Why you shouldn't change lanes when stuck in a traffic jam? Mathematicians explain

It seems that traffic jams always have some obvious cause: an accident, road repairs, or a narrowing of the carriageway. However, as Randi Hertzell, Associate Professor of Applied Mathematics at the University of Warwick, notes in an article for The Conversation, this is not always the case. Quite often, traffic jams literally arise out of nowhere — without any obstacles ahead.

Scientists call such traffic jams phantom jams. These are waves of deceleration and acceleration that propagate backward through the stream of cars, even though all cars continue to move only forward.

In 2008, Japanese researchers conducted a simple experiment. Twenty-two drivers drove on a circular track, trying to maintain a consistent speed and safe distance. There were no obstacles on the road.

Nevertheless, after just a few minutes, the flow began to alternately speed up and almost completely stop. The reason was very small differences in people's reactions: one driver pressed the brake a little harder than necessary, the next reacted with a slight delay and braked even more sharply, then the third did the same.

Thus, a small disturbance gradually amplified and propagated backward through the flow until drivers hundreds of meters behind were forced to come to a complete stop, although none of them understood how it all began.

The Mathematics of Traffic

Instead of modeling the behavior of each driver individually, mathematicians view traffic as a continuous flow. They borrow ideas from hydrodynamics (fluid mechanics), where the movement of vehicles is analyzed similarly to how water flows through a pipe.

For this, a simple relationship is used between the number of cars (ρ), their average speed (v), and the road's capacity (q=ρv).

Initially, everything seems logical: the more cars on the road, the more pass through a certain section. But only up to a certain point. When the traffic flow becomes too dense, drivers are forced to slow down. As a result, the overall capacity of the road no longer grows but begins to decrease.

The equation shows that there is a certain optimal traffic density that maximizes the number of cars passing through the road per hour. If this point is passed, each new car only reduces the road's efficiency — and increases the time required for everyone to reach their destination.

This same mathematical model explains why constant lane changes rarely bring benefits.

Changing lanes creates a small disturbance that adjacent drivers must react to. If many drivers behave similarly, these disturbances accumulate and increase the likelihood of phantom traffic jams. As a result, what seems like a reasonable decision to one driver ultimately worsens traffic conditions for everyone else.

In 2018, scientists repeated the Japanese experiment, but with one important change. One of the cars, previously driven by a human, was replaced by a self-driving car programmed to accelerate and decelerate as smoothly as possible.

The result was unusual: this single vehicle (less than 5% of the total traffic) was enough to smooth out the stop-and-go wave across the entire circle, improve overall flow, and reduce fuel consumption for every driver in the group.

So how should one behave?

As the author notes, today applied mathematicians are developing general mathematical models that can be used to create intelligent transportation systems. Their task is to ensure that cars, exchanging data with each other and with road infrastructure, act in concert, helping to improve traffic flow.

For this, intelligent systems will analyze data from cameras, sensors, and connected cars, predict the occurrence of traffic jams, and determine how the road network should react — by changing traffic light operations, introducing adaptive speed limits, suggesting alternative routes, or coordinating the movement of self-driving cars.

Meanwhile, the author advises adhering to three simple rules that genuinely help make traffic smoother:

• maintain a safe distance;
• accelerate and brake as smoothly as possible;
• do not succumb to the temptation to constantly switch to the adjacent lane in hopes of gaining a few seconds.

According to mathematical models, it's not aggressive driving that helps you get there faster, but rather behavior that supports the stability of the entire traffic flow.

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