A ray approximation to a PDE that arises in the study of tandem queues

We use singular perturbation methods to analyze a diffusion equation that a rose in studying two tandem queues. Denoting by p(n(1), n(2)) the probabili ty that there are n(1) customers in the first queue and n(2) customers in t he second queue, we obtain the approximation p(n(1),n(2)) similar to epsilo n(2)P(X,Y) = epsilon(2)P(epsilon n(1), epsilon n(2)), where epsilon is a sm all parameter. The diffusion approximation P satisfies an elliptic PDE with a nondiagonal diffusion matrix and boundary conditions that involve both n ormal and tangential derivatives. We analyze the boundary value problem usi ng the ray method of geometrical optics and other singular perturbation tec hniques. This yields the asymptotic behavior of P(X,Y) for X and/or Y large .