Whatever the traffic is doing, the jam itself moves backwards along the road at a remarkably consistent speed. The cars go forwards; the jam goes the other way, and it does so at roughly walking-pace-times-ten no matter how fast the free-flowing traffic is.
It falls straight out of two numbers. Cars sitting in a queue are packed about one vehicle length plus a small gap apart — call it 7 m. When the queue starts moving, each driver pulls away roughly one headway after the car in front — call it 1.5 s. The back of the jam therefore eats into the queue at 7 m every 1.5 s:
wave speed = (vehicle length + jam gap) ÷ start-up headway = 7 ÷ 1.5 ≈ 4.7 m/s ≈ 17 km/h, backwards.
Nothing in that depends on the speed limit, the weather or how many cars are queued. It is a property of how tightly cars pack and how quickly people react, which is why the number turns up again and again on real motorways.
It scales with what is in the queue. A jam full of lorries has a much longer packing distance, so its wave runs faster backwards. Tailgating drivers with a short start-up headway also speed the wave up, which is the opposite of the intuition that impatient driving clears a jam.
And it only describes the back of a jam that is already there. It says nothing about whether a jam will form in the first place — that is a stability question, and it is what the simulator above is actually for.
Take the two numbers the rule is built from: queued cars packed 7 m apart, and a start-up headway of 1.5 s. The back of the jam releases one car every 1.5 s and each release frees 7 m of road, so the wave runs at 7 ÷ 1.5 = 4.7 m/s — about 17 km/h, backwards, whatever the traffic ahead is doing.
Now make everyone tailgate: a 1.0 s headway. The wave becomes 7 ÷ 1.0 = 7 m/s, or 25 km/h. Impatient driving does not clear the jam; it sends its back edge upstream faster.
Both are measured in seconds and both are about following the car in front, so they get muddled constantly. They do opposite things.
Preferred gap is where you choose to sit: the steady-state distance you leave, expressed as the time it would take to cover it. It sets how much road the traffic uses, and therefore how many cars fit before the road is full.
Reaction time is delay in the feedback loop: how long after the car in front changes speed before you do anything about it. It does not change where you sit in steady traffic at all. What it changes is whether steady traffic is stable.
This is the same phenomenon as dead time in a control loop, and it behaves the same way: add enough delay to a feedback loop and a disturbance that used to die away starts to grow instead. A line of cars is a chain of such loops, each one feeding the next.
Density does it too, and independently. Worth being straight about, because it is easy to credit the wrong cause: the 22-car default here is packed tightly enough to collapse even with instantaneous drivers. Reaction time then makes it markedly worse — average speed falls by roughly half again as the delay goes from nothing to 1.4 s — but it is not what tipped it over. For the clean experiment use the Delay alone preset: 16 cars on the same road, stable for as long as you care to run it at a 0.8 s reaction, and jamming when you raise the reaction past about 1.2 s with every other setting untouched.
Which is the uncomfortable finding: the jam is not caused by a bad driver. It is caused by ordinary drivers with ordinary reflexes, and the only thing anyone can do about it is leave a bigger gap.
Because of two numbers that have nothing to do with the speed limit. Queued cars sit about 7 m apart, and each driver pulls away roughly 1.5 s after the one in front, so the back of the jam eats into the queue at 7 m every 1.5 s — about 4.7 m/s, or 17 km/h, backwards. It is a property of how tightly cars pack and how quickly people react, which is why the figure turns up on every motorway.
No, and they do opposite things. Your preferred gap is where you choose to sit in steady traffic; it sets how many cars fit on the road. Reaction time is delay in the feedback loop, and it does not change where you sit at all — it changes whether steady traffic is stable. It is dead time in a control loop, and it behaves like it.
Nobody’s. It is caused by ordinary drivers with ordinary reflexes in traffic that is dense enough, and the only thing any one of them can do about it is leave a bigger gap.
Density, on the default here: the 22-car setting is packed tightly enough to collapse even with instantaneous drivers. Reaction time then makes it markedly worse — average speed falls by roughly half again as the delay goes from nothing to 1.4 s — but it is not what caused it. Use the delay-only preset for the clean experiment.
There is no junction, obstacle or bad driver on this ring. The jam is produced by the interaction of perfectly ordinary drivers, which is why it is called a phantom jam. Sugiyama and colleagues did exactly this with 22 real cars on a 230 m circle in 2008 and got the same result within a couple of minutes.
Density and delay each cause a jam on their own. The default 22-car ring is past the stability limit on density alone: it collapses even with instant reflexes. Load the Delay alone preset for the clean experiment: 16 cars on the same road stay steady at a 0.8 s reaction, and jam when you raise it past about 1.2 s with nothing else touched.
The gap term in the model is squared, so being 20% too close pushes much harder than being 20% too far away pulls. A line of drivers therefore amplifies a slowdown on the way back but smooths an acceleration, which is why waves grow travelling upstream.
Every car drives forwards the whole time, yet the jam travels the other way: it is not an object but a pattern in where the cars happen to be, and new cars keep arriving at its back faster than old ones leave the front. Queued cars pack about 7 m apart and each pulls away about 1.5 s after the one in front, so the back of the queue eats backwards at 7 ÷ 1.5 ≈ 4.7 m/s ≈ 17 km/h, whatever the free-flow speed.
Hold everything else fixed and add cars. Below a threshold the ring settles to a steady speed forever; above it, no amount of care prevents the collapse. That threshold is what a traffic engineer means by capacity.
Increasing the preferred gap makes the traffic stable at higher density: the extra space is repaid because nobody has to stop. The instinct to close the gap is what breaks it, and closing it is why the wave you are sitting in exists.
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