Occupation frequencies for irreducible finite semi-Markov processes with reliability applications

A finite irreducible semi-Markov process Y= is considered whose state space S is partitioned into two subsets, A1 and A2. In reliability applications, these two subsets of S usually stand for the working and failed sets of system states respectively. denoting by the number of visits of Y to A 1 during [0, t], closed form expressions are obtained for the Laplace transforms of the family of vector-valued functions h(t; m) = The Nag Fortran Library implementation of a modified version of weeks' numerical Laplace transform inversion scheme is used to assess the practical applicability of the formulae. The examples examined are: (1) the Markov model of a parallel power transmission system; (2) the semi-Markov model of a two-unit repairable system of transformers. From earlier work, explicit expression are available for h(t; m) in terms of matrix exponentials under the Markov assumption, in which case the results based on a MatLab implementation of the exact formulae closely agree with those obtained by numerical Laplace transform inversion. The results for the semi-Markov example are validated by simulation. They show that the new approach is both accurate and computationally less expensive than the alternative.