### Speed limits, congestion, and following distances

The highways in a metropolitan area are congested. At times of extreme congestion, cars move at a crawl. At low congestion, speed is constrained only by the speed limit. Suppose city planners want to maximize traffic flow -- the number of cars that drive over a particular section of highway in a given amount of time, say in a minute. What is the ideal speed limit for maximizing this measure of traffic flow? On first glance, it seems like completely removing the speed limit would do the trick. But is that really obvious? At higher speeds, cars must keep larger following distances. The faster the traffic, the thinner the traffic, so it is not clear if the number of cars passing a given point is higher or lower at high speeds.

Some standard assumptions:

Kinetic energy = a constant times the square of velocity.

Work = force times distance.

The maximum force applied by brakes does not vary with speed.

How drivers choose following distances completely determines the solution to the problem. The calculation details are messy and difficult to post to a blog, but here are the conclusions.

Case 1: Suppose that drivers tailgate to the extreme, staying a constant distance behind the car in front regardless of speed. Then you get the greatest traffic flow by posting no speed limit, and letting cars go as fast as possible.

Case 2: Suppose drivers follow a two-second rule (or any set number of seconds), staying two seconds behind the car in front. Then the speed of the cars has no effect on the level of traffic flow!

Case 3: Drivers stay back the same distance that it would require to come to a complete stop. (Most drivers don't follow this rule, because even if there is a wreck ahead, the wrecking cars do not normally come to a stop instantaneously, so the tailing cars have more time to slow down.) Then you can maximize traffic flow by making the speed limit as "low as possible." The vagueness of this answer is necessary because my model does not take into account the length of a car. This particularly matters at the extremes: If a car were zero feet long, then you can get arbitrarily high traffic flow by putting the speed limit sufficiently close to zero. I suspect that the length-of-car factor begins to outweigh the slow-is-efficient factor at around 40 mph, when the stopping distance is at least 5 car-lengths.

Some standard assumptions:

Kinetic energy = a constant times the square of velocity.

Work = force times distance.

The maximum force applied by brakes does not vary with speed.

How drivers choose following distances completely determines the solution to the problem. The calculation details are messy and difficult to post to a blog, but here are the conclusions.

Case 1: Suppose that drivers tailgate to the extreme, staying a constant distance behind the car in front regardless of speed. Then you get the greatest traffic flow by posting no speed limit, and letting cars go as fast as possible.

Case 2: Suppose drivers follow a two-second rule (or any set number of seconds), staying two seconds behind the car in front. Then the speed of the cars has no effect on the level of traffic flow!

Case 3: Drivers stay back the same distance that it would require to come to a complete stop. (Most drivers don't follow this rule, because even if there is a wreck ahead, the wrecking cars do not normally come to a stop instantaneously, so the tailing cars have more time to slow down.) Then you can maximize traffic flow by making the speed limit as "low as possible." The vagueness of this answer is necessary because my model does not take into account the length of a car. This particularly matters at the extremes: If a car were zero feet long, then you can get arbitrarily high traffic flow by putting the speed limit sufficiently close to zero. I suspect that the length-of-car factor begins to outweigh the slow-is-efficient factor at around 40 mph, when the stopping distance is at least 5 car-lengths.

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