Where do traffic jams come from when nothing’s wrong?

It is natural to assume every jam has a culprit somewhere up ahead. Often it does. But in 2008 a team of Japanese physicists led by Yuki Sugiyama showed that you don’t need one. They put 22 cars on a circular track 230 metres round, on a flat open field, and asked the drivers to cruise at a steady 30 km/h. There was nothing on the track to cause a jam. Within minutes, one formed anyway: a cluster of stopped cars, while the cars outside it drove at about 40 km/h. The cluster crept backwards round the track at roughly 20 km/h, and it did not go away.

Such a jam is called a phantom jam. To see how one forms, we need a model of drivers simple enough to understand completely.

A road made of boxes

In 1992 Kai Nagel and Michael Schreckenberg proposed a model of motorway traffic that fits on an index card. Cut the road into cells 7.5 m long, about the space one car takes up in a queue, bumper to bumper. Each cell holds one car or none. Time moves in steps of one second. A car’s speed is a whole number of cells per step, from 0 up to 5. One cell per second is 27 km/h, so the top speed of 5 is 135 km/h.

Every second, every driver does the same four things:

1. Speed up by one, unless already at the top speed.
2. Don’t hit anyone. If the gap to the car ahead (the number of empty cells) is smaller than your speed, slow down to the gap.
3. Dawdle. With probability p, slow down by one more for no reason: a glance at the phone, a slightly late reaction, easing off over a crest.
4. Move forward by your speed.

That third rule is the whole story. Take it away (set p to zero) and every car settles into a perfectly smooth, jam-free pattern whenever there is room for one. Leave it in, and the small hesitations start to pile up.

Here is what happens. A driver dawdles and slows down. The driver behind, who was following at a comfortable gap, finds the gap shrinking and must slow down too, and usually a little more, because they react a step late. The next driver slows more still. If the cars are far apart, the slowdown is absorbed by the spare room and dies out. If they are close together, there is no spare room, and the slowdown grows car by car until someone has to stop. Now there is a jam.

Once a car is stopped, it can only leave when the car in front has pulled away. Cars leave from the front of the jam one at a time, while new cars arrive at the back. So the jam loses cars at its front and gains them at its back: it moves backwards, even though every car in it is moving forwards, or waiting to.

Seeing the jam move

A ring is hard to read over time. The next figure unrolls a 1.5 km ring into a straight line, so cars drive left to right (and a car leaving the right edge comes back on the left). Each row of dots is the road at one moment; the next second is drawn just below it, so time runs down the page. This is a space-time diagram, the tool traffic engineers use for exactly this question.

The stripes lean left by roughly one car length every second or two. In this model, with a dawdle probability between 0.1 and 0.25, a jam’s front edge moves upstream at about 15 to 23 km/h (see the checks in the colophon). That is close to what is measured on real motorways, where stop-and-go waves travel against the traffic at about 15 km/h, give or take 5, almost regardless of the road. Sugiyama’s jam on the test track moved back at roughly 20 km/h.

This explains the experience of driving through a phantom jam. You meet it, crawl through it, and leave it at the front. By the time you are out, the jam is several hundred metres behind where you entered it, and there is nothing to see, because the jam was never a place. It was a wave passing through the cars.

More cars, less traffic

How many cars does it take? To answer that, we need two numbers. Density is how many cars there are per kilometre of lane. Flow is how many cars pass a fixed point per hour; it is what a road is for. On a ring, flow is simply density times average speed:

flow = density × average speed

so 20 cars per kilometre at 100 km/h makes 2,000 cars an hour. Figure 3 runs the model at many densities, lets each settle for five minutes of model time, measures the flow over the next ten, and plots the result. Traffic engineers call this curve the fundamental diagram.

The density at the peak is the critical density. Below it, traffic is free: a dawdle is absorbed. Above it, the road is congested and jams form on their own. With p = 0.25 the model peaks at about 1,850 cars per hour at 15 to 17 cars per kilometre: one car every 60 m or so. A real motorway lane carries at most about 2,400 cars an hour in ideal conditions, and often nearer 2,000 in practice. Sugiyama’s group, looking at motorway data, put the critical density near 25 vehicles per kilometre. (Their test track was much denser, 22 cars in 230 m, about 96 per kilometre, because everyone was driving at only 30 km/h, so the two numbers are not directly comparable.)

Note the cruel asymmetry. Below the critical density, adding cars adds flow. Above it, adding cars makes everyone slower and moves fewer people. With p = 0.25, a lane at 50 cars per kilometre in this model carries about 1,400 cars an hour, less than one at 15 (about 1,800).

Can one driver help? A team led by Raphael Stern and Daniel Work (paper published 2018) repeated the ring experiment on a 260 m track at the University of Arizona with more than 20 cars, then switched one of them to a computer that held a smooth, steady speed instead of chasing the car in front. The waves died out. Compared with the runs where waves were present, they measured up to 40% less fuel burned and up to 15% more traffic flow. The calm car works by leaving a gap large enough to soak up each slowdown before it reaches the car behind. (This page’s model is too crude to show that fairly: with speeds in steps of 27 km/h and no overtaking, a capped car mostly acts as a slow lead car.)

So the answer to the question is: when the road is busy enough, nothing needs to go wrong. The ordinary, tiny unevenness of human driving, amplified from car to car, is enough. Sugiyama’s drivers were told to drive steadily, and they tried; the jam came anyway.