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Stuck in traffic? Maths can get you on your way

Mathematics may not be the first thing your mind turns to when you are caught in a traffic jam. Yet mathematics holds the key to understanding how traffic congestion develops, and how to prevent it. Perhaps…

An expanding population means more cars on the road than ever before. So how can maths help ease traffic congestion? Mr. T in DC

Mathematics may not be the first thing your mind turns to when you are caught in a traffic jam. Yet mathematics holds the key to understanding how traffic congestion develops, and how to prevent it.

Perhaps one of the best known (and most surprising) mathematical results concerning how traffic flows around a network is Braess’s paradox. In the context of a road network, this is the seemingly counter-intuitive result that even without an increase in traffic, building a new road can actually make every single journey slower.

This can arise because the distribution of cars in a network is determined by the individual decisions of many drivers, each acting to reduce their own personal travel time instead of working as a group to reduce travelling times overall. Such behaviour often results in suboptimal use of the road network.

In fact, closing roads can even improve traffic congestion.

This highlights that there is a lot of very interesting, challenging, and important mathematics that is involved in efficiently transporting human beings around a modern city.

For example, how should public transport be scheduled to minimise people’s travel times? How should traffic signals be operated to reduce congestion? When and where should we build new roads?

Maths holds the answers


As populations continue to increase, tackling the transportation problem becomes ever more challenging. Mathematics has recently yielded some significant wins, however.

For example, mathematical control of the traffic lights at the entrances to Melbourne’s M1 freeway has already increased travel speeds on the M1 by 25% during the morning peak hour. And this was achieved without the cost of building extra lanes.

As regular commuters know all too well however, significant challenges remain. The term “rush hour” now seems quaintly old-fashioned; morning traffic congestion in Melbourne lasts from 6.30am until 9.30am.

The annual cost of congestion to Victoria is estimated to rise from A$3 billion to A$6 billion by 2020. In addition to these economic costs, there is also the negative impact congestion has on the environment, and on people’s quality of life.

Luckily, there are researchers around the world working on these issues.

A new field: jamology

One of these researchers is Katsuhiro Nishinari from the University of Tokyo. Nishinari played a pioneering role in trying to understand the fundamental behaviour of traffic flows from the perspective of theoretical physics and coined the term “jamology”.

His recent work focuses on studying flows of “self-driven particles”. Self-driven particles are individual, autonomous agents who each follow simple rules, governing their behaviour. Understanding the kinds of behaviour that can emerge from large collections of self-driven particles is an active area of theoretical physics, which can be applied from ants to motorists.

It has been found that there are many common features in the way jams form in flows of such self-driven particles, regardless of whether the particles are insects or people. To physicists, the onset of jamming is an example of a “phase transition”, similar in many ways to the phase transition that a pot of water undergoes when it turns to steam.

You can watch a presentation on jamology by Nishinari in the video below.

Dive into a carpool

On a more immediately practical level, much recent effort within the mathematical community has been expended on improving the efficiency of the transport networks we currently have. Breakthroughs in this direction are due in part to technological advances, which allow detailed information on traffic conditions to be constantly collected and processed.

Even more crucial, however, has been the development of improved mathematical models and algorithms to process and harness the new data.

London Permaculture

One novel idea gaining traction is real-time collaborative transport, which is essentially carpooling for the 21st century.

Using smartphone technologies, an app can efficiently match passengers with vehicles using optimisation algorithms and smart user interfaces. Platforms to accommodate real-time ride-sharing are currently being developed at the University of Melbourne by Stephan Winter and collaborators.

The vehicles used in such schemes could be private cars, but they could also be buses that are routed adaptively, to make sure there’s a bus when and where you need it. A group led by Monash University’s Mark Wallace is currently developing scheduling algorithms capable of controlling such adaptive bus schemes.

An even broader vision would be to schedule all transport. If drivers notified the transport system each time they started a journey, it could then schedule road use to balance out traffic across the road system and minimise congestion. Simulations suggest even with a small percentage of drivers using the system, users could halve their travel times.

So the next time you find yourself in bumper-to-bumper traffic, instead of leaning on your horn, try pondering some mathematics. Not only will it save your sanity, it could also save you a lot of time.

Mark Wallace will be presenting an overview of many of above topics at a free public lecture on Tuesday June 18 2013: Cheap solutions to the transport problem from 5:30pm-7pm in Theatre S3, Building 25, of Monash University’s Clayton campus.

The lecture forms part of an international program dedicated to the Mathematics of Planet Earth.

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21 Comments sorted by

  1. Harrison Morrow


    Any chance the lecture will be online somewhere for those of us who cannot make it?

