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Section: Scientific Foundations

Keywords : Fluid Limits, Functional Limit Theorems, Statistical Physics.

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 :

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.

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.   [4] , a conveniently 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.

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 dis 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 [25] for example. In statistical physics, these methods are quite classical, see Comets [19] .

Multi-Class Networks . The state space of the Markov processes encountered up to now were embedded into some finite dimensional vector space. For Im1 ${J\#8712 \#8469 }$ , J$ \ge$2 and j= 1 ,... J, $ \lambda$j and $ \mu$j are positive real numbers. It is assumed that JPoissonnian arrivals flows arrive at a single server queue with rate $ \lambda$j for j= 1 ,..., Jand customers from the jth flow require an exponentially distributed service with parameter $ \mu$j . All the arrival flows are assumed to be independent. The service discipline is FIFO.

A natural way to describe this process is to take the state space of the finite strings with values in the set {1, ..., J} , i.e. Im2 ${\#119982 =\#8746 _{n\#8805 0}{1,...,J}^n,}$ with the convention that {1, ..., J} 0 is the set of the null string. If n$ \ge$1 and Im3 ${{x=(}x_1,...,x_n{)\#8712 \#119982 }}$ is the state of the queue at some moment, the customer at the kth position of the queue comes from the flow with index xk , for k= 1 , ..., n. The length of a string Im4 ${x\#8712 \#119982 }$ is defined by Im5 ${\#8214 x\#8214 }$ . Note that Im6 ${\#8214 ·\#8214 }$ is not, strictly speaking, a norm. For n$ \ge$1 , there are Jn vectors of length n ; the state space has therefore an exponential growth with respect to that function. Hence, if the string valued Markov process ( X( t)) describing the queue is transient then certainly the length Im7 ${\#8214 X(t)\#8214 }$ converges to infinity as tgets large. Because of the large number of strings with a fixed length, the process ( X( t)) itself has, a priori, infinitely many ways to go to infinity. Bramson [18] has shown that complicated phenomena could indeed occur. It turns out that the « classical » fluid limits methods of the finite dimensional case cannot be used in such a setting. This is probably one of the most challenging question in the domain to be able to propose new methods to tackle the problems due to the infinite dimension of the state space. Dantzer and Robert  [1] derives results in this direction. See also the corresponding chapter of Robert  [5] .

Goals

The general goals are, in some way, contained in the previous sections. They will consist in developing scaling techniques in the various cases encountered in sampling problems or tree algorithms where the traffic will be supposed to be close to saturation. The following fundamental questions will be analyzed :


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