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A general class of shock models with dependent inter-arrival times

Author

Listed:
  • 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
    University of Strathclyde)

Abstract

We introduce and study a general class of shock models with dependent inter-arrival times of shocks that occur according to the homogeneous Poisson generalized gamma process. A lifetime of a system affected by a shock process from this class is represented by the convolution of inter-arrival times of shocks. This class contains many popular shock models, namely the extreme shock model, the generalized extreme shock model, the run shock model, the generalized run shock model, specific mixed shock models, etc. For systems operating under shocks, we derive and discuss the main reliability characteristics (namely the survival function, the failure rate function, the mean residual lifetime function and the mean lifetime) and study relevant stochastic comparisons. Finally, we provide some numerical examples and illustrate our findings by the application that considers an optimal mission duration policy.

Suggested Citation

  • Dheeraj Goyal & Nil Kamal Hazra & Maxim Finkelstein, 2023. "A general class of shock models with dependent inter-arrival times," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(3), pages 1079-1105, September.
    • Handle: RePEc:spr:testjl:v:32:y:2023:i:3:d:10.1007_s11749-023-00867-w
    DOI: 10.1007/s11749-023-00867-w
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    References listed on IDEAS

    as
    1. 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.
    2. Dheeraj Goyal & Nil Kamal Hazra & Maxim Finkelstein, 2022. "On the general $$\delta $$ δ -shock model," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(4), pages 994-1029, December.
    3. Gut, Allan & Hüsler, Jürg, 2005. "Realistic variation of shock models," Statistics & Probability Letters, Elsevier, vol. 74(2), pages 187-204, September.
    4. 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.
    5. Fatih Tank & Serkan Eryilmaz, 2015. "The distributions of sum, minima and maxima of generalized geometric random variables," Statistical Papers, Springer, vol. 56(4), pages 1191-1203, November.
    6. Jozef L. Teugels & Petra Vynckier, 1996. "The structure distribution in a mixed Poisson process," International Journal of Stochastic Analysis, Hindawi, vol. 9, pages 1-8, January.
    7. A-Hameed, M. S. & Proschan, F., 1973. "Nonstationary shock models," Stochastic Processes and their Applications, Elsevier, vol. 1(4), pages 383-404, October.
    8. Omid Shojaee & Majid Asadi & Maxim Finkelstein, 2021. "On Some Properties of $$\alpha $$ α -Mixtures," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(8), pages 1213-1240, November.
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    Cited by:

    1. 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).

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