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Details on the Mathematical Model

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

As long as there's no obstruction (another car or a red light) in front of a car, it will travel at the speed limit of c kilometres per hour.
If a car gets within a distance of L metres from an obstruction (where, to avoid having to think about the car length as a separate factor, we measure distances from the back of the car), it will reduce its speed proportionally to that distance.
If a car gets within a distance of l metres from an obstruction, it will stop altogether.

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:

           -----     |
          | car |    X red light, or another car  (Direction of travel: -->) 
           -----     | 
          |          |
          |<-------->|
          |     x    |

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

v=0 when x=l
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