Rescaled Markov Processes and Fluid Limits

Abstract

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 of Chapter 2 for dimension 1 cannot be easily generalized to higher dimensions. Note that the natural order on the real line plays an important role for Wiener-Hopf methods. The example of queueing networks seen in Chapter 4 for which the stationary distribution has a product form should be seen as an interesting exception, but an exception. In the same way as in Chapter 3, it is possible nevertheless to get some insight on the behavior of these processes through some limit theorems. In this chapter, limit results consist 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 (see the rigorous definition below). As it will be seen, 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[RS92] for example. In statistical physics, these methods are quite classical, see Comets[Com91].