Boundedness and Global Stability of a Nonlinear Congestion Control With Delays

摘要

This note concerns the global dynamics of a “dual” nonlinear congestion control with time-delays. It has been shown that such control laws are able to maintain local dynamic stability around their unique equilibrium point for networks of arbitrary delay, capacity and topology. We focus here on the case of a single bottleneck network, shared by traffic sources with heterogeneous delays. We first prove that, for any parameter values, an a priori bound can be given on the system state after a certain amount of time. Next, we prove global exponential stability of the delay-differential equations under constraints on its gain parameter and the delays; these are more restrictive than conditions for linear stability, but only mildly so in the homogeneous delay case. The proof is based on a detailed investigation and bounding of the system trajectories.