The extremal behaviour over regenerative cycles for Markov additive processes with heavy tails

We consider the reflection of an additive process with negative drift controlled by a Markov chain on a finite state space. We determine the tail behaviour of the distribution of the maximum over a regenerative cycle in the case with subexponential increments. Based on this, the asymptotic distribution of the running maximum is derived. Applications of the results to Markov modulated single server queueing systems are given.