A Simple Discrete Approximation for the Renewal Function

Background: The renewal function is widely useful in the areas of reliability, maintenance and spare components inventory planning. Its calculation relies on the type of the probability density function of component failure times which can be, regarding the region of the component lifetime, modelled either by the exponential or by one of the peak-shaped density functions. For most peak-shaped distribution families the closed form of the renewal function is not available. Many approximate solutions can be found in the literature, but calculations are often tedious. Simple formulas are usually obtained for a limited range of functions only. Objectives: We propose a new approach for evaluation of the renewal function by the use of a simple discrete approximation method, applicable to any probability density function. Methods/Approach: The approximation is based on the well known renewal equation. Results: The usefulness is proved through some numerical results using the normal, lognormal, Weibull and gamma density functions. The accuracy is analyzed using the normal density function. Conclusions: The approximation proposed enables simple and fairly accurate calculation of the renewal function irrespective of the type of the probability density function. It is especially applicable to the peak-shaped density functions when the analytical solution hardly ever exists.