Global Stability for a Single Link and a Single Flow

In the previous two chapters, we obtained stability conditions for various congestion control and congestion indication algorithms by linearizing around the equilibrium point. However, such an analysis does not guarantee convergence to equilibrium, starting from an arbitrary initial condition. In this chapter, we consider the global stability problem. However, little is known about the global stability of such controllers in general. Therefore, we confine ourselves to the special case of proportionally-fair controllers for a single link accessing a single link and derive conditions for global stability. While the global stability question is open for general topology networks, the analysis in this chapter and extensive simulations of various controllers in many papers in the congestion-control literature suggest that the conditions obtained from a linear analysis may indeed be sufficient conditions for stability even if the initial conditions lie in a large region of attraction around the equilibrium point.