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 transportation 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.
We tend to think traffic jams need a trigger: an accident, roadwork 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 backward through traffic even though every vehicle is moving forward.
In 2008, 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 traveled 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 backward until drivers hundreds of meters (hundreds of yards) behind are forced to stop completely, even though nobody can identify what caused the queue in the first place.
Rather than modeling every driver individually, mathematicians treat traffic as a continuous flow — borrowing ideas from fluid dynamics, where the movement of vehicles is analyzed much like the flow of water through a pipe.
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 the number of vehicles, reducing the overall flow.
The equation shows there is an optimal traffic density that maximizes the number of vehicles passing along the road each hour. Beyond that point, adding more cars reduces the efficiency of the road — and increases the time it takes everyone to get to their destination.
The same mathematics explains why constantly changing lanes is rarely worthwhile.
A lane change creates a small disturbance that neighboring drivers must react to. If many drivers behave in the same way, these disturbances accumulate and increase the likelihood of traffic waves. So, what feels like a clever decision for one driver can ultimately make conditions worse for everyone.
Perhaps the most striking demonstration came in 2018, when researchers repeated the 2008 circular-track experiment with one key difference. They replaced one of the human-driven vehicles with a single autonomous car programmed to accelerate and brake smoothly.
Remarkably, that one vehicle — less than 5% of the traffic — was enough to dampen the stop-start wave around the entire circuit, improving traffic flow and reducing fuel consumption for every driver involved.
Traffic congestion is unlikely ever to disappear completely. Population growth, increasing travel demands and the unpredictability of human behavior will always put pressure on our roads.
But mathematics is helping us move away from reacting to congestion toward preventing it.
Applied mathematics shows the fastest way to your destination isn't to drive more aggressively. It's to help keep the entire system stable.