A two-phase queuing system with access control and correlated arrival processes

A two-phase queuing system with access control is studied whose first phase consists of an infinite number of servers, while the second one consists of a finite number of servers. To the first phase, a Markovian input flow of queries arrives. On completion of the first-phase service, the customer either leaves the system or is passed to the second phase. In addition, a Markovian input flow of queries that do not require any first-phase service arrives to the second phase. An algorithm for finding a steady-state distribution of the system’s state probabilities is given, and some problems of the system’s paramaters optimization are considered.