poisson process. Different criteria of system failure (termination) are discussed and the corresponding probabilities of failure (accident)-free performance are derived. The described analytical approach is based on deriving integral equations for each setting and solving these equations through the Laplace transform. Some approximations are analyzed and further generalizations and applications are discussed.

Suggested Citation

  • Finkelstein, Maxim & Marais, Francois, 2010. "On terminating Poisson processes in some shock models," Reliability Engineering and System Safety, Elsevier, vol. 95(8), pages 874-879.
    • Handle: RePEc:eee:reensy:v:95:y:2010:i:8:p:874-879
    DOI: 10.1016/j.ress.2010.03.003
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    References listed on IDEAS

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    1. Montoro-Cazorla, Delia & Pérez-Ocón, Rafael & del Carmen Segovia, Maria, 2009. "Replacement policy in a system under shocks following a Markovian arrival process," Reliability Engineering and System Safety, Elsevier, vol. 94(2), pages 497-502.
    2. Chen, Jinyuan & Li, Zehui, 2008. "An extended extreme shock maintenance model for a deteriorating system," Reliability Engineering and System Safety, Elsevier, vol. 93(8), pages 1123-1129.
    3. van der Weide, J.A.M. & Pandey, M.D. & van Noortwijk, J.M., 2010. "Discounted cost model for condition-based maintenance optimization," Reliability Engineering and System Safety, Elsevier, vol. 95(3), pages 236-246.
    4. Tang, Ya-yong & Lam, Yeh, 2006. "A [delta]-shock maintenance model for a deteriorating system," European Journal of Operational Research, Elsevier, vol. 168(2), pages 541-556, January.
    5. Gut, Allan & Hüsler, Jürg, 2005. "Realistic variation of shock models," Statistics & Probability Letters, Elsevier, vol. 74(2), pages 187-204, September.
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    Cited by:

    1. Ye, Kewei & Wang, Han & Ma, Xiaobing, 2023. "A generalized dynamic stress-strength interference model under δ-failure criterion for self-healing protective structure," Reliability Engineering and System Safety, Elsevier, vol. 229(C).
    2. Cha, Ji Hwan & Finkelstein, Maxim, 2019. "Stochastic modeling for systems with delayed failures," Reliability Engineering and System Safety, Elsevier, vol. 188(C), pages 118-124.
    3. Rodríguez, Joanna & Lillo, Rosa E. & Ramírez-Cobo, Pepa, 2015. "Failure modeling of an electrical N-component framework by the non-stationary Markovian arrival process," Reliability Engineering and System Safety, Elsevier, vol. 134(C), pages 126-133.
    4. Ji Cha & Maxim Finkelstein & Francois Marais, 2014. "Survival of systems with protection subject to two types of external attacks," Annals of Operations Research, Springer, vol. 212(1), pages 79-91, January.
    5. Maxim Finkelstein & Gregory Levitin, 2018. "Optimal Mission Duration for Partially Repairable Systems Operating in a Random Environment," Methodology and Computing in Applied Probability, Springer, vol. 20(2), pages 505-516, June.
    6. Xufeng Zhao & Toshio Nakagawa, 2016. "Over-time and over-level replacement policies with random working cycles," Annals of Operations Research, Springer, vol. 244(1), pages 103-116, September.
    7. Gregory Levitin & Maxim Finkelstein, 2017. "A new stress–strength model for systems subject to stochastic shocks," Journal of Risk and Reliability, , vol. 231(2), pages 172-179, April.
    8. Zhao, Xufeng & Qian, Cunhua & Nakagawa, Toshio, 2013. "Optimal policies for cumulative damage models with maintenance last and first," Reliability Engineering and System Safety, Elsevier, vol. 110(C), pages 50-59.
    9. Maxim Finkelstein & Gregory Levitin, 2020. "On missions’ quality of performance for systems with partially or completely observable degradation," Journal of Risk and Reliability, , vol. 234(5), pages 676-685, October.
    10. Maxim Finkelstein & Gregory Levitin, 2018. "Optimal mission duration for systems subject to shocks and internal failures," Journal of Risk and Reliability, , vol. 232(1), pages 82-91, February.
    11. Cha, Ji Hwan & Finkelstein, Maxim, 2016. "On some properties of shock processes in a ‘natural’ scale," Reliability Engineering and System Safety, Elsevier, vol. 145(C), pages 104-110.

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