We use singular perturbation methods to analyze a diffusion equation that arose in studying two tandem queues. Denoting by p(n1, n2) the probability that there are n1 customers in the first queue and n2 customers in the second queue, we obtain the approximation p(n1, n2)∼ɛ2P(X, Y)=ɛ2P(ɛn1, ɛn2), where ɛ is a small parameter. The diffusion approximation P satisfies an elliptic PDE with a nondiagonal diffusion matrix and boundary conditions that involve both normal and tangential derivatives. We analyze the boundary value problem using the ray method of geometrical optics and other singular perturbation techniques. This yields the asymptotic behavior of P(X, Y) for X and/or Y large.