Sample-path based performance sensitivity construction of semi-Markov systems

In this note, we study the sample-path-based construction of performance sensitivities of discrete time semi-Markov chains (SMCs). With a sample path of SMCs, we show that the performance sensitivities of SMCs can be constructed intuitively by the first principle. This results directly extends the potential structure of discrete time Markov chains to semi-Markov models by introducing an extended perturbation realization factor.