Some Results for Uniformizable Semi-Markov Processes

It is well known that finite Markov chains (M.C.s) in continuous time are uniformizable. That is, a finite M. C. in continuous time can be treated as an M. C. in discrete time with random Poisson transition epochs. In this paper, we see to what extent generalization of the uniformization to a class of semi-Markov Processes (S.M.P.s) is possible. A necessary condition under which S.M.P.s are uniformizable is provided. It is shown that, an S.M.P. with dwell-time distributions depending only on the current state is uniformizable if and only if the distributions are compound geometric distributions having the same base distribution. It is also shown that if the distributions are of generalized phase type then an S.M.P. being uniformizable implies that it is an M.C. in continuous time. Some properties that are shared by a uniformizable S.M.P. and the associated M.C. in discrete time are also discussed.