Photo by Nabeel Syed on Unsplash.

Most people can relate to (or feel) the frustration caused by being stuck in traffic, waiting in a queue to board a plane, or circling the parking lot to find a space. Routes that could take 30 minutes can turn into hours, congested aisles of passengers cause bottlenecks, or while on your third-round around the parking lot you see someone behind you take the only space. In this post, I share some of the interesting math behind common jams you might find yourself in.

In her article, Can a city ever be traffic jam-free?, Katia Moskvitch highlights the environmental, health, and economic implications of traffic jams.

“Jams are not only frustrating, they are also a major contributor to air pollution, and that’s bad not just for our climate, but everybody’s health too. According to researchers at the Harvard Center for Risk Analysis, congestion in the 83 largest urban areas in the United States caused more than 2,200 premature deaths in 2010 and added $18bn to public health costs. Then there is the economic cost of lost hours (both work and leisure) and delayed shipments. Drivers in the 10 most-congested cities in the United States sit around 42 hours in traffic jams every year, wasting more than $121bn in time and fuel while doing so.”- Katia Moskvitch

With such high implications, you can see why traffic modeling has become a big part of applied mathematics research. From the same article, I loved this quote by Gabor Orosz (University of Michigan) which illustrates how traffic flows can be understood through analogies ( such  as fluid and gas flow, to the movement of birds and skiers) but still, “although such analogies may help scientists to gain some understanding, it is becoming more and more obvious that traffic flows like no other flow in the Newtonian universe”. I became more curious about the math behind traffic modeling after reading Mathematicians have solved traffic jams, and they’re begging cities to listen by Arianne Cohen. This article summarizes some of the key points of the work by Alexander Krylatov  and Victor Zakharov (St. Petersburg University) whose research tackles traffic modeling from an optimization perspective. Along with Tero Tuovinen, they are also authors of the book Optimization Models and Methods for Equilibrium Traffic Assignment which gives new approaches,  algorithms, methods, prospective implementations developed by the authors on the problem of traffic assignment.  Cohen highlights that four ideas that could reduce traffic jams are the following,

1. All drivers need to be on the same navigation system. Cars can only be efficiently rerouted if instructions come from one center hub. One navigation system rerouting some drivers does not solve traffic jams.
2. Parking bans. Many urban roads are too narrow and cannot be physically widened. Traffic-flow models can indicate where parking spots should be turned into lanes.
3. Green lanes. For cities that want to increase electric car use, special lanes should be created for electric cars, providing an incentive for their use.
4. Digital twins. Traffic demands and available infrastructure can only be balanced with digital modeling that creates an entire “twin” of existing roadways. The software will be “an extremely useful thought tool in the hands of transport engineers.”

After reading the article, I was curious to see if other perspectives on these matters were out there. In response to Cohen’s article, Daniel Herriges writes that human behavior is a strong factor in traffic congestion that is difficult (if not impossible) to account for with models.

“As long as we build a growing city around roads for cars, it’s a pretty sure bet that people in their cars are going to find ways to fill up those roads. We can’t build or network-engineer our way out of congestion, but we can bankrupt ourselves trying. There’s a better way to deal with traffic—and “deal with” does not mean “solve.” It is to make our places resilient to congestion, so that if and when it happens, it doesn’t destroy our quality of life. This means 15-minute neighborhoods: more destinations within walking distance of home. It means a range of ways to get around so nobody is forced into just one option, and a well-connected street network so there are many paths from A to B.”  – Daniel Herriges

The two perspectives are fascinating! This is not the first time that traffic models have appeared around the internet as the solution to traffic jams. Many researchers have tackled versions of these questions using different areas of math. For example, back in 2007  “Traffic jam mystery solved by mathematicians”.

In Traffic Modelling: Is Beating Traffic a Zero-Sum Game? Paul Sobocinski asks if self-driving cars that stick to one lane lead to less time on the road than humans switching lanes? He finds through simulations that opportunistic lane changing (i.e. weaving through lanes of traffic to shorten a commute) is not a zero-sum game. In fact,

“Opportunistic lane changing can benefit all drivers on the road if exercised judiciously. This means not changing lanes too frequently (i.e. adhering to a reasonable minimum time in lane), and only changing lanes if it saves a significant amount of time (i.e. the time saved in the new lane is 90% or higher). What do the results tell us about how to be a better driver? To state it simply: Be patient. Change lanes, but not frivolously. Everybody wins. Experienced drivers will likely not find this conclusion surprising.” – Paul Sobocinski

Following the same spirit, Jenna Marshall explains in Where to park your car, according to math the research of physicists Paul Krapivsky (Boston University) and Sidney Redner (Santa Fe Institute) which ordinary differential equations and simulations to find the best parking space (i.e. the one that lets you spend the least amount of time in the lot). As conveniently shown in the video below, they consider three strategies: meek (i.e. grabs the first space available), opportunistic (i.e. gambles on finding a space right next to the entrance), and prudent (i.e. drives past the first available space, betting finding another other space further in).

So, what is the answer? Being more prudent. However, the authors also acknowledge the limitations of their work.

“If you really want to be an engineer you have to take into account how fast people are driving, the actual designs of the parking lot and spaces — all these things,” he remarks. “Once you start being completely realistic, [every parking situation is different] and you lose the possibility of explaining anything.”- Sidney Redner

Finally, if you are a frequent flier you may have wondered about the best way to board an airplane. In Mathematician crunches the numbers to find most efficient way to board a plane, CBC radio interviews Eitan Bachmat whose work with Rami Pugatch (Ben-Gurion University), Sveinung Erland, Vidar Frette (Western Norway University), and Jevgenijs Kaupužs (University of Liepaja) tackles the airplane boarding policies using a Lorentzian-geometry-based analysis. As explained by Bachman,

“In our latest studies, we’ve been looking at random boarding versus if you have two groups of people — some which are slower and some which are faster. For example, people without luggage — they’re supposed to be the fast group. And people with luggage, the slow group. A lot of the times it happens they board first people who have children and need assistance. That would be a slow group.So, if you have a fast and slow group, what we found is that you should board the slow passengers first, which is kind of counterintuitive and surprising, I think.”- Eitan Bachmat

When asked about the mathematics behind this project his layman explanation what very insightful!

“OK. So I’ll try my best to keep it really simple. So the same math can describe very different things. I can say three plus three equals six, and three apples plus three apples equals six apples. Or, it could be three houses plus three houses equals six houses. Apples and houses have nothing in common. But, sort of, the math that describes the situation is the same. And what turned out, and that was very surprising, is that when you and 200, or 300, other people board the airplane, in terms of the mathematics, you’re doing a quite complicated computation in relativity theory about the aging of some free-falling particle and some model of the universe.” –  – Eitan Bachmat

Next time you are in a jam, you can rest easy knowing that a lot of cool mathematics is happening behind the scene. Do you have suggestions of topics or blogs you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter (@MissVRiveraQ)