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 $p_{ij}$ in a queueing system consisting of two single-server queues, each of capacity $N\ge 3$, 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 $p_{ij}$ and $p_{nm}$. If relations between $p_{i-1,N}$ and $p_{i,N}$ for $i=3,5,7,\ldots$ are known, then the blocking probability can be computed recursively. Using the known results for the asymptotic behavior of $p_{ij}$ as $i,j \rightarrow \infty$, the authors illustrate this idea by a simple numerical example.