Three Approaches in the Study of Recurrent Markovian and Semi-Markovian Processes

We present three classical methods in the study of dynamic and stationary characteristic of processes of Markovian or Semi-Markovian type which possess points of regeneration. Our focus is on the stationary distributions and conditions of its existence and use. The first approach is based on detailed probability analysis of time dependent passages between the states of the process at a given moment. We call this approach Kolmogorov approach. The second approach uses the probability meaning of Laplace-Stieltjes transformation and of the probability generating functions/ Some additional arteficial excrement construction is used to show how derive direct relationships between these functions and how to find them explicitly. The third approach obtains relationships between the stationary characteristics of the process by use of so called “equations of equilibrium”. The input flow in each state must be equal to the respective output flow from that state. In such a way no accumulations should happen on each of that states when process gets its equilibrium. In all the illustrations of the these approaches we analyze a dynamic Marshal-Olkin reliability model with dependent components functioning in parallel. Results on this example are new. © 2020, Springer Nature Switzerland AG.