Strong Law of Large Numbers for Nonlinear Semi-Markov Reward Processes

We obtain a Strong Law of Large Numbers (SLLN) for the reward process {Z(t), t≥0}, the cumulative reward gained by operating a Semi-Markov system during the time interval [0, t]. The important and striking point in this study is leaving the usual assumption that rewards for each state are of constant rates. In most of applications this frequently used assumption is not realistic, therefore we deal with reward functions of general forms. The SLLN obtained is in the sense that Limt→∞Z(t)/t = α, a.s. for some real value α. Mild conditions for this SLLN are existence of sojourn times means and integrability of reward functions with respect to sojourn time distributions. As it has been shown, the parameter α coincides with the shift parameter in asymptotic representation of E[Z(t)], t→∞.