A Simple Approximation to the Renewal Function

This article presents a simple, easy to understand approximation to the renewal function; the approximation is easy to implement on a personal computer. The key idea is that, for small values of time, the renewal function is almost equal to the Cdf of the inter-renewal time, whereas for larger values of time an asymptotic expansion — depending upon only the first and second moment of the inter-renewal time — can be used. The relative error is typically smaller than a few percent for Weibull inter-renewal times. The simple approximation method works very well with one term if not too much accuracy is required (eg, in the block replacement problem) or if the inter-renewal (failure) distribution is not exactly known (eg, only the first two moments are known). Although the accuracy of the simple approximation can be improved by increasing the number of terms, we do not advocate this strategy, since then speed and simplicity are lost. If high accuracy is required it is better to use another approximating method (eg, power series expansion or cubic splines method).