  We analyze a second order, linear, elliptic PDE with mixed boundary
conditions. This problem arose as a limiting case of a Markov-modulated
queueing model for data handling switches in communications networks. We use
singular perturbation methods to analyze the problem. In particular we use the
ray method to solve the PDE in the limit where convection dominates diffusion.
We show that there are both interior and boundary caustics, as well as a cusp
point where two caustics meet, an internal layer, boundary layers and a corner
layer. Our analysis leads to approximate formulas for the queue length (or
buffer content) distribution at the switch.
