Traffic Flow Models on a Network of Roads

Abstract

Macroscopic models of traffic flow on a network of roads can be formulated in terms of a scalar conservation law on each road, together with boundary conditions, determining the flow at junctions. Some of these intersection models are reviewed in this note, discussing the well posedness of the resulting initial value problems. From a practical point of view, one can also study traffic patterns as the outcome of many decision problems, where each driver chooses his departure time and route to destination, in order to minimize the sum of a departure and an arrival cost. For the new models including a buffer at each intersection, one can prove: (i) the existence of a globally optimal solution, minimizing the total cost to all drivers, and (ii) the existence of a Nash equilibrium solution, where no driver can lower his own cost by changing his departure time or the route taken to reach destination.