## A Mathematical Model for Moving Bottlenecks in Road Traffic

Serious traffic gridlocks, like the jam on Beijing’s national expressway a few months ago which brought vehicles to a halt for days, are a real-world issue needing attention. Unfortunately, such standstills are not uncommon in Beijing, or in other cities around the world.

Such incidents motivate the analysis of traffic to minimize similar events and provide insight into road design and construction, such as where to install traffic lights and toll booths, how many lanes to build, and where to construct an overpass or a tunnel. The goals of these analyses are to relieve congestion in high traffic areas, reduce the risk of accidents, and manage safety and security of motorists.

Not surprisingly, vehicular traffic flow has been tackled by mathematicians, engineers and physicists alike.  Mathematical approaches to study traffic are usually based on the speed, density and flow of vehicles on a given roadway. In a paper published this month in the SIAM Journal on Mathematical Analysis, authors Corrado Lattanzio, Amelio Maurizi and Benedetto Piccoli propose a mathematical model of vehicular traffic based on the study of a moving bottleneck caused by a slow-moving vehicle within the flow of cars. The effect of moving bottlenecks on flow of traffic is an important factor in evaluating travel times and traveling paths for commuters.

Many different mathematical models have been proposed to study traffic, including models that use second-order equations for mass and momentum, multipopulation models that factor in the varying characteristics of different kinds of vehicles, and dynamic models that consider traffic flows.

Most of the models so far proposed, however, solve the problem of a single vehicle independently of the entire traffic flow, and so are not completely coupled. An example is a PDE-ODE model that used a partial differential equation to model the flow of traffic while using an ordinary differential equation to determine the position of a single vehicle. Since both could be solved independently, the system did not take into account the influence of the single car on the entire traffic flow.

The paper by Lattanzio et al provides a fully coupled, multi-scale model in which the microscopic position of a single car is taken together with the macroscopic car density on the road. In this micro-macro model, the dynamics of a moving bottleneck caused by a slow-moving vehicle on a street are used to study the effects of disruptions on the flow of traffic. Mathematically, the problem is solved using the fractional step method. In successive time steps, a PDE is first solved for the density of traffic and then the ODE is solved for the position of the slow-moving vehicle.

By solving the bottleneck problem in a coupled fashion, better transportation designs can be made in anticipation of such inevitable traffic congestion.

Source:
Moving Bottlenecks in Car Traffic Flow: A PDE-ODE Coupled Model
Corrado Lattanzio, Amelio Maurizi, and Benedetto Piccoli
SIAM Journal on Mathematical Analysis, 43 (2011), pp 50-67
Pub date 4 January 2011
© Society for Industrial and Applied Mathematics

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