More and more people take part in marathons these days – over 30,000 people will run the London Marathon this weekend, for instance. But it’s not just the 26 miles and 385 yards that could be a daunting prospect. “I have to admit to being completely frustrated by the congestion and for 18-19 miles was just dodging people and being held up,” one participant grumbled after the 2012 London Marathon. “I had to overtake a lot of people and ended up with bruised forearms from all the elbows,” said another.
How do such crowding problems arise, and could they be reduced? Some researchers believe that we can find the answers through a more familiar system in which jams appear – road traffic flow. Martin Treiber, of the Technical University of Dresden in Germany, has previously developed models for traffic flow, and now he has reported modifications that capture the essential details of sporting events such as marathons.
One of the first attempts to model traffic flow was made in the 1950s by James Lighthill and his collaborator Gerard Whitham of Manchester University. They considered the traffic as a kind of liquid flowing down a pipe, and looked at how the flow changes as the fluid gets denser. At first the flow rate increases as the density increases, since you simply get more stuff through in the same period of time. But if the density becomes too high, there’s a risk of blockages or jams, and the flow rate plummets.
Treiber’s model of a marathon invokes this same principle that the flow rate first increases and then decreases as the density of runners increases, thanks to an abrupt switch from free to congested flow. He assumes that there is a range of different preferred speeds for different runners, which each sustains throughout the race. With just these ingredients, Treiber can calculate the flow rate of runners, knowing the “carrying capacity” at each point on the route. For example, when the route narrows at bottlenecks, so that the maximum flow rate is lower, the model predicts how congestion might develop and spread elsewhere.
This allows Treiber to figure out how congestion might depend on the race conditions – for example, for different starting procedures. Some marathons start by letting all the runners set off at once (which means those at the back have to wait until those in front have moved forward). Others assign runners to various groups according to ability, and let them start in a series of waves.
Treiber has applied the model to the annual Rennsteig half-marathon in central Germany, which attracts around 6,000 participants. The traditional route had to be altered in 2013, because the police were no longer willing to close a road to ensure that runners could cross safely. It could pass either over a 60m wooden hiking bridge or through a tunnel. Treiber used his model to predict the likely congestion incurred in the various options. The model predicted that a mass start would risk an overload of runners if the bridge were to be used, but so, to a lesser degree, would wave starts (which the Rennsteig uses). Only by moving the starting point further back from the bridge could the danger be avoided – and even then, if some of the numbers assumed in the model were only slightly inaccurate, there was still a risk of jams at the bridge. On the other hand, no dangerous congestion seemed likely for the tunnel route. The run organisers consulted Treiber’s team, and eventually chose this option. They also adopted their recommendation for a staggered start with delays of about 150 seconds between waves.
Other mass events like cross-country skiing are more complicated to model, partly because the speed of the skiers can depend quite dramatically on the slope of the course, especially when it is uphill. Treiber built this explicitly into his model for an annual 90km race in Sweden called the Vasaloppet, which draws around 15,000 participants. His computer simulations have predicted that massive jams, delaying participants by up to 40 minutes, would form where the route has a steep uphill gradient – just as is seen in the real event. The Vasaloppet has a mass start – but Treiber says that if organisers adopted a wave start, with five-minute delays between waves, all the jams would disappear. Whether those in charge will accept this “wisdom for the crowd” remains to be seen.