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The Math of Traffic Jams

June 24, 2009

Whenever I’m caught in a traffic jam I do three things: I try to relax, I make sure there’s some good music playing in my car stereo, and I start contemplating the mathematics of traffic jams.

In my geeky mind I start thinking about how the jam accumulates when a critical mass of vehicles slows down at the same time, and then how the road can still be jammed long after the original cause has been cleared; or how a jam seems to always form in certain points on certain roads and then clear at another specific point down the road, despite neither of the two points being remarkably different that those around them.

You’d often think something like “oh sure, the jam forms around this interchange because everybody’s pouring in from the east trying to get to work,  and so by that interchange it should clear because obviously everybody who joined earlier wanted to get to Route 4, right?”, but it’s rarely like that. It’s more like “the jam starts at this non-remarkable point and then suddenly stops at this other non-remarkable point, and it seems to have nothing to do with cars joining the route or leaving it via interstates”.

I often think about how interesting it might be to model the cars (or more correctly — their drivers) in a computer simulation, and then observe, from outside the system, how the jam is formed and then cleared. If only I knew how to model it correctly.

Why am I telling you all this? Because it turns out some other people had been sharing my thoughts on the subject, only they were knowledgeable enough to actually model everything and run a simulation:

Poor little fellas, stuck in a traffic jam on a road that doesn’t even lead anywhere, and it’s all for our sick entertainment.

Anyway, now they claim to know how to solve “traffic jam equations” and thus how to design roads to better avoid potential traffic jams. I’ll drink to that.

More info in this BoingBoing post which is in turn a summary of this article from Wired. Worth a read if you’re a traffic jam aficionado, like me.

  1. In the rare occasion I get in a traffic jam (having a scooter and all), I think about this odd phenomenon: When the road gets narrower and lanes disappear, the traffic actually drives faster than on a huge road with many lanes.

    Think about it (hint – it’s a lot like fluid mechanics).

    • Avish permalink

      I don’t know much about fluid mechanics. Is it because there’s less lane changing (or “spreading across the huge pipe” if we’re talking fluids)?

  2. Imagine a single point where a huge 10-lanes road becomes 2-lanes. In any given moment, only 2 cars from the large road can be added to the smaller road – so most of the cars on the large road simply wait. In a steady state, this means the rate of movement on the larger road is slower.

    Think of water in a large tube squeezed to a small tube.

    • Avish permalink

      Won’t that make the rate of traffic (which I take as the number of cars that cross an arbitrary point on the road in a given period of time) on both roads equal to the rate on the narrower road? So the wider road isn’t slower than the narrower, it’s just slower than what it could have been.

      Oh, but that makes the average speed per car about 5x slower (right?). So from inside the system it feels slower. Nice.

  3. FireAphis permalink

    While taking Artificial Life class, we were asked to model traffic. Students got to similar simulation, but it’s neater to show it in an endless circle :)

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