Stochastic macroscopic traffic modeling

Jean-Patrick Lebacque

This presentation focuses on stochastic aspects of macroscopic traffic models. Macroscopic traffic flow models are mainly constructed based on two approaches: the kinetic (Boltzmann-like) and the hydrodynamic analogy. The resulting models express themselves as systems of conservation laws. Both approaches rely on the continuum hypothesis and thus on low resolution approximations of traffic flow. The Avogadro number of traffic flow is of the order of 180 vh/km/lane. Nevertheless macroscopic models describe correctly many macroscopic features of traffic dynamics: formation of waves (acceleration/deceleration), interaction between local traffic supply and demand, aggregate behavior of traffic flow through intersections, etc. Thus these models are rather well adapted to the description, simulation and management of traffic flow on large networks. The behavior of drivers is constrained by many factors: regulations, capacity of infrastructures, vehicle capabilities, and local environment. This is why traffic flow is predictable at a macroscopic level with a reasonable level of precision. Still many stochastic factors affect the dynamics of traffic: the variability of driver behavior and vehicle capabilities, random (and essentially inaccessible to measurement) events affecting drivers, random elements in the environment … Further, at a microscopic level, driver behavior and the dynamics of vehicles depend on numerous parameters, which macroscopic models neglect. Finally, driver behavior is determined by factors which cannot be readily measured, such as OD and path choice. All these elements constitute error factors which contribute to discrepancies between models and measurements. Random components can and should be used to take into account and model these error factors. As a result, macroscopic models with stochastic components should provide a better representation of the actual dynamics of traffic flow and should be better suited for applications such as traffic management and control.
The presentation will start with a brief review of the principles of macroscopic traffic modeling, followed by a description of some early efforts of stochastic modeling at a macroscopic level such as the works of Weits, Kühne etc as well as stochastic elements in kinetic models. A second part of the presentation will be concerned with more recent approaches, based on lagrangian or eulerian random perturbations of traffic flow, or random perturbations of discretized macroscopic traffic flow models.
The last part of the presentation will be devoted to two new approaches. The first is based the first on the GSOM model family. The idea is that the stochastic perturbation affects individual behavioral attributes and thus affects the traffic dynamics without inducing diffusion. The resulting model is expressed as a system of stochastic conservation equations. Elements of resolution will be given. The second is based on modeling traffic as an exclusion process and is closely related to the LWR and GSOM models. Both models will also be considered in a network setting: some elements of intersection modeling will be given in a stochastic context.

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