The joint distribution of sojourn times in finite semi-Markov processes

Abstract

Let Y={ Y t : t⩾0} be a semi-Markov process whose state space S is finite. Assume that Y is either irreducible and S is then partitioned into two classes A 1, A 2, or that Y is absorbing and S is partitioned into A 1, A 2, A 3, where A 3 is the set of all absorbing states of Y. Denoting by T A i,j the jth sojourn of Y in A i , i=1, 2, we determine the Laplace transform of the joint distribution of T={T A i,j :i=1, 2; j=1,…, m} . This result is derived from a recurrence relation for the Laplace transform of T . The proof of the recurrence relation itself is based on what could be called a ‘generalized renewal argument’. Some known results on sojourn times in Markov and semi-Markov processes are also rederived using our main theorem. A procedure for obtaining the Laplace transform of the vector of sojourn times in special cases if S is partitioned into more than two non-absorbing classes is also considered.