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A Gamma–normal series truncation approximation for computing the Weibull renewal function

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  • Jiang, R.

Abstract

This paper presents a series truncation approximation for computing the Weibull renewal function. In the proposed model, the n-fold convolution of the Weibull Cdf is approximated by a mixture of the n-fold convolutions of Gamma and normal Cdfs. The mixture weight can be optimally determined and fitted into a very accurate linear function of Weibull shape parameter β. Major advantages of the proposed model include:(a)The proposed model and its parameters can be directly written out. Using the proposed model, the renewal density and variance functions can be easily evaluated.(b)The proposed model includes Gamma and normal series truncation models as its special cases. It is easy to be implemented in Excel. The series converges fairly fast.(c)Over the range of β∈(0.87,8.0), the maximum absolute error is smaller than 0.01; and over β∈(3.0,8.0), the maximum absolute error is smaller than 0.0037.(d)The model can be easily extended to non-Weibull case with some additional work.

Suggested Citation

  • Jiang, R., 2008. "A Gamma–normal series truncation approximation for computing the Weibull renewal function," Reliability Engineering and System Safety, Elsevier, vol. 93(4), pages 616-626.
    • Handle: RePEc:eee:reensy:v:93:y:2008:i:4:p:616-626
    DOI: 10.1016/j.ress.2007.03.026
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    References listed on IDEAS

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    1. 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.
    2. J J A Moors & L W G Strijbosch, 2002. "Exact fill rates for (R, s, S) inventory control with gamma distributed demand," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 53(11), pages 1268-1274, November.
    3. Murthy, D. N. P. & Blischke, W. R., 1992. "Product warranty management -- III: A review of mathematical models," European Journal of Operational Research, Elsevier, vol. 63(1), pages 1-34, November.
    4. Michael Tortorella, 2005. "Numerical Solutions of Renewal-Type Integral Equations," INFORMS Journal on Computing, INFORMS, vol. 17(1), pages 66-74, February.
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    Cited by:

    1. Jiang, R., 2009. "An accurate approximate solution of optimal sequential age replacement policy for a finite-time horizon," Reliability Engineering and System Safety, Elsevier, vol. 94(8), pages 1245-1250.
    2. 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.
    3. Brezavšček Alenka, 2013. "A Simple Discrete Approximation for the Renewal Function," Business Systems Research, Sciendo, vol. 4(1), pages 65-75, March.
    4. Jiang, R., 2020. "A novel two-fold sectional approximation of renewal function and its applications," Reliability Engineering and System Safety, Elsevier, vol. 193(C).
    5. Wang, Zhaoqiang & Hu, Changhua & Wang, Wenbin & Zhou, Zhijie & Si, Xiaosheng, 2014. "A case study of remaining storage life prediction using stochastic filtering with the influence of condition monitoring," Reliability Engineering and System Safety, Elsevier, vol. 132(C), pages 186-195.

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