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.

# 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
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).
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