Revisiting joint stationary distribution in two finite capacity queues operating in parallel
The paper revisits the problem of the computation of the joint stationary probability distribution p<sub>ij</sub> in a queueing system consisting of two single-server queues, each of capacity N ≥ 3, operating in parallel, and a single Poisson flow. Upon each arrival instant, one customer is put simultaneously into each system. When a customer sees a full system, it is lost. The service times are exponentially distributed with different parameters. Using the approach based on generating functions, the authors obtain a new system of equations of a smaller size than the size of the original system of equilibrium equations (3N - 2 compared to (N + 1)<sup>2</sup>). Given the solution of the new system, the whole joint stationary distribution can be computed recursively. The new system gives some insights into the interdependence of p<sub>ij</sub> and p<sub>nm</sub>. If relations between p<sub>i-1,N</sub> and p<sub>i,N</sub> for i = 3, 5, 7,... are known, then the blocking probability can be computed recursively. Using the known results for the asymptotic behavior of p<sub>ij</sub> as i, j → ∞, the authors illustrate this idea by a simple numerical example.