Markov - Modulated Birth-Death Processes

We consider the birth-death process, the parameters of which are determined according to the external environment. The latter is described by a continuous-time Markov chain. In the literature, such processes are designated as Markov-modulated. They allow making such processes time-dependent. Therefore, stochastic models that are based on them prove to correspond closer to the real processes of the Internet, insurance systems, etc. The article expounds the methodology for computing stationary state probabilities for the considered processes. As an example, we analyze a one-line queueing system functioning in a random environment. The numerical data demonstrate the presence of a dependence that significantly alters the efficiency factors of the system.