Title:

Fluid limits for congestion control in networks

In the Internet, congestion control mechanisms such as TCP are required in order to provide useful services. Propagation delays in the network affect any congestion control scheme, by causing a delay between an action and the controller's reaction, which can lead to undesirable instabilities. This problem is fundamental since, despite the steady increase in speed of networking technologies, the delays imposed by the finite speed of light provide a lower bound on the delays. We should like to understand the dynamical behaviour of the congestion control, for example to determine whether it is stable or not. Working with a model of a network carrying packet traffic, we consider the limit of a sequence of such networks, suitably rescaled, as the bandwidth tends to infinity. We approach this through the technique of the fluid limit, a functional analogue of the weak law of large numbers. First we show how a measure valued process formalism describes the evolution of such a system. Then we prove tightness conditions for a sequence of such systems as the capacity increases. Further, the timescale of the flow dynamics is shown to be separated from the timescale of the queues. The major difficulties in obtaining the fluid limit equations arise from the closed loop feedback nature of the congestion control and the absence of an exogenous driving process. We prove that for a variety of scalable congestion control schemes, with suitable initial conditions, the scaled transmission rate of a user converges weakly to a limit process which satisfies certain differential equations. These equations though subtly different from those of heuristic origin previously analysed, are still amenable to analysis through standard techniques. For example, the methods of control theory allow parameters to be chosen which give a suitable stable operating region.
