Section: Scientific Foundations

Scaling of Markov Processes

The growing complexity of communication networks makes it more difficult to apply classical mathematical methods. For a one/two-dimensional Markov process describing the evolution of some network, it is sometimes possible to write down the equilibrium equations and to solve them. When the number of nodes is more than 3, however, this kind of approach is not possible in general. The key idea to overcome these difficulties is to consider the system in limit regimes.

• By considering the asymptotic behavior of the probability of some events as in the case of large deviations at a logarithmic scale or for heavy tailed distributions, or by looking at Poisson approximations to describe a sequence of events associated with them.

• By taking some parameter of the model and look at the behavior of the system when it approaches some critical value _c . In some cases, even if the model is complicated, its behavior simplifies as _c : some of the nodes grow according to a classical limit theorem while the remainder reach an equilibrium which can be described.

• By changing the time scale and space scale with a common normalizing factor N and letting N go to infinity. This leads to functional limit theorems as discussed below.

This list of possible renormalization procedures is, of course, not exhaustive. This methodology has become more and more developed over the last ten years. Its advantages lie in its flexibility to various situations and to the interesting theoretical problems it has raised since then.

Fluid Limits

A fluid limit scaling is a particularly important means to scale a Markov process. It is related to the first order behavior of the process and, roughly speaking, 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 significantly complicates any investigation: analytical tools such as Wiener-Hopf techniques for dimension 1 cannot be easily generalized to higher dimensions. It is nevertheless possible to gain insight into the behavior of these processes through some limit theorems. The considered limiting procedure 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 second order stochastic fluctuations disappear. In “good” cases, a fluid limit is a deterministic function, obtained as the solution of some ordinary differential equation. As 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 [32] , for example. In statistical physics, these methods are quite classical, see Comets [27] .