How traffic jams form out of nothing.

Watch a busy motorway from above on a Wednesday afternoon and you will see something that, from a thousand feet, behaves like a fluid. Cars enter the frame, accelerate to cruise, drift between lanes, and exit. Most of the time the flow is laminar. Then, sometimes for no reason any passenger could ever describe, the cars compress. A wave of brake lights propagates backward through the platoon and a stationary jam of fifty cars sits where no obstacle exists. After a minute or an hour the jam dissolves at the front faster than it accretes at the back, and the wave of brake lights vanishes upstream. Nobody honked. Nothing crashed. The road is no narrower than it was a minute ago. Nothing was on fire.

This essay is about that phenomenon. It is called a phantom traffic jam, and the central claim is the one that surprises people most when they first see the simulation: phantom jams are not a malfunction of the road or the drivers. They are an emergent property of any system in which cars are dense, their reactions are noisy, and they cannot pass each other. The minimum ingredients are remarkably few. We will install them one at a time.

The figure below is a circular road. There are no junctions, no roadworks, no slow lorries, no weather. There is one lane, a fixed number of cars, and a single human imperfection: with a small probability each tick, each driver brakes for no reason. That is the entire model. Press play and watch what happens.

The behaviour you are watching has three robust features that the rest of the essay will explain. First, the jam moves backward against the direction of traffic. Second, the cars leaving the jam at the front are well separated and moving freely, so flow is not destroyed, only paused. Third, the jam is born without a cause that any single driver could identify. Each of these is a consequence of the model rather than an accident of it.

In 1992 Kai Nagel and Michael Schreckenberg published a paper that resembles, in its proportions, a small joke. Three pages of model, one page of figures, two pages of discussion. The model is a cellular automaton. The road is a one dimensional array of cells. Each cell is either empty or occupied by a car carrying an integer velocity from zero up to a fixed maximum. At every discrete timestep, every car updates in four short rules, applied in this order, to every car in parallel.

1. Acceleration. If the current velocity is below the maximum, increase it by one. Drivers want to go faster.
2. Slowing. If the gap to the next car ahead is smaller than the current velocity, reduce velocity to the size of that gap. Drivers do not want to crash.
3. Randomisation. With probability p, if the velocity is positive, reduce it by one. A glance at the radio. A daydream. A passenger speaking.
4. Movement. Each car advances by its current velocity.

Read the rules and notice what is not in them. There is no roadworks, no slow lorry, no junction, no rain, no driver typing on a phone. There is one human imperfection, rule three, and one safety constraint, rule two. What follows from those four rules, when the road is moderately full, is the phenomenon part one described.

The figure below applies the rules one at a time so the mechanism is visible. The road is a short straight segment of forty cells with six cars on it. Press each rule in order, then press next tick to repeat. After half a dozen ticks the cars settle into a pattern that you can recognise as the precursor of the rings you watched a moment ago.

The four rules are the minimum mathematical content of the model. Each by itself is innocuous. Acceleration alone is a fleet of identical drivers all flooring it. Slowing alone is a safety rule that says nobody crashes. Randomisation alone is a Bernoulli trial with no spatial structure. Movement alone is a clock. The phenomenon only appears once all four are running together at a density that is not too low. At low density the gap in rule two is rarely binding, so the rules act independently. At high density the gap is binding most of the time, and the noise from rule three propagates backward through the chain. That backward propagation is the wave you saw in part one.

Traffic engineers have measured the relationship between density and flow on every motorway with a sensor since the 1930s. The plot is universal in shape. Flow rises with density at first, reaches a maximum at a critical density, then falls. The free-flow branch on the left is the regime in which drivers do not interact with each other. The congested branch on the right is the regime in which they do. The critical density is where the maximum capacity of the road is realised, and it is also, not coincidentally, where the Nagel and Schreckenberg model begins to produce jams from noise.

The figure below generates the diagram from first principles. The simulator is run at thirty different densities, each for a long enough time that the average flow stabilises. The blue dots are the measurements. The reader can drag the vertical density line to pick a point on the curve and watch the corresponding simulation run on the small ring at the right.

Two consequences of this diagram are worth holding on to. First, more cars do not always mean more throughput. Above the critical density, adding a car removes throughput. Adding the next car removes more. The marginal contribution of a vehicle on a motorway near capacity is negative. Second, the critical density is a function of driver behaviour. Smoother drivers, modelled here as smaller p, raise the critical density. They do not eliminate the phenomenon, but they push it later, and the same road carries more vehicles before collapsing.

So far we have shown the road at one moment. To see jams as a coherent object, plot the road over time. The horizontal axis is position. The vertical axis is time, advancing downward. Each row is one tick. A bright pixel is a car. A dark pixel is empty. The result is a diagram of trajectories: each car traces a diagonal line that slopes down and to the right as the car moves forward in time.

The signature of a jam, in this diagram, is the opposite slope. A jam is a region of bright pixels that runs down and to the left. It is the boundary at which cars come to a stop and then leave again, and that boundary moves backward along the road at a characteristic speed. In the standard parameterisation the speed is approximately one fifth of a cell per tick, which in real units is about five kilometres per hour against the flow of traffic. This number is robust. It is a property of the model, not of the road.

One last observation about the diagram. The fact that the wave speed is approximately constant means jams behave like solitons. A jam born in one place propagates through the road with a fixed character, and when two jams meet they pass through each other rather than merging. Looking at the diagram is the cleanest way to convince yourself that the jam is a coherent object, not a name we put on a temporary clustering.

Given the model, three interventions are worth considering. The first is to reduce p, the rate at which drivers brake for no reason. This is the regime promised by adaptive cruise control and, eventually, by autonomous vehicles. The second is to reduce vmax, the speed limit. This is the regime enforced by overhead variable speed limit gantries, which is why those gantries appear in advance of congestion and not at the location of it. The third is the absence of perturbations, an idealisation of perfect drivers who never spontaneously brake. Each of the three can be tested in the figure below, side by side against the vanilla model under identical noise.

The first intervention, lowering p, is the strongest. Adaptive cruise control is one of the most consequential safety technologies of the last twenty years not because it prevents collisions, although it does, but because it removes the random component of human braking. Fleets of cars with adaptive cruise control have been shown empirically to absorb the perturbations that would otherwise grow into jams, and the effect appears even when only a fifth of cars on the road are equipped.

The second intervention, lowering vmax, is counterintuitive. Drivers presented with a variable speed limit ahead of congestion frequently object on the grounds that the limit is being lowered when the road is empty. The point of the limit is that the road is about to be busy. By the time it is busy, the limit is too late. The space ahead of the gantry is where the intervention has to live.

The third intervention, eliminating random braking entirely, is the limit case. Real human drivers do not achieve it. Coordinated platoons of autonomous vehicles, on a dedicated lane, approximately do. The figure suggests, and the literature confirms, that the throughput of such a lane can be roughly twice that of a human lane at the same speed limit.