Renewal Function under Prefailure Lives Distributed as a Mixture of n Exponential Distributions: Obtaining the Parameters of Mixtures Using the Method of Moments

This paper deals with the problems of the theory of reliability of technical systems for the case when the prefailure lives of the restored (replaced) elements are distributed as a mixture of distributions. For a simple renewal process, when the prefailure lives are distributed as a mixture of exponential distributions, a method for obtaining the renewal function (average number of failures in the range from 0 to t) is presented. For the case n = 3, its explicit formula is written out. The case of n = 2 was considered in [1]. For mixtures of two and three distributions, such as exponential, Erlang, Rayleigh, and Maxwell distributions, an algorithm to obtain point estimates of the unknown parameters of the mixture in an explicit form is presented. In this work, the studies begun in [1, 2] are continued.