Statistical forecasting of high-way traffic jam at a bottleneck

Abstract

Maintenance works on high-ways usually require installation of bottlenecks that disturb traffic. The article presents a new mathematical model for analysis and forecasting of traffic jam evolution in front of a bottleneck. The model is comprised of two partial differential equations for the mean velocity and density of cars. The first equation describes relaxation of velocity to its equilibrium value determined by a new fundamental diagram. The second is the continuity equation and describes adaptation of the density to the input traffic flow that is forecast statistically. Numerical treatment of the model yields distributions of traffic variables that exhibit characteristic properties of jam evolution. The performance of the method is demonstrated by forecasting the jam that would develop during rush-hour if a bottleneck were installed on a high-way close to Ljubljana. Beside the model a new method is presented for approximate prediction of jam length based upon input flow and bottleneck capacity that is specified by the fundamental diagram. The corresponding computer programs represent a new tool by which experts can analyze properties of bottlenecks in order to optimize them.