An analytical method for the analysis of inhomogeneous continuous Markov processes with piecewise constant transition intensities

The article deals with an inhomogeneous Markov process with finitely many discrete states, continuous time, and piecewise constant transition intensities. For the first time, analytical expressions are presented that describe both the transient and steady-state modes of the random process. To solve this problem, the fundamental matrix of the Kolmogorov system of differential equations is found in closed form in terms of elementary functions. In addition, an inhomogeneous process with periodically varying transition intensities is considered. For this case, the conditions for the existence of a steady-state mode are presented. Results of numerical calculations are provided for processes without jumps, with jumps, and with periodic jumps in the transition intensities.