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Scaling of Markov Processes
As the complexity of communication networks increases (and, consequently, the algorithms regulating them), the classical mathematical methods used to estimate the stationary behavior, the transient behavior show more and more their limitations. 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, this kind of approach is not, in general, possible. The key idea to overcome these difficulties is to consider limiting procedures for the system:
- by considering the asymptotic behavior of the probability of some events like it is done for large deviations at a logarithmic scale or for heavy tailed distributions, or looking at Poisson approximations to describe a sequence of events associated to them.
- by taking some parameter of the model and look at the behavior of the system when it approaches some critical value. In some cases, even if the model is complicated, its behavior simplifies in the neighborhood of the critical parameter: some of the nodes grow according to some classical limit theorem and the rest of the nodes reach some equilibrium which can be described.
- by changing the time scale and the space scale with a common normalizing factor N and let N goes to infinity. This leads to functional limit theorems, see below.
The list of possible renormalization procedures is, of course, not exhaustive. But for the last ten years, this methodology has become more and more developed. Its advantages lies in its flexibility to various situations and also to the interesting theoretical problems it has raised since then.
