A Markov Modulated Poisson Process (MMPP) M(t) defined on a Markov chain J(t) is a pure jump process where jumps of M(t) occur according to a Poisson process with intenstiy λi whenever the Markov chain J(t) is in state i. M(t) is called strongly renewal (SR) if M(t) is a renewal process for an arbitrary initial probability vector of J(t) with full support on P={i:λi>0}. M(t) is called weakly renewal (WR) if there exists an initial probability vector of J(t) such that the resulting MMPP is a renewal process. The purpose of this paper is to develop general characterization theorems for the class SR and some sufficiency theorems for the class WR in terms of the first passage times of the bivariate Markov chain [J(t),M(t)]. Relevance to the lumpability of J(t) is also studied.
