Analysis of a counting process associated with a semi-Markov process: number of entries into a subset of state space

Let N(t) be a finite semi-Markov process on .. and let X(t) be the associated age process. Of interest is the counting process M(t) for transitions of the semi-Markov process from a subset G of .. to another subset B where .. = B . G and B . G = .. By studying the trivariate process Y(t) =[N(t), M(t), X(t)] in its state space, new transform results are derived. By taking M(t) as a marginal process of Y(t), the Laplace transform generating function of M(t) is then obtained. Furthermore, this result is recaptured in the context of first-passage times of the semi-Markov process, providing a simple probabilistic interpretation. The asymptotic behavior of the moments of M(t) as t . . is also discussed. In particular, an asymptotic expansion for E[M(t)] and the limit for Var [M(t)]/t as t . . are given explicitly.