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 pij in a queueing system consisting of two single-server queues, each of capacityN ≥ 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)2). 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 pij and pnm. If relations between pi-1,N and pi,N for i = 3, 5, 7, · · · are known, then the blocking probability can be computed recursively. Using the known results for the asymptotic behavior of pij as i, j → ∞, the authors illustrate this idea by a simple numerical example.