Section: Research Program
Probabilistic modeling for large transportation systems
Participants : Guy Fayolle, JeanMarc Lasgouttes.
This activity concerns the modeling of random systems related to ITS, through the identification and development of solutions based on probabilistic methods and more specifically through the exploration of links between large random systems and statistical physics. Traffic modeling is a very fertile area of application for this approach, both for macroscopic (fleet management [46], traffic prediction) and for microscopic (movement of each vehicle, formation of traffic jams) analysis. When the size or volume of structures grows (leading to the socalled “thermodynamic limit”), we study the quantitative and qualitative (performance, speed, stability, phase transitions, complexity, etc.) features of the system.
In the recent years, several directions have been explored.
Traffic reconstruction
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.
While the widelyused macroscopic traffic flow models are well adapted to highway traffic, where the distance between junction is long (see for example the work done by the NeCS team in Grenoble), our focus is on a more urban situation, where the graphs are much denser. The approach we are advocating here is modelless, and based on statistical inference rather than fundamental diagrams of road segments. Using the Ising model or even a Gaussian Random Markov Field, together with the very popular Belief Propagation (BP) algorithm, 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 ways the make the belief propagation algorithm more efficient:

find the best way to inject realvalued data in an Ising model with binary variables [50];

build macroscopic variables that measure the overall state of the underlying graph, in order to improve the local propagation of information [47];

make the underlying model as sparse as possible, in order to improve BP convergence and quality [49].
Exclusion processes for road traffic modeling
The focus here is on road traffic modeled as a granular flow, in order to analyze the features that can be explained by its random nature. This approach is complementary to macroscopic models of traffic flow (as done for example in the Opale team at Inria), which rely mainly on ODEs and PDEs to describe the traffic as a fluid.
One particular feature of road traffic that is of interest to us is the spontaneous formation of traffic jams. It is known that systems as simple as the NagelSchreckenberg model are able to describe traffic jams as an emergent phenomenon due to interaction between vehicles. However, even this simple model cannot be explicitly analyzed and therefore one has to resort to simulation.
One of the simplest solvable (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.
The aspect that we have particularly studied is the possibility to let the speed of vehicle evolve with time. To this end, we consider models equivalent to a series of queues where the pair (service rate, number of customers) forms a random walk in the quarter plane ${\mathbb{Z}}_{+}^{2}$.
Having in mind a global project concerning the analysis of complex systems, we also focus on the interplay between discrete and continuous description: in some cases, this recurrent question can be addressed quite rigorously via probabilistic methods.
We have considered in [43] some classes of models dealing with the dynamics of discrete curves subjected to stochastic deformations. It turns out that the problems of interest can be set in terms of interacting exclusion processes, the ultimate goal being to derive hydrodynamic limits after proper scaling. A seemingly new method is proposed, which relies on the analysis of specific partial differential operators, involving variational calculus and functional integration. Starting from a detailed analysis of the Asymmetric Simple Exclusion Process (ASEP) system on the torus $\mathbb{Z}/n\mathbb{Z}$, the arguments a priori work in higher dimensions (ABC, multitype exclusion processes, etc), leading to systems of coupled partial differential equations of Burgers' type.
Random walks in the quarter plane ${\mathbb{Z}}_{+}^{2}$
This field remains one of the important "violon d'Ingres" in our research activities in stochastic processes, both from theoretical and applied points of view. In particular, it is a building block for models of many communication and transportation systems.
One essential question concerns the computation of stationary measures (when they exist). As for the answer, it has been given by original methods formerly developed in the team (see books and related bibliography). For instance, in the case of small steps (jumps of size one in the interior of ${\mathbb{Z}}_{+}^{2}$), the invariant measure $\{{\pi}_{i,j},i,j\ge 0\}$ does satisfy the fundamental functional equation (see [45]):
$Q(x,y)\pi (x,y)=q(x,y)\pi \left(x\right)+\tilde{q}(x,y)\tilde{\pi}\left(y\right)+{\pi}_{0}(x,y).$  (1) 
where the unknown generating functions $\pi (x,y),\pi \left(x\right),\tilde{\pi}\left(y\right),{\pi}_{0}(x,y)$ are sought to be analytic in the region $\{(x,y)\in {\u2102}^{2}:x<1,y<1\}$, and continuous on their respective boundaries.
The given function $Q(x,y)={\sum}_{i,j}{p}_{i,j}{x}^{i}{y}^{j}1$, where the sum runs over the possible jumps of the walk inside ${\mathbb{Z}}_{+}^{2}$, is often referred to as the kernel. Then it has been shown that equation (1) can be solved by reduction to a boundaryvalue problem of RiemannHilbert type. This method has been the source of numerous and fruitful developments. Some recent and ongoing works have been dealing with the following matters.

Group of the random walk. In several studies, it has been noticed that the socalled group of the walk governs the behavior of a number of quantities, in particular through its order, which is always even. In the case of small jumps, the algebraic curve $R$ defined by $\left\{Q\right(x,y)=0\}$ is either of genus 0 (the sphere) or 1 (the torus). In [Fayolle2011a], when the drift of the random walk is equal to 0 (and then so is the genus), an effective criterion gives the order of the group. More generally, it is also proved that whenever the genus is 0, this order is infinite, except precisely for the zero drift case, where finiteness is quite possible. When the genus is 1, the situation is more difficult. Recently [44], a criterion has been found in terms of a determinant of order 3 or 4, depending on the arity of the group.

Nature of the counting generating functions. Enumeration of planar lattice walks is a classical topic in combinatorics. For a given set of allowed jumps (or steps), it is a matter of counting the number of paths starting from some point and ending at some arbitrary point in a given time, and possibly restricted to some regions of the plane. A first basic and natural question arises: how many such paths exist? A second question concerns the nature of the associated counting generating functions (CGF): are they rational, algebraic, holonomic (or Dfinite, i.e. solution of a linear differential equation with polynomial coefficients)?
Let $f(i,j,k)$ denote the number of paths in ${\mathbb{Z}}_{+}^{2}$ starting from $(0,0)$ and ending at $(i,j)$ at time $k$. Then the corresponding CGF
satisfies the functional equation
where $z$ is considered as a timeparameter. Clearly, equations (2) and (1) are of the same nature, and answers to the above questions have been given in [Fayolle2010].

Some exact asymptotics in the counting of walks in ${\mathbb{Z}}_{+}^{2}$. A new and uniform approach has been proposed about the following problem: What is the asymptotic behavior, as their length goes to infinity, of the number of walks ending at some given point or domain (for instance one axis)? The method in [Fayolle2012] works for both finite or infinite groups, and for walks not necessarily restricted to excursions.
Discreteevent simulation for urban mobility
We have developed two simulation tools to study and evaluate the performance of different transportation modes covering an entire urban area.

one for collective taxis, a public transportation system with a service quality provided will be comparable with that of conventional taxis (system operating with or without reservations, doortodoor services, well adapted itineraries following the current demand, controlling detours and waits, etc.), and with fares set at rates affordable by almost everyone, simply by utilizing previously wasted vehicle capacity;

the second for a system of selfservice cars that can reconfigure themselves into shuttles, therefore creating a multimodal public transportation system; this second simulator is intended to become a generic tool for multimodal transportation.
These two programs use a technique allowing to run simulations in batch mode and analyze the dynamics of the system afterward.