    1. Tim Garoni

      Senior Lecturer / ARC Future Fellow, School of Mathematical Sciences at Monash University

      In reply to Harrison Morrow

      Yes indeed, Martin Burd has done some very interesting work with Majid Sarvi et al on pedestrian dynamics.

  2. Aden Date

    Service Learning Coordinator at University of Western Australia

    While I'm sure this wasn't the author's intentions, the existential implications of comparing people to particles is too good to ignore. Suddenly, comparisons to worker ants and rats in a maze seem flattering.

    In seriousness, it would be absolutely wonderful to see this sort of technology applied to the public transport network - I think it would be quite achievable to pilot Smartphone technology with late-night buses in nightlife districts.

    1. Phil Dolan


      In reply to Aden Date

      'the existential implications of comparing people to particles is too good to ignore. Suddenly, comparisons to worker ants and rats in a maze seem flattering.'

      Yes, especially after seeing all the recent political polling.

  3. Riddley Walker


    I was on my way to the talk but I got held up by traffic.

  4. Riddley Walker


    It's a good article and some interesting links, thanks.

    However there is another intrinsic mathematical problem with traffic as I describe here:

    4 million people in Melbourne and the road system is now at, or nearly at, full capacity.
    Projected numbers – 8 million.
    24 people driving cars to work.

    24 people driving cars to work.

    But doubling the road space will not prevent gridlock.

    This is because each person driving on the road takes up 20 – 200 square metres of space. So another…

    Read more
    1. Riddley Walker


      In reply to Riddley Walker

      sorry, the lines
      24 people driving cars to work. should have been deleted they are from something else...

    2. Riddley Walker


      In reply to Riddley Walker

      This is not so clear on the page expressed as numbers, but if you graph it out the car "line" goes through the roof pretty quickly, the others grow much more slowly.

  5. Adam Butler

    Engineer and Data Analyst

    Nice work Tim, any chance you can speak with the boffins at the NSW government and let them know about Braess' Paradox?
    Though I fear that even a rudimentary explanation would go over their asphalt loving heads. Perhaps its the fumes.

  6. Riddley Walker


    Also very interesting is Miranda Blogg's powerpoint on "Smart traffic" systems.

    travel speeds increased by 25.9% from 48.9 km to
    66 km/h in peak hour traffic on the M1 in Melbourne.

    On-ramp throughput increased by 9% in peak hour traffic

    However one only needs to read the comments sections in the Age on any traffic related topic to see that drivers would see this kind of management not as an increase in EVERYBODY'S speeds and improved travel time, but as a REDUCTION in their…

    Read more
  7. Geoff Russell

    Computer Programmer, Author

    It's always seemed obvious to me that a bunch of drivers intent at optimising their own trips can (and in practice often will) lead to suboptimal flows ... but that's after a maths degree and 30 years on the scheduling/rostering/timetabling software business.

    So why do some many economists/politicians/political pundits these days insist on the optimality of an unfettered free market for everything from climate change to water use?

  8. Alan O'Neill

    Freelance Consultant / Inventor at freelance consultant

    Nice article thanks. The flaw in the adaptive car pooling, public transport etc is the verification of the transport request..the though of so many drunks with just as many apps having fun with adaption doesnt bear thinking about.

    The critical issues in transport as in packet routing is keeping the system and buffers at low enough utilization to prevent the phase change occuring. Road layouts (flat, hierarchies, branches, power forms etc), types of intersections and of course intermittent disruptions…

    Read more
    1. Riddley Walker


      In reply to Alan O'Neill

      All good, but you still come back to the fundamental problem of the size of a car compared to the number of people moved.

      Inefficiency is built into the system, unless you get cars the size of bicycles. If oyu care to, have a look at my post above, re this.

      Well there are motorbikes and scooters, but I don't spend my time thinking about them.

  9. Doug Rankin


    All this may be true to a point. However demand over time will always exceed the resource available (unless priced out of reach). So in effect making the transport system more efficient will mean people travel further thus defeating the point in the first place.

  10. Graeme Jensen

    Postgraduate Student at The University of Melbourne

    A fascinating article! Would like to see more like this!

  11. Iain Cummings

    logged in via Facebook

    Another bit of very simple maths that you would think motorists could easily understand but many don't. - Every bicycle is one less car causing traffic congestion. It always astounds me how a minority of motorists get annoyed at cyclists and accuse them of holding up traffic, when the amount of time a car gets stuck behind a bike is usually very small but the time a car gets stuck behind queues of other cars is far greater.

    Another simple bit of maths, the average speed of cars in many areas is…

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