When you’re caught in a traffic jam, you probably don’t know why. Is there an accident ahead of you? In most of the cases, the answer is no. And all the electronic devices installed in your car can’t help you. You’re stuck for a while — until the traffic gradually improves. Now, European mathematicians have solved the traffic jam mystery. The mathematical model they’ve developed shows that traffic jams are mostly caused by a single driver who brakes too much when faced to an unexpected event. Of course, the driver behind him also slows down, and so does the next one, until the road is totally blocked. But read more…
This mathematical model has been developed by Dr Gábor Orosz of the Dynamical Systems & Control research institute of the School of Engineering, Computing and Mathematics at the University of Exeter, UK. Orosz, who also maintains a personal homepage, worked on this project with Gábor Stépán of the Department of Applied Mechanics at the Budapest University of Technology and Economics (BUTE) and who leads the research group on ‘Dynamics of Machines and Vehicles.’
Here is a description of what did the two mathematicians. “The team developed a mathematical model to show the impact of unexpected events such as a lorry pulling out of its lane on a dual carriageway. Their model revealed that slowing down below a critical speed when reacting to such an event, a driver would force the car behind to slow down further and the next car back to reduce its speed further still. The result of this is that several miles back, cars would finally grind to a halt, with drivers oblivious to the reason for their delay. The model predicts that this is a very typical scenario on a busy highway (above 15 vehicles per km). The jam moves backwards through the traffic creating a so-called ‘backward travelling wave’, which drivers may encounter many miles upstream, several minutes after it was triggered.”
And here is how the team thinks it’s possible to remedy to this situation. “‘When you tap your brake, the traffic may come to a full stand-still several miles behind you. It really matters how hard you brake — a slight braking from a driver who has identified a problem early will allow the traffic flow to remain smooth. Heavier braking, usually caused by a driver reacting late to a problem, can affect traffic flow for many miles,’ said Orosz. The research team now plans to develop a model for cars equipped with new electronic devices, which could cut down on over-braking as a result of slow reactions.”
For more information, this research work has been published in the Proceedings of the Royal Society A under the title “Subcritical Hopf bifurcations in a car-following model with reaction-time delay” (Volume 462, Number 2073, Pages 2643-2670, September 8, 2006). Here is a link to the abstract which starts like this. “A nonlinear car-following model of highway traffic is considered, which includes the reaction-time delay of drivers. Linear stability analysis shows that the uniform flow equilibrium of the system loses its stability via Hopf bifurcations and thus oscillations can appear. The stability and amplitudes of the oscillations are determined with the help of normal-form calculations of the Hopf bifurcation that also handles the essential translational symmetry of the system.”
Like certainly most of you, I’ve never heard about the Hopf bifurcations before today. But I found lots of details on this Wikipedia page.
You also might want to read the full paper from Orosz and Stépán (PDF format, 28 pages, 338 KB). But if I except the introduction and the conclusion, this is not something I’d recommend to anyone without a math degree.
Here is an excerpt from the conclusions of this paper. “Stop-and-go traffic jams can develop for large enough perturbations even when the desired uniform flow is linearly stable. These perturbations can be caused, for example, by a slower vehicle (such us a lorry) joining the inner lane flow for a short-time interval via changing lanes. It is essential to limit these unwanted events, for example, by introducing temporary regulations provided by overhead gantries. Still, if a backward travelling wave shows up without stoppings, it either dies out by itself or gets worse ending up as a persistent stop-and-go travelling wave. In order to dissolve this undesired situation, an appropriate control can be applied using temporary speed limits given by overhead gantries that can lead the traffic back ‘inside’ the unstable travelling wave and then to reach the desired uniform flow.” And be warned again: read the paper only if you’re good in maths!
Sources: University of Exeter, December 19, 2007; and various websites
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