Stationary Characteristics of the two-node Tandem Queueing System with Poisson Arrivals and General Renovation

We consider the two-node tandem queueing system with finite capacity queues in both nodes and Poisson input flows. There is one server in each node and the service times are assumed to be i.i.d. random variables, having Erlang distributions with different parameters. General renovation mechanism is assumed to be implemented in each node. It implies that the queue is controlled upon customers' departure instants. Upon quitting the $1^{st}$ node a customer pushes out $i$ customers from its queue with the given probability distribution $\{q^{(1)}_i, 0 \le i \le N_1-1\}$, with $N_1$ being the $1^{st}$ node capacity. Pushed-out customers leave the system and do not have any further effect on it. Upon quitting the $2^{nd}$ node a customer pushes out customers from its queue with another given probability distribution $\{q^{(2)}_i, 0 \le i \le N_2-1\}$, where $N_2$ is the $2^{nd}$ node capacity. The analytic method, based on well-known matrix analytic technique, is being briefly discussed, which allows one to compute the main stationary performance characteristics of the model including loss probabilities.