10. Semi-Markov Processes with General State Spaces and Multi-step Artificial Regeneration

This chapter delves into the intricate world of semi-Markov processes with general state spaces, focusing on those that admit multi-step artificial regeneration. The text begins by describing the thinning procedure for these processes and the relationships between recurrence conditions for a semi-Markov process and its thinned version. It then introduces the main multi-step splitting condition and the corresponding algorithm, which allows for the construction of a regenerative extended semi-Markov process. The chapter provides explicit upper bounds for power and exponential moments of the regenerative time and the duration of the transition period, which are essential for understanding the stability and convergence properties of these processes. Additionally, it presents ergodic theorems with explicit upper bounds for convergence rates, offering a comprehensive framework for analyzing the long-term behavior of semi-Markov processes. The chapter concludes with a discussion on the relationships between stationary distributions of the initial semi-Markov process and its accompanying Markov process with analogous distributions for the corresponding extended semi-Markov process. This detailed exploration provides valuable insights into the behavior of complex systems and offers practical tools for their analysis.