Asymptotic analysis by the saddle point method of the Anick–Mitra–Sondhi model

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 et al. 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 tends to infinity. We give detailed asymptotic results for the joint distribution as well as the associated marginal and conditional distributions. In particular, simple conditional limits laws are obtained. These show how the buffer content behaves conditioned on the number of active sources and vice versa. Numerical comparisons show that our asymptotic results are very accurate even for N=20.