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 Lévy processes Xi(·), 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 (formule) can be computed. It turns out that these follow, for given nonnegative a and q, from a system of linear equations, which has a unique positive solution. Several extensions are considered as well.