A multivariate Markov modulated Poisson process M(t)=[M1(t),...,NK(t)] governed by a Markov chain {J(t):t≥0} on 𝒩={0,1,...,N} is introduced where jumps of Mk(t) occur according to a Poisson process with intensity λ(k,i) whenever the Markov chain J(t) is in state i, 1•k•K, 0•i•N. Of interest to the paper is the time-dependent joint distribution of the multivariate process [M(t),J(t)]. In particular, the Laplace transform generating function is explicitly derived and its probabilistic interpretation is given. Asymptotic expansions of the cross moments and covariance functions of M(t) are also discussed.
