Queueing system with finite memory and jump intensity of the arrival process

We consider a single-server queueing system (QS) with finite memory and exponential service with parameter $\mu$. The QS is fed by a doubly stochastic Poisson flow whose intensity $\lambda(t)$ is a jump process. We propose a matrix and a functional-analytical method to study stationary and nonstationary characteristics of the QS; we also prove existence and uniqueness of a stationary regime and stabilization of a nonstationary regime of the QS. Results of numerical analysis are discussed.