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On the general $$\delta $$ δ -shock model

Author

Listed:
  • Dheeraj Goyal

    (Indian Institute of Technology Jodhpur)

  • Nil Kamal Hazra

    (Indian Institute of Technology Jodhpur)

  • Maxim Finkelstein

    (University of the Free State
    University of Strathclyde)

Abstract

The $$\delta $$ δ -shock model is one of the basic shock models which has a wide range of applications in reliability, finance and related fields. In existing literature, it is assumed that the recovery time of a system from the damage induced by a shock is constant as well as the shocks magnitude. However, as technical systems gradually deteriorate with time, it takes more time to recover from this damage, whereas the larger magnitude of a shock also results in the same effect. Therefore, in this paper, we introduce a general $$\delta $$ δ -shock model when the recovery time depends on both the arrival times and the magnitudes of shocks. Moreover, we also consider a more general and flexible shock process, namely, the Poisson generalized gamma process. It includes the homogeneous Poisson process, the non-homogeneous Poisson process, the Pólya process and the generalized Pólya process as the particular cases. For the defined survival model, we derive the relationships for the survival function and the mean lifetime and study some relevant stochastic properties. As an application, an example of the corresponding optimal replacement policy is discussed.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:testjl:v:31:y:2022:i:4:d:10.1007_s11749-022-00810-5
    DOI: 10.1007/s11749-022-00810-5
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    References listed on IDEAS

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    1. 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.
    2. Ranjkesh, Somayeh Hamed & Hamadani, Ali Zeinal & Mahmoodi, Safieh, 2019. "A new cumulative shock model with damage and inter-arrival time dependency," Reliability Engineering and System Safety, Elsevier, vol. 192(C).
    3. A-Hameed, M. S. & Proschan, F., 1973. "Nonstationary shock models," Stochastic Processes and their Applications, Elsevier, vol. 1(4), pages 383-404, October.
    4. Maxim Finkelstein, 2008. "Failure Rate Modelling for Reliability and Risk," Springer Series in Reliability Engineering, Springer, number 978-1-84800-986-8, June.
    5. Ji Hwan Cha & Maxim Finkelstein, 2018. "Point Processes for Reliability Analysis," Springer Series in Reliability Engineering, Springer, number 978-3-319-73540-5, June.
    6. 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.
    7. 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.
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