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Millions of people will take to the roads this holiday season, only to end up spending frustrating hours sitting in traffic jams. Congestion costs drivers time, fuel and patience – while also increasing pollution and placing huge pressure on transport networks.

If you’ve ever found yourself staring enviously at the lane next to you, convinced it’s moving faster, you’re not alone. Most of us instinctively believe that changing lanes will get us home sooner. But mathematics suggests this intuition is usually wrong.

As an applied mathematician, much of my research is driven by one question: how can we predict and control the behaviour of complex systems operating under uncertainty? And this can apply as much to holiday traffic as it does to emerging quantum technologies.

We tend to think traffic jams need a trigger: an accident, roadworks or a lane closure. Yet some of the most frustrating queues have no obvious cause at all. These are known as phantom traffic jams—stop-start waves that travel backwards through traffic even though every vehicle is moving forwards.

In 2008, Japanese researchers asked 22 drivers to drive around a circular track while maintaining a constant speed and safe distance from the vehicle in front. Within minutes, tiny and unavoidable differences in braking and reaction times grew into a stop-start wave that travelled continuously around the circuit, despite there being no obstacle anywhere on the road.

The explanation is surprisingly simple. One driver brakes slightly more than necessary. The driver behind reacts a little later and brakes a little harder. The next driver does the same.

In this way, the initial tiny disturbance grows as it travels backwards until drivers hundreds of metres behind are forced to stop completely, even though nobody can identify what caused the queue in the first place.

The mathematics behind traffic

Rather than modelling every driver individually, mathematicians treat traffic as a continuous flow – borrowing ideas from fluid dynamics, where the movement of vehicles is analysed much like the flow of water through a pipe.

One of the simplest relationships in traffic flow theory is q = ρv, where q is the traffic flow (the number of vehicles passing a point each hour), ρ is the traffic density (amount of cars on the road), and v is their average speed.

This deceptively simple equation explains a counterintuitive phenomenon. Initially, adding more vehicles obviously increases the overall traffic flow, as more vehicles pass along the road.

But once the road becomes too crowded, everyone is forced to slow down. Eventually, this reduction in speed outweighs the increase in number of vehicles such that the overall flow is reduced.

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