A tandem retrial queueing system with two Markovian flows and reservation of channels

We consider a tandem queueing system with single-server first station and multi-server second station. The input flow at Station 1 is described by the BMAPBMAP (batch Markovian arrival process). Customers from this flow are considered as non-priority customers. Customers of an arriving group, which meet a busy server, go to the orbit of infinite size. From the orbit, they try their luck in exponentially distributed random time. Service times at Station 1 are independent identically distributed random variables having an arbitrary distribution. After service at Station 1 a non-priority customer proceeds to Station 2.The service time by a server of Station 2 is exponentially distributed. Besides customers proceeding from Station 1, an additional MAPMAP flow of priority customers arrives at Station 2 directly, not entering Station 1. If a priority customer meets a free server upon arrival, it starts service immediately. Else, it leaves the system forever. It is assumed that a few servers of Station 2 are reserved to serve the priority customers only.We calculate the stationary distribution and the main performance measures of the system. The problem of optimal design is numerically investigated.