Semi-Markov Processes with General State Spaces and One-Step Artificial Regeneration

In Chap. 9 , we present ergodic theorems with explicit upper bounds for convergence rates for semi-Markov processes with general state spaces admitting so-called one-step artificial regeneration. This method is based on adding a special indicator component to a semi-Markov process in such a way that the extended process is a regenerative semi-Markov process with a distributional atom. We introduce semi-Markov processes and their accompanying Markov processes admitting one-step artificial regeneration, describe the semi-Markov variant of the method of artificial regeneration, and construct using this method the corresponding extended semi-Markov process and its accompanying Markov process. Upper bounds for power and exponential moments of the regenerative time and the duration of the transition period for semi-Markov processes with distributional atoms are translated to extended semi-Markov processes resulting from the splitting algorithm applied to semi-Markov processes admitting one-step artificial regeneration. Finally, representations for the corresponding stationary distributions of extended semi-Markov processes and initial semi-Markov processes and ergodic theorems with explicit power and exponential upper bounds for convergence rates for semi-Markov processes and their accompanying Markov processes admitting one-step artificial regeneration are given.