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Two approximations of renewal function for any arbitrary lifetime distribution

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  • R. Jiang

    (Changsha University of Science and Technology)

Abstract

The renewal functions (RFs) of most distribution functions do not have closed-form expressions while such expressions are desired for the optimization problems involved RF. Many efforts have been made to develop approximations of RF. However, it seems that no RF approximation is accurate enough in the entire time range. In this paper, we propose two RF approximations. The first approximation is obtained through smoothly connecting two limiting relations and fairly accurate in the entire time range. The second approximation has the same function form as the first part of the first approximation but the model parameter is determined in a different way so as to achieve higher accuracy for small to moderate time range. The expressions of the proposed approximations are simple and applicable for any arbitrary lifetime distribution. Their accuracy is analyzed and, the appropriateness and usefulness are illustrated by a numerical example.

Suggested Citation

  • R. Jiang, 2022. "Two approximations of renewal function for any arbitrary lifetime distribution," Annals of Operations Research, Springer, vol. 311(1), pages 151-165, April.
    • Handle: RePEc:spr:annopr:v:311:y:2022:i:1:d:10.1007_s10479-019-03356-2
    DOI: 10.1007/s10479-019-03356-2
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    References listed on IDEAS

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    1. Renyan Jiang, 2015. "Introduction to Quality and Reliability Engineering," Springer Series in Reliability Engineering, Springer, edition 127, number 978-3-662-47215-6, March.
    2. Constantine, A. G. & Robinson, N. I., 1997. "The Weibull renewal function for moderate to large arguments," Computational Statistics & Data Analysis, Elsevier, vol. 24(1), pages 9-27, March.
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    4. Jiang, R., 2018. "Performance evaluation of seven optimization models of age replacement policy," Reliability Engineering and System Safety, Elsevier, vol. 180(C), pages 302-311.
    5. Parsa, Hossein & Jin, Mingzhou, 2013. "An improved approximation for the renewal function and its integral with an application in two-echelon inventory management," International Journal of Production Economics, Elsevier, vol. 146(1), pages 142-152.
    6. Jiang, R., 2010. "A simple approximation for the renewal function with an increasing failure rate," Reliability Engineering and System Safety, Elsevier, vol. 95(9), pages 963-969.
    7. Behzad Ghodrati, 2011. "Efficient Product Support—Optimum and Realistic Spare Parts Forecasting," Springer Series in Reliability Engineering, in: Lotfi Tadj & M.-Salah Ouali & Soumaya Yacout & Daoud Ait-Kadi (ed.), Replacement Models with Minimal Repair, pages 225-269, Springer.
    8. Daley, D.J. & Vesilo, Rein, 1997. "Long range dependence of point processes, with queueing examples," Stochastic Processes and their Applications, Elsevier, vol. 70(2), pages 265-282, October.
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    Cited by:

    1. Asadi, Majid, 2023. "On a parametric model for the mean number of system repairs with applications," Reliability Engineering and System Safety, Elsevier, vol. 234(C).

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