Moments-Based Approximation to the Renewal Function

Abstract

Numerous approximations to the renewal function that have been proposed so far are based on the knowledge of the inter-arrival time distribution function F(t) of the renewal process. This article proposes a simple and easy to evaluate approximation to the renewal function based only on the first three moments of F(t) but not on the functional form of F(t). In this sense, the approximation is nonparametric and yields exact renewal function for several important distributions like K-2 (also called Coxian-2 distribution) and mixture of two exponentials. An iterative procedure to improve the approximation is also proposed and is shown to converge to the renewal function. For the application of the iterative procedure, a new method of fitting a K-2 distribution to match the given moments is developed. Comparisons of the present method with the benchmark approximations available are made. As an application, an optimal replacement problem is used to test the approximation.