Economic Rewards in Non-homogeneous Semi-Markov Systems

In this paper, a general reward model for a non homogeneous semi-Markov system will be provided. Both discrete and continuous time models will be studied considering reward to be a random variable of the economic type associated with the state occupancies and transitions. Basic equations will be exhibited and the main formulas for the expected reward that the system will generate in some interval of time (with and without discounting) will be determined for the discrete and continuous cases and a method is provided in order to find it in closed analytic form in relation with the basic parameters of the system. Also, results will be obtained for the expected reward per time period for the steady state.