A simple approximation for the renewal function with an increasing failure rate

This paper proposes a simple approximation for the renewal function of a failure distribution with an (equivalently) increasing failure rate. The approximation is a linear combination of the cumulative distribution and hazard functions, and the coefficients are functions of the shape parameter of the distribution. The approximation is applied to the Weibull, gamma and lognormal distributions, and it is shown that the approximation is accurate for t up to a certain value of larger than the characteristic life. The approximation is useful for maintenance policy analysis and optimization where the renewal function needs to be evaluated.