**Question: **

What should the speed limit be for cars on the Lions Gate Bridge in

Vancouver, British Columbia, during rush hour traffic, in order to maximize

the flow of traffic?

**Solution:**

This is of course an “open-ended” problem. The answer depends on one’s

assumptions. Here we make some simple-minded assumptions:

(a) The source of cars to the bridge is excellent.

(b) All cars move at the same speed v and have the same length.

(c) The separation distance between each car is the same.

*(Note: How to find such simplifying assumptions. There is no clear cut answer. You just have start with some assumptions, as is true in mathematical modelling/applied mathematics, and then refine your assumptions to get a tractable problem).*

Let q be the traffic flow rate, i.e., the flux of cars, corresponding to the number of cars per hour passing a given point or the capacity of the bridge;

let be the density of cars, i.e., the number of cars per mile of road; let be the effective space occupied by each car.

equals l plus the spacing between each car.

Then, and

For a given bridge one can determine by aerial photographs and q by traffic counts. The problem now is to determine the density and find the optimal speed , maximizing

[].

**Model 1**. This model is based on radio and television advertisements in British Columbia which recommended that drivers should space out one car length for every 10 mph of speed, that is, .

corresponding to . The ratio is called headway.

Hence,

Hence, is a monotonically increasing function of v. Now,

This implies that there should be no speed limit!

For a typical Vancouver car such as a 1983 Ford Fairmont, miles. Hence the optimal flow rate is cars per hour.

Note that when mph, q is 1950 cars per hour.

Note that when mph, q is approximately 2450 cars per hour.

Since for a given situation one is able to determine q and p from simple measurements, a traffic engineer is interested in the flow vs. density curve. For Model I,

Experimentally it appears that figure will show the shape of a typical curve representing flow vs. density on a throughway.

It is observed that the optimal flow rate during rush hours for each lane of traffic on the Lions Gate Bridge is about 1600 to 1800 cars/hour.

**My suggestion — try two other models on your own. You are welcome to discuss them on the blog. **

More later,

Nalin Pithwa