Semi-Regenerative Processes with Unbounded Rewards

A semi-regenerative process (SRP) is combined with a reward structure such that the accumulated reward during [0, t] is the sum of a functional of the SRP and a functional of the embedded Markov renewal process (MRP). For the expected (discounted) return a Markov renewal equation (MRE) is given. The solution of the MRE is investigated using bounding functions in case the expected (discounted) return between two successive regeneration times of the MRP is not bounded as is the rule in many applications. Explicit limiting results for t → ∞ are given for the expected return in case of discounting and the return rate in case of no discounting.