Two approximations of renewal function for any arbitrary lifetime distribution

  • R. JiangEmail author
S.I. : Reliability Modeling with Applications Based on Big Data


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.


Renewal function Approximation Limiting relation Weibull distribution Lognormal distribution 



The research was supported by the National Natural Science Foundation of China (No. 71771029).


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Automotive and Mechanical EngineeringChangsha University of Science and TechnologyChangshaPeople’s Republic of China

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