On the Positive Recurrence of Finite Regenerative Stochastic Models

We consider a general approach to establish the positive recurrence (stability) of regenerative stochastic systems. The approach is based on the renewal theory and a characterization of the remaining renewal time of the embedded renewal process generated by regeneration. We discuss how this analysis is simplified for some classes of the stochastic systems. The general approach is then illustrated by the stability analysis of a k-out-of-n repairable system containing n unreliable components with exponential lifetimes. Then we extend the stability analysis to the system with non-exponential lifetimes.