Renewal characterization of Markov modulated Poisson processes

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 λ i whenever the Markov chain J ( t ) is in state i. M ( t ) is called strongly renewal ( S R ) if M ( t ) is a renewal process for an arbitrary initial probability vector of J ( t ) with full support on P = { i : λ i > 0 } . M ( t ) is called weakly renewal ( W R ) 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 S R and some sufficiency theorems for the class W R 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.