Operator analysisof Kolmogorov-Chapman type equations with differential operator

We study the stationary mode of a queuing system (QS) with an infinite storage device, one service device, and exponential service with intensity $µ$. The input of QS receives a doubly stochastic Poisson flow, the intensity $\lambda(t)$ of which varies over the interval [$\alpha, \beta$] and is a diffusion process with the zero drift coefficient $\alpha = 0$, diffusion coefficient $b > 0$ and elastic boundaries $\alpha, \beta$. In this paper, two models of a stationary queuing system QS are constructed in the form of systems of differential equations with respect to the characteristics of the applications number. We obtain the necessary and su cient conditions for the existence and uniqueness of the stationary mode of the QS, the solution of the system of differential equations with respect to the characteristics of the number of applications of the stationary QS, and the non-negativity of the characteristics of the QS. The probability of downtime is estimated. Using the methods of operator analysis, a stationary generating function of the solution in the form of a convergent power series is found.