The phantom traffic jam: a slowdown that travels backwards

What braking waves look like, car by car#

On 13 April 2005, seven cameras on the roof of a 30-storey building beside the I-80 at Emeryville filmed every vehicle on about 500 metres of motorway: six lanes and an on-ramp. From the footage, researchers on the US NGSIM programme extracted the position of each vehicle ten times a second and published it as open data.

Below is the quarter-hour from 17:00, played eight times faster: the road seen from above and, beneath it, the lane you pick, drawn through time.

Follow a red patch on the road. Cars enter it on the left and leave on the right, so they move forwards. The patch, though, moves left: backwards, towards the cars that have not reached it yet. On the drawing beneath, the same patch is a band sloping down to the right.

The waves enter the filmed stretch from ahead, already formed, so the drawing does not show what started them: a lane change, a car joining from the ramp, or just someone braking. It does show what happens next: the wave runs back through the queue without fading. A jam with no cause at all, the phantom jam in its purest form, appears on the ring further down.

Why the wave travels backwards at 18 km/h#

When the queue ahead starts moving, it does not move all at once. Each driver sees the car in front move, and only then lifts off the brake. On the I-80, across 803 measured departures, each driver pulled away 1.8 seconds after the one in front (the median), and stopped cars stood 7.4 m apart, front bumper to front bumper.

So every 1.8 seconds the queue releases one place, and that place is 7.4 m further back. 7.4 m every 1.8 s comes to about 14.8 km/h. Measured directly from the cars’ speeds, the wave moved at 17.1–18.9 km/h in the three measurements taken on different lanes and intervals. The two methods do not give the same number: part of the gap comes from the thresholds chosen for the moment a car “pulls away”, part from 7.4 m and 1.8 s being medians taken separately. The order of magnitude is the same: the wave travels backwards at roughly the speed at which the queue empties.

This speed is not a quirk of one Californian motorway. Yuki Sugiyama’s team notes that about 20 km/h is the usual value measured on real motorways. It comes from two driver habits: how long each driver waits before moving off, and how much room they leave when they stop.

What one touch of the brake does: a ring of 22 cars#

In 2008 a team of Japanese physicists led by Yuki Sugiyama had 22 drivers circle a 230 m track at about 30 km/h, evenly spaced. There was no junction and no narrowing. In the picture taken three minutes later, five cars were already bunched into a queue, which moved backwards round the track at about 20 km/h.

The ring below is a model of that experiment, with 22 cars on 300 m. Each driver reacts to the car in front 0.8 s late, and the settings are chosen so the queue empties much as it did on the I-80: in the model, 1.7 s and 8 m from one departure to the next (on the I-80, 1.8 s and 7.4 m), so a wave of 17 km/h. Your car is the one in the circle. At first every driver drives perfectly evenly, so nothing happens until you brake.

Try the same thing with 14 cars on the ring. With more room between them, a 1.5 s touch of the brake dies out about 23 s after you lift your foot. With 16 or more, the same touch leaves a queue that is still there three minutes later. Traffic behaves this way in models and in experiments: below a certain density a brake light goes unnoticed; above it, the same brake light starts a queue.

Then press “Ordinary drivers”: each driver works the pedal a little unevenly, as people do. In six runs of the model, a queue formed on its own after 30–43 s, without anyone braking on purpose.

One car that does not brake for every brake light#

A team from US and French institutions repeated Sugiyama’s experiment with 21 cars on a 260 m track (published in 2018). One car was autonomous, programmed to hold a steady speed and let the gap ahead grow and shrink, instead of braking for every move of the car in front. In the best interval of the first experiment, compared with the period when everyone drove themselves, the spread of all the cars’ speeds fell by 80.8%, fuel use by 42.5%, and the number of cars passing a point on the track rose by 14.1%. In the second experiment the same car was driven by a person who simply held a steady speed: the spread of speeds fell by 49.5%. In the third, by 54.7%.

The “Drive as the buffer” button on the ring above does the same with your car. In six runs of five minutes each, your car never stopped (before, 146 stops in total), and the five cars behind it stopped 80% less often and braked hard 98% less. In the model the effect fades after a few cars: the rest of the ring stopped 12% more often. In the real experiment the effect reached the whole ring. The model does not reproduce that difference.

On the road, this is a simple habit: when you see brake lights ahead, ease off the accelerator early and let the gap in front shrink slowly, instead of closing up fast and braking. As a rule, the cars behind you will brake less; on the American track, one driver doing just that halved the spread of everyone’s speeds.

The data, the model and what they do not show#

The data. NGSIM I-80 trajectories (US Department of Transportation): eastbound, at Emeryville, California, 13 April 2005, 16:00–16:15 and 17:00–17:30, one position every tenth of a second. The speeds in the file are known to be imprecise, so the speeds here come from positions smoothed over 1.1 s. The replay above uses 1,969 trajectories between 17:00 and 17:15, at two positions a second.

Wave speed. On lanes 2 and 3, the cars’ average speed was computed in cells of 10 m by one second. For two points 80 or 100 m apart, the delay with which a slowdown reaches the rear point after the front one gives the wave speed. 55 pairs had a clear match (correlation above 0.6). Windows with fewer than five pairs were set aside, among them lane 3 between 17:00 and 17:15 and the whole 16:00–16:15 interval, when, according to the dataset’s fact sheet, congestion was only building up. Median: 18 km/h, half the values between 16.9 and 18.5 km/h.

Departures. A car “pulls away” at the last moment it is below 5 km/h before it passes 15 km/h. Each departure is paired with that of the car behind (3–15 m further back, within the next five seconds). The 14.8 km/h here is a ratio of two medians and depends on these thresholds, which is why it differs from the 18 km/h measured directly.

The wave traced in the top drawing is the line with a slope of 18 km/h along which the average lane-2 speed is below 8 km/h in most cells (98%). A car counts if it crosses that line below 5 km/h, within six seconds of it.

The model. The Intelligent Driver Model (Treiber, Hennecke and Helbing, 2000), with a 0.8 s reaction delay and a small random variation in acceleration. Settings: desired speed 50 km/h, time gap to the car in front 1.2 s, minimum gap 1.5 m, acceleration 0.8 m/s², comfortable braking 1.5 m/s². The model was chosen to come close to the I-80 departures; it does not reproduce everything, for instance the buffer car’s effect on the whole ring.

Sources

1. US Department of Transportation, Next Generation Simulation (NGSIM) Vehicle Trajectories and Supporting Data, I-80. data.transportation.gov
2. FHWA, NGSIM I-80 dataset fact sheet (FHWA-HRT-06-137): site, cameras, periods. fhwa.dot.gov
3. Sugiyama et al., “Traffic jams without bottlenecks”, New Journal of Physics 10, 033001 (2008). iopscience.iop.org
4. Stern et al., “Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments”, Transportation Research Part C 89 (2018). arxiv.org
5. Treiber, Hennecke and Helbing, “Congested traffic states in empirical observations and microscopic simulations”, Physical Review E 62 (2000). arxiv.org