17 March, 2005: Speed demons

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A brief comment (and tangential pointless graph update to) something from the Sunday newspapers. Derek Turner, apparently a `jam buster', has told a number of newspapers that speed cameras are a cause of traffic congestion on motorways, because many drivers -- travelling, presumably, above the speed limit -- brake sharply on seeing a camera.

You might wonder why this behaviour causes congestion. Drivers, instructed by the Highway Code to drive two seconds behind the car in front in busy traffic, typically in fact drive about one second behind. On those grounds you'd expect the flow of traffic in one lane of a busy motorway to be about one vehicle per second, independent of speed.

(You might want to think about why this isn't true before reading on....)

... one way to look at this is because real drivers keep a certain distance behind the back of the car in front, and the length of a car is significant compared to the distance it travels in about one second (~4m versus 31m at 70mph). As traffic gets slower, the space taken up by the cars becomes more significant relative to the gaps, and so in a naive model you'd expect a flow of f(v) = v / (v t + L) cars per unit time, where v is the speed of traffic, t the time between cars, and L the length of a car. So for reasonable figures you get something like the following simple picture:

Flow rate as a function of vehicle speed

(The horizontal dotted line is the `ideal' one vehicle per second rate.)

Now, suppose that drivers, proceeding illegally along the motorway at 80mph (right vertical blue line) observe a speed camera. Keen to remain unpunished for their transgression, they break sharply, reducing their speed below the legal limit to 60mph (left vertical blue line). Once past the camera, they accelerate again.

That means that cars on the part of the road just after the camera are travelling at 60mph, and so the rate of vehicles passing through that region is smaller than that in the part before the camera -- 0.87 cars per second as against 0.90 cars per second. Cars enter the region of slow-moving traffic more quickly than they leave it, so that the region must get bigger, creating a congested stretch of road before the camera. This, we surmise, is responsible for the effect Derek Turner mentioned in last Sunday's Observer.

So, how important is this?

Well, a simple calculation based on the above numbers shows that the length of the congested stretch of road will expand at about 2.6 mph (the rate is (f(v1) - f(v2)) (v2 t + L), in case you were wondering). So, suppose that a motorway is busy for three hours during the `rush hour'; you'd expect each speed camera to be associated with a stretch of congested road of size about eight miles. In the congested region, cars travel at 60mph; outside it at 80mph. Each speed camera therefore causes a delay of approximately two minutes in each driver's journey.

This does not seem like very much to me, especially since there aren't -- so far as I understand -- all that many speed cameras on most bits of motorway.

(Of course, one can always try to put a value on this. Using the same assumptions as above, on a busy three-lane motorway, about 10,000 pass in an hour. In three hours the average traveller is delayed by one minute. Suppose that travellers' time is worth, on average, 8 per hour -- the current median wage, roughly. That means that, each rush hour, each speed camera `costs' about 1,200 in wasted travellers' time. I don't regard this as a very sensible way to look at this problem, but if somebody from the ABD wants to turn it into one of their trademark silly press releases -- if, indeed, they have not already done something similar -- I will be very amused.)

Quite independently of this, you might ask whose fault this wasted time is. But, like most `moral' discussions, that's not very likely to be productive, and in any case I've given my opinions on motorway speed limits here before.

(I should say that there are lots of other subtleties which I've ignored here, but which will, no doubt, be brought up in the comments.)

Copyright (c) 2005 Chris Lightfoot; available under a Creative Commons License.