Derivation of Kolmogorov – Chapman type equations with integrated operator

In the work the authors derived equations of Kolmogorov – Chapman type with the integral operator of theoretical and applied importance in the differential equations theory and various applications, for example, of the queueing theory, the population evolution theory, etc. In the work we consider a class of queueing systems with exponential service on one technician device, the input is supplied twice stochastic Poisson flow whose intensity is an spasmodic process at intervals of constancy, distributed according to the exponential law. Models of queueing ystems can have the infinite or the final storage device including zero capacity (queueing system with refusals).