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Fluid Limits
A fluid limit scaling is a particular important way of scaling a Markov process. It is related to the first order behavior of the process, roughly speaking, it amounts to a functional law of large numbers for the system considered.
It is in general quite difficult to have a satisfactory description of an ergodic Markov process describing a stochastic network. When the dimension of the state space d is greater than 1, the geometry complicates a lot any investigation : Analytical tools such as Wiener-Hopf techniques for dimension~1 cannot be easily generalized to higher dimensions. It is possible nevertheless to get some insight on the behavior of these processes through some limit theorems. The limiting procedure investigated consists in speeding up time and scaling appropriately the process itself with some parameter. The behavior of such rescaled stochastic processes is analyzed when the scaling parameter goes to infinity. In the limit, one gets a sort of caricature of the initial stochastic process which is defined as a fluid limit.
A fluid limit keeps the main characteristics of the initial stochastic process while some stochastic fluctuations of second order vanish with this procedure. In “good cases”, a fluid limit is a deterministic function, solution of some ordinary differential equation. As it can be expected, the general situation is somewhat more complicated. These ideas of rescaling stochastic processes have emerged recently in the analysis of stochastic networks, to study their ergodicity properties in particular. See Rybko and Stolyar [3] for example. In statistical physics, these methods are quite classical, see Comets [4].
An Example of Scaling Methods: TCP
In our past work, the Congestion Avoidance Algorithm of the TCP protocol has been analyzed by using such a technique. The equilibrium of the one -dimensional Markov chain associated to this algorithm is not known for the moment. A large number of papers have been written on this famous AIMD Algorithm. But either it was, in some way, idealized or approximations were used without justifications. In a series of papers, Dumas et al. [2], Guillemin et al. [1], a rescaled (time and space) Markov process has been analyzed in the limit when the loss rate of packets of some long connection was converging to 0. It provided a rigorous analysis to the scaling properties of this important algorithm of TCP.
