Overview | Part I | Part II | Part III | Model | Computer Simulation | Graphical Version | PostScript version | U of T Math Network Home

Refer to the competition introduction and overview for more information on where the above links will take you. If your browser supports graphics, you should try the graphical version for better pictures and formulas.

Part III: Find the Best Light Timings

Prerequisites: None.
Mathematical Reasoning/Computer Experimentation balance: primarily CE.

Below are 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!

The Three Scenarios

In each scenario, c=40, l=4, and L=20; the only difference is in the values for the densities d[1], ..., d[10]: the number of cars per minute trying to use each lane.

Scenario 1. (d[1], ..., d[10]) = (10, 5, 5, 10, 15, 10, 8, 8, 5, 10)

Scenario 2. (d[1], ..., d[10]) = (5, 15, 2, 15, 2, 10, 1, 8, 10, 5)

Scenario 3. (d[1], ..., d[10]) = (7, 20, 15, 15, 12, 6, 10, 10, 2, 12)

How to Time the Lights

To time the lights, you need to pick, for each light, two numbers: how many seconds it lets east-west traffic through, and how many seconds it lets north-south traffic through. Let's call these numbers Aew and Ans for light A, Bew and Bns for light B, and so on.

You might also get more efficient results by making the lights turn green at different times, so that a lane with heavy traffic can encounter all green lights (for instance, making the second light in the lane turn green just before cars released by the first light reach it).

The scenario begins just as light A starts its east-west green cycle. To stagger the lights, you can specify how far into the scenario lights B through F begin their east-west green cycle. Let's call these numbers B0 through F0. For example, setting B0 = 15 means that light B will begin an east-west green cycle 15 seconds after the start of the scenario.

That gives a total of 17 numbers: Aew, Ans, ..., Few, Fns, B0, ..., F0. Your job is to choose values of these numbers which will lead to the smallest average waiting time for cars over a 10-minute period, for the scenario you chose.

You may want to refer back to the map in the introduction to see where each of the ten lanes is located relative to the six lights, think about the density values given, and try to think about what kind of light timings might be most efficient for the given density values.

To try out your timing values on the computer simulation, go to the "Predefined Scenarios" section, enter your timing values, and select "Start Simulation". (The yellow stage of the light is set automatically at 5 seconds. For instance, if you entered "Aew= 30", east-west traffic would see light A green for 25 seconds and yellow for 5 seconds).

You will see a score telling you how efficient your timings were. You will also see the best score that anybody else has achieved; see if you can beat that high score!

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

Overview | Part I | Part II | Part III | Model | Computer Simulation | Graphical Version | PostScript version | U of T Math Network Home