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 intensity 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.