Traffic generated by a semi-Markov additive process

We consider a semi-Markov additive process $A(\cdot)$, i.e., a Markov additive process for which the sojourn times in the various states have general (rather than exponential) distributions. Letting the L\'evy processes $X_i(\cdot)$, which describe the evolution of $A(\cdot)$ while the background process is in state $i$, be increasing, it is shown how double transforms of the type $\int_0^\infty e^{-qt}\,{\mathbb E} [e^{-\alpha A(t)}] {\rm d}t$ can be computed. It turns out that these follow, for given $\alpha\ge 0$ and $q>0$, from a system of linear equations, which has a unique positive solution. Several extensions are considered as well.