Fundamental Diagrams of 1D-Traffic Flow by Optimal Control Models

Traffic on a circular road is described by dynamic programming equations associated to optimal control problems. By solving the equations analytically, we derive the relation between the average car density and the average car flow, known as the fundamental diagram of traffic. First, we present a model based on min-plus algebra, then we extend it to a stochastic dynamic programming model, then to a stochastic game model. The average car flow is derived as the average cost per time unit of optimal control problems, obtained in terms of the average car density. The models presented in this article can also be seen as developed versions of the car-following model. The derivations proposed here can be used to approximate, understand and interprete fundamental diagrams derived from real measurements.Comment: 17 pages

Piecewise linear car-following modeling

We present a traffic model that extends the linear car-following model as well as the min-plus traffic model (a model based on the min-plus algebra). A discrete-time car-dynamics describing the traffic on a 1-lane road without passing is interpreted as a dynamic programming equation of a stochastic optimal control problem of a Markov chain. This variational formulation permits to characterize the stability of the car-dynamics and to calculte the stationary regimes when they exist. The model is based on a piecewise linear approximation of the fundamental traffic diagram.Comment: 19 pages, 3 figure

About Dynamical Systems Appearing in the Microscopic Traffic Modeling

Motivated by microscopic traffic modeling, we analyze dynamical systems which have a piecewise linear concave dynamics not necessarily monotonic. We introduce a deterministic Petri net extension where edges may have negative weights. The dynamics of these Petri nets are well-defined and may be described by a generalized matrix with a submatrix in the standard algebra with possibly negative entries, and another submatrix in the minplus algebra. When the dynamics is additively homogeneous, a generalized additive eigenvalue may be introduced, and the ergodic theory may be used to define a growth rate under additional technical assumptions. In the traffic example of two roads with one junction, we compute explicitly the eigenvalue and we show, by numerical simulations, that these two quantities (the additive eigenvalue and the growth rate) are not equal, but are close to each other. With this result, we are able to extend the well-studied notion of fundamental traffic diagram (the average flow as a function of the car density on a road) to the case of two roads with one junction and give a very simple analytic approximation of this diagram where four phases appear with clear traffic interpretations. Simulations show that the fundamental diagram shape obtained is also valid for systems with many junctions. To simulate these systems, we have to compute their dynamics, which are not quite simple. For building them in a modular way, we introduce generalized parallel, series and feedback compositions of piecewise linear concave dynamics.Comment: PDF 38 page