An improved approximation for the renewal function and its integral with an application in two-echelon inventory management

A simple but effective approximation is proposed to compute the renewal function, M(t), and its integral. The asymptotic approximation of the renewal function and its integral, which are widely used in decision makings involving a renewal process, may not perform well when t is not large enough. To overcome the inaccuracy of the asymptotic approximation, we propose a modified approximation that computes the renewal function and its integral based on the probability distribution function of inter-renewal time when the distribution function is known or based on its mean and standard deviation when the distribution function is unknown. The proposed approximation provides closed form expressions, which are important in decision makings, for the renewal function and its integral for the entire range of t rather than numerically computes them for given values of t. Extensive numerical experiments on commonly used distributions are conducted and demonstrate better performance of the proposed approximations compared to the asymptotic approximation. The new approximations are further applied to a case study of a two-echelon inventory system and result in better solutions compared to the reported results based on the asymptotic approximation.