In Chap. 10 , we present ergodic theorems with explicit upper bounds for convergence rates for semi-Markov processes with general state spaces, which do not admit a one-step artificial regeneration but do admit a multi-step artificial regeneration. We present the multi-step splitting condition and describe the multi-step splitting algorithm for a semi-Markov process. This algorithm lets us construct a regenerative extended semi-Markov process with a distributional atom for the thinned version of a semi-Markov process. Upper bounds for power and exponential moments of the regenerative time and the duration of the transition period for extended semi-Markov processes resulting from the one-step splitting algorithm are translated to extended semi-Markov processes resulting from the multi-step splitting algorithm. 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 multi-step artificial regeneration are given.