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On the point process with finite memory and its application to optimal age replacement

Author

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  • Amy Langston
  • Maxim Finkelstein
  • Ji Hwan Cha

Abstract

There has been extensive study of various repair models in the literature, mostly under the assumption that these repairs are minimal or imperfect/better than minimal. Although this is often a realistic assumption, it may not be sufficient to model instances where the repair is worse than minimal. The generalized Polya process (GPP) that has been used to describe this type of repair takes into account all previous events/repairs, which is not often the case in practice. Therefore, in this paper, we define a new process with finite memory that starts as the GPP but, after a certain number of events or elapsed time, becomes the non-homogeneous Poisson process of repairs (minimal repairs). The corresponding age replacement policy is defined and the optimal solutions that minimize the long-run expected cost rate are analyzed. The detailed numerical examples illustrate our findings.

Suggested Citation

  • Amy Langston & Maxim Finkelstein & Ji Hwan Cha, 2024. "On the point process with finite memory and its application to optimal age replacement," Journal of Risk and Reliability, , vol. 238(6), pages 1184-1194, December.
    • Handle: RePEc:sae:risrel:v:238:y:2024:i:6:p:1184-1194
    DOI: 10.1177/1748006X231205903
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    References listed on IDEAS

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    1. Tanwar, Monika & Rai, Rajiv N. & Bolia, Nomesh, 2014. "Imperfect repair modeling using Kijima type generalized renewal process," Reliability Engineering and System Safety, Elsevier, vol. 124(C), pages 24-31.
    2. Liu, Xingheng & Finkelstein, Maxim & Vatn, Jørn & Dijoux, Yann, 2020. "Steady-state imperfect repair models," European Journal of Operational Research, Elsevier, vol. 286(2), pages 538-546.
    3. Toledo, Maria Luíza Guerra de & Freitas, Marta A. & Colosimo, Enrico A. & Gilardoni, Gustavo L., 2015. "ARA and ARI imperfect repair models: Estimation, goodness-of-fit and reliability prediction," Reliability Engineering and System Safety, Elsevier, vol. 140(C), pages 107-115.
    4. Badía, F.G. & Berrade, M.D. & Cha, Ji Hwan & Lee, Hyunju, 2018. "Optimal replacement policy under a general failure and repair model: Minimal versus worse than old repair," Reliability Engineering and System Safety, Elsevier, vol. 180(C), pages 362-372.
    5. Hongzhou Wang & Hoang Pham, 2006. "Reliability and Optimal Maintenance," Springer Series in Reliability Engineering, Springer, number 978-1-84628-325-3, December.
    6. Maxim Finkelstein, 2008. "Failure Rate Modelling for Reliability and Risk," Springer Series in Reliability Engineering, Springer, number 978-1-84800-986-8, December.
    7. Toshio Nakagawa, 2005. "Maintenance Theory of Reliability," Springer Series in Reliability Engineering, Springer, number 978-1-84628-221-8, December.
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    1. Amy Langston & Maxim Finkelstein & Ji Hwan Cha, 2025. "A bathtub pattern for repairable systems: From better-than-minimal to minimal and worse-than-minimal repairs," Journal of Risk and Reliability, , vol. 239(5), pages 1000-1015, October.

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