Analysis of multivariate Markov modulated Poisson processes

A multivariate Markov modulated Poisson process M(t) = [M₁(t),...,M_K(t)] governed by a Markov chain {J(t):t ≥ 0} on N = {0, 1,...,N} is introduced where jumps of M_k(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.