A Moment Based Approximation for Expected Number of Renewals for Non-Negligible Repair

This paper focuses on the renewal function which is simply the mathematical expectation of number of renewals in a stochastic process. Renewal functions are important, and they have various applications in many fields. However, obtaining an analytical expression for the renewal function may be very complicated and even impossible. Therefore, researchers focused on developing approximation methods for them. The purpose of this paper is to explore the renewal functions for non-negligible repair for the most common reliability underlying distributions using the first four raw moments of the failure and repair distributions. This article gives the approximate number of cycles, number of failures and the resulting availability for particular distributions assuming Mean Time to Repair is not negligible and that Time to Restore, or repair has a probability density function denoted as r(t). When Mean Time to Repair is not negligible and Time to Restore has a probability density function denoted as r(t), the expected number of failures, cycles and the resulting availability were obtained by taking the Laplace transforms of corresponding renewal functions. An approximation method for obtaining the expected number of cycles, number of failures and availability using raw moments of failure and repair distributions are provided. Results show that the method produces very accurate results for especially large values of time t.