Sojourn times with finite time‐horizon in finite semi‐markov processes

Let Y = {Yt:t ≥ 0} be a semi‐Markov process with finite state space S. Assume that Y is either irreducible and S is then partitioned into two classes A and B, or, that Y is absorbing and S is partitioned into A, B and C, where C is the set of all absorbing states of Y. Denote by TA, m(t) the mth sojourn time of Y in A during [0, t]. TA, m(t) is thus defined as the duration in [0, t] of the mth visit of Y to A if A is visited by Y during [0, t] at least m times; TA, m(t) = 0 otherwise. We derive a recurrence relation for the vectors of double Laplace transforms gm**(T1,T2) = {gm**(T1, T2;S):sSC}, m = 1,2,… which are defined by \documentclass{article}\pagestyle{empty}\begin{document}$$ g_m ^{**} (\tau _1,\tau _2;s) = \;\int_0^\infty {\int_0^\infty {\exp (- \tau _1 t_1}} - \;\tau _2 t_2)p(T_{A, m} (t_1)\;\tau _2 \;|\;Y_0 \; = \;s)\;{\rm d}t_1 \;{\rm d}t_2 $$\end{document} with T1, T2, Re(T1), Re(T2) > 0. This result is then applied to alternating renewal processes. Symbolic Laplace transform inversion with MAPLE is used to obtain the first two moments of TA, m(t). The assumed holding time distributions are exponential and Erlang respectively. This paper is a continuation of some of the author's recent work on the distribution theory of sojourn times in a subset of the finite state space of a (semi‐)Markov process where the time horizon t = + ∞. The practical importance of considering a finite time horizon for semi‐Markov reliability models has been discussed recently by Jack (1991), Jack and Dagpunar (1992), and Christer and Jack (1991).