Exponential approximation to Weibull renewal with decreasing failure rate

A simple yet accurate approximation method is proposed to solve the Weibull renewal function (RF) when the shape parameter β is between 0 and 1. The idea is to find a simple mixture model, constructed by two exponential functions, to approximate the Weibull distribution function. Parameters of the mixture model can be determined using minimum mean square errors (MMSEs) or moment match. Now finding the Weibull RF becomes the problem to find the RF for the mixture model. The mixture model allows the Laplace transform that significantly simplifies the computation of the RF. Numerical data show that the new analytical RF model is mathematically accurate and computationally convenient to approximate the actual Weibull RF for 0<β<1. The primary contribution of this paper is that the Weibull RF with decreasing failure rate can be directly approximated by the combination of two exponential functions.