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 𝒩 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 𝒩 to another subset B where 𝒩 = 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.