State space splitting of a finite markov process and some discussions on related counting processes

In this paper, state space of a time-homogeneous Markov process is split into several ordered subspaces. Then, there are three kinds of transitions between states—transitions from a higher-order subspace to a lower-order subspace, transitions within the same subspace, and transitions from a lower-order subspace to a higher-order subspace. Considering time interval omission problem and first passage time considered, we define some related counting processes for the Markov process and discuss their associated probabilities, expectations and generating functions by using Laplace transform. The main results are presented in matrix forms. The relationships among counting processes and their special cases are also discussed briefly. Some simple numerical examples are presented to illustrate the established results. These results will be useful in different problems arising in reliability, economics and social science fields.