Competition Introduction and Overview
  Part I: When are Traffic Lights Feasible?
  Part II: What Happens when a Light Turns Green?
  Part III: Find the Best Light Timings
  Details on The Mathematical Model Used
  Run the Computer Simulation
  Switch to graphical version (better pictures & formulas)
  Access printed version in PostScript format (requires PostScript printer)
  Go to University of Toronto Mathematics Network Home Page

Keep The Traffic Moving!

Welcome to a competition that was held in 1996 by the UNIVERSITY OF TORONTO MATHEMATICS NETWORK. Although that competition is over, you are still invited to try your hand at exploring mathematics in action in this real life situation!

The scenario

Everybody knows how frustrating it is to sit at a red light. Worse than that, a light which is red too long can cause cars to back up endlessly, leading to gridlock and other chaos. It is the job of the traffic engineer to time things so that the traffic has the least amount of waiting to do.

In this competition, you are the traffic engineer. You control the timings of the lights at the six intersections shown in the picture below. Each road has two lanes, one in each direction. The lanes are each 4 metres wide (so each road is 8 metres wide), and the spacing between the roads is as shown. For convenience we've numbered the lanes from 1 to 10, indicated their directions, and labelled the lights A through F.

            Lane   Lane  Lane   Lane  Lane   Lane                      /\
             5 | | | 6    7 | | | 8    9 | | |10       /\              ||
             ||| | |/\    ||| | |/\    ||| | |/\       || 20 m       North
 <-          \/| | |||    \/| | |||    \/| | |||       \/
Lane 1 --------     --------     --------     --------
       --------  A  --------  B  --------  C  --------
Lane 2 --------     --------     --------     --------
 ->            | | |        | | |        | | |         /\
               | | |        | | |        | | |         || 32 m  
 <-            | | |        | | |        | | |         \/
Lane 3 --------     --------     --------     --------
       --------  D  --------  E  --------  F  --------
Lane 4 --------     --------     --------     --------
 ->            | | |        | | |        | | |         /\
               | | |        | | |        | | |         || 20 m
       <------>| | |<------>| | |<------>| | |<------> \/
        72 m          72 m         72 m         72 m

You can control these lights yourself through a computer simulation available on our web site.

What You (the Traffic Engineer) Already Know

You have measured the traffic densities d[1], ..., d[10]: the number of cars per minute trying to use each lane.

You have also found that the traffic behaviour (the way a car speeds up or slows down when faced with an obstacle in front, such as a red light or another car) follows a certain mathematical model governed by three numbers c, l, and L (the model is described in the section Details on the Mathematical Model Used).

What You Need to Figure Out

This competition has three parts; we encourage you to test your mathematical skills on any or all of them! Parts I and II require you to do some serious mathematical thinking, but in Part III all you have to do is time the lights and see if you can beat the current "high scores" on the computer. So, if you're not in a serious mathematical mood right now, you can skip to part III.

To get the full details on each part you can use the links at the top and bottom of each page. Here's a brief description of them:

  1. How can you use the information you have (the thirteen numbers d[1], ..., d[10], c, l, and L) to decide whether traffic lights are even feasible, or whether a more drastic solution is required, such as widening the roads or building overpasses?

    In the detailed discussion of this part, we'll give you hints on how basic algebra can be used to come up with a simple set of inequalities that tells whether or not traffic lights would be capable of handling the traffic flow. See if you can figure out this formula. You can test your answer on the computer simulation.

  2. What happens when a long light of cars is waiting at a red light, and the line turns green? If you've ever been near the back of such a line, you've probably sat in frustration wishing the cars in front would get a move on! But is it only their slowness in reaction?

    In the detailed discussion of this part, we'll explain how ideas of calculus and " differential equations" can be used to answer the question, and challenge you to find out how long it takes for the line to get moving even if everybody's reaction time is instantaneous and there's no limit on the cars' acceleration power. You may be surprised at the answer, and less inclined to blame the cars in front of you next time you're kept waiting! Note: the mathematics in this part is quite challenging and requires a knowledge of calculus. But you can still watch what happens on the computer simulation even if you can't solve the problem mathematically.

  3. Find the best light timings for three specific situations. In the detailed discussion of this part, we give you three different situations (three different sets of numbers for d[1], ..., d[10], c, l, and L). Your job: time the traffic lights in a way that causes the least amount of delay. This part of the competition is a free-for-all: you can use any combination of mathematical reasoning or pure experimentation. The computer simulation lets you enter timing values, watch what happens, and get a score for how well your timings did. It keeps track of the high score, so see if you can beat it and get a new high score!

Use the links above or below to get more information on each of these three competition questions and on the mathematical model used, or to access the computer simulation.

Good luck!

This page last updated: April 12, 1997
Original Web Site Creator / Mathematical Content Developer: Philip Spencer
Current Network Coordinator and Contact Person: Joel Chan -

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