Mathematical models of traffic flow on a network of roads

A basic PDE model of traffic flow on a single road was introduced by Lighthill and Witham (1955) and Richards (1956). It consists of a scalar conservation law for the vehicle density. In recent years, research has focused on

• Models of traffic flow on a network of roads, supplementing the conservation laws on the various roads with suitable boundary conditions describing the flow at intersections.
• Optimization problems, where drivers have costs depending on the times of departure and of arrival. It is here assumed that drivers can choose the time of departure and the route taken to destination.

In this setting, one seeks either (i) a globally optimal solution, minimizing the sum of all costs to all drivers, or (ii) a Nash equilibrium solution, where no driver can lower his/her own cost by changing departure time or choosing a different path to destination.