Refer to the competition introduction and overview for more information on where the above buttons will take you.

Details on the Mathematical Model

We're assuming that traffic behaves according to the following model:

Is this model realistic? Not really; it does not take into account drivers' reaction times, limits to how fast a car can brake or accelerate, and other factors. But, although it's not completely realistic, the inadequacies of the model don't make a whole lot of difference to the basic nature of the answers you get. Working with a more realistic model requires much, much harder mathematical calculations, and the final answers aren't all that much different! So, we'll stick with this simple model.

Let's be a little more precise about how this model works. Let x denote the distance in metres between two cars (measured from the back of one to the back of another), or between the back of one car and a red light up ahead.

This is indicated below:


The speed v of the car is:

c, if x >= L
between 0 and c, if l < x < L
0, if x <= l.
In the second case, l < x < L, we're assuming the car changes speed proportionally to changes in distance, so that v is given by a "linear" function of x (the graph of v as a function of x is a stratight line). This means that v = m x + b for two constants m and b. These constants can be determined from the two conditions
  1. v=0 when x=l
  2. v=c when x=L;
from these, you can solve for m and b in terms of the three fundamental numbers c, l, and L.

In summary, then, we're assuming that the speed of a car is given by the following function of the distance x to the closest obstacle in front of it:

v = c, if x >= L
v = m x + b, if l < x < L
v = 0, if x <= l.
where m and b can be expressed in terms of c, l, and L (which we leave for you to do). These three numbers, therefore, completely determine the traffic behaviour (according to our model).
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