A novel two-fold sectional approximation of renewal function and its applications

Abstract

The renewal function (RF) has many applications such as reliability analysis, maintenance policy optimization and inventory planning. The RFs of most distribution functions do not have closed-form expressions while such expressions are desired for most of applications. Several models that aim to approximate RF over the entire time range have been developed in the literature, but their accuracy is not high enough. To fill this gap, this paper proposes a two-fold sectional approximation, which is obtained through smoothly connecting two limiting relations. The proposed approximation is simple, applicable for ordinary lifetime distributions (e.g., Weibull and lognormal distributions), and accurate for the distributional parameters in the usual range. The variance of renewals derived from the approximation is fairly accuracy and the integral of the approximation has a closed-form expression for the Weibull distribution. The approximation is useful for solving the optimization problems that involve RF or/and its integral, such as optimization of block replacement policy.