Analysis of a finite queue with Markov flow and arbitrary servicing that depends on the number of system customers

Consideration is given to the queuing system with one server, finite-length buffer, and Markov flow of customers. The servicing limes have arbitrary distributions that depend on the number of customers in the system al the instant when servicing of the current customer begins. For the stationary distribution of slates of the Markov linear-wise process that describes the system behavior, recurrent matrix formulas have been obtained. For the stationary state probabilities of the system that is considered at the instants of customer arrival and servicing completion, expressions have been obtained as well.