Two approximations of renewal function for any arbitrary lifetime distribution

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 t...

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Bibliographic Details
Published in:Annals of operations research 2022-04, Vol.311 (1), p.151-165
Main Author: Jiang, R.
Format: Article
Language:English
Subjects:
Accuracy
Approximation
Approximation theory
Business and Management
Combinatorics
Distribution functions
Inventory control
Mathematical optimization
Methods
Operations research
Operations Research/Decision Theory
Optimization
Reliability (Engineering)
S.I. : Reliability Modeling with Applications Based on Big Data
Theory of Computation
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Summary: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.
ISSN:0254-5330
1572-9338
DOI:10.1007/s10479-019-03356-2