Section: Scientific Foundations
Managing the system (via probabilistic modeling)
Participants : Oumar Baba Diakhate, Guy Fayolle, Cyril Furtlehner, JeanMarc Lasgouttes, Jennie Lioris, Victorin Martin.
The research on the management of the transportation system is a natural continuation of the research of the Preval team, which joined IMARA in 2007. For many years, the members of this team (and of its ancestor Meval) have been working on understanding random systems of various origins, mainly through the definition and solution of mathematical models. The traffic modelling field is very fertile in difficult problems, and it has been part of the activities of the members of Preval since the times of the Praxitèle project.
Following this tradition, the roadmap of the group is to pursue basic research on probabilistic modelling with a clear slant on applications related to LaRA activities. A particular effort is made to publicize our results among the traffic analysis community, and to implement our algorithms whenever it makes sense to use them in traffic management. Of course, as aforementioned, these activities in no way preclude the continuation of the methodological work achieved in the group for many years in various fields: random walks in Z_{ + }^{n} ([3] , [4] , [7] ), large deviations ([2] , [1] ) birth and death processes on trees, particle systems. The reader is therefore encouraged to read the recent activity reports for the Preval team for more details.
In practice, the group explores the links between large random systems and statistical physics, since this approach proves very powerful, both for macroscopic (fleet management [6] ) and microscopic (carlevel description of traffic, formation of jams) analysis. The general setting is mathematical modelling of large systems (mostly stochastic), without any a priori restriction: networks [5] , random graphs or even objects coming from biology. When the size or the volume of those structures grows (this corresponds to the socalled thermodynamical limit), one aims at establishing a classification based on criteria of a twofold nature: quantitative (performance, throughput, etc) and qualitative (stability, asymptotic behavior, phase transition, complexity).
Exclusion processes
One of the simplest basic (but non trivial) probabilistic models for road traffic is the exclusion process. It lends itself to a number of extensions allowing to tackle some particular features of traffic flows: variable speed of particles, synchronized move of consecutive particles (platooning), use of geometries more complex than plain 1D (cross roads or even fully connected networks), formation and stability of vehicle clusters (vehicles that are close enough to establish an adhoc communication system), twolane roads with overtaking.
Most of these generalizations lead to models that are obviously difficult to solve and require upstream theoretical studies. Some of them models have already been investigated by members of the group, and they are part of wide ongoing research.
Message passing algorithms
Large random systems are a natural part of macroscopic studies of traffic, where several models from statistical physics can be fruitfully employed. One example is fleet management, where one main issue is to find optimal ways of reallocating unused vehicles: it has been shown that Coulombian potentials might be an efficient tool to drive the flow of vehicles. Another case deals with the prediction of traffic conditions, when the data comes from probe vehicles instead of static sensors. Using some famous Ising models together with the Belief Propagation algorithm very popular in the computer science community, we have been able to show how realtime data can be used for traffic prediction and reconstruction (in the spacetime domain).
This new use of BP algorithm raises some theoretical questions about the properties of the Bethe approximation of Ising models

how do the stability of the BP fixed points relate the the minima of the free energy?

what is the effect of the various extensions to BP (fractional, treereweighted, regionbased,...) of these fixed points?

what is the behaviour of BP in the situation where the underlying data have many different statistical components, representing a variety of independent patterns?