Some renewal-theoretic investigations in the theory of sojourn times in finite semi-Markov processes

Abstract

In this note, an irreducible semi-Markov process is considered whose finite state space is partitioned into two non-empty sets A and B. Let MB(t) stand for the number of visits of Y to B during the time interval [0, t], t > 0. A renewal argument is used to derive closed-form expressions for the Laplace transform (with respect to t) of a certain family of functions in terms of which the moments of MB(t) are easily expressible. The theory is applied to a small reliability model in conjunction with a Tauberian argument to evaluate the behaviour of the first two moments of MB(t) as t →∞.