| We consider a fluid queue where the input process consists of N identical sources that turn on and off at exponential waiting times. The server works at the constant rate c and an on source generates fluid at unit rate. This model was first formulated and analyzed by Anick, Mitra and Sondhi. We obtain an alternate representation of the joint steady state distribution of the buffer content and the number of on sources. This is given as a contour integral that we then analyze in the limit N □□. We also use singular perturbation methods to analyze the problem, with appropriate scalings of the two state variables. In particular, the ray method and asymptotic matching are used. We give detailed asymptotic results for the joint distribution. Numerical comparisons show that our asymptotic results are very accurate even for N = 20.; Next we analyze a second order, linear, elliptic PDE with mixed boundary conditions. This problem arises as limiting case of a Markov-modulated M/G/1 queuing model. We employ singular perturbation methods to study the problem for small values of a parameter, and then we use the ray method to solve the PDE in the limit where convection dominates diffusion. We show that there are interior and boundary caustics, a cusp point where two caustics meet, as well as internal, boundary and corner layers. Our analysis leads to approximate formulas for the queue length (or buffer content) distribution. |
