Traffic generated by a semi-Markov additive process

We consider a semi-Markov additive process A(.)-that is, a Markov additive process for which the sojourn times in the various states have general (rather than exponential) distributions. Letting the Levy processes X-i(.), which describe the evolution of A(.) while the background process is in state i, be increasing, it is shown how double transforms of the type integral(infinity)(0) e(-qt) E[e(-alpha A(t)) dt] can be computed. It turns out that these follow, for given nonnegative alpha and q, from a system of linear equations, which has a unique positive solution. Several extensions are considered as well.