Elementary Calculus and traffic flow problem

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 $\rho$ be the density of cars, i.e., the number of cars per mile of road; let $l^{*}$ be the effective space occupied by each car.

$l^{*}$ equals l plus the spacing between each car.

Then, $\rho = 1/l^{*}$ and $q=\rho v$

For a given bridge one can determine $\rho$ by aerial photographs and q by traffic counts. The problem now is to determine the density $\rho = \rho (v)$ and find the optimal speed $v=v_{opt}$ , maximizing

$q(v)$ [ $q=q(v_{opt})$ ].

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, $l^{*}=l[1+v/10]$ .

corresponding to $\rho = \frac {1}{l(1+v/10)}$ . The ratio $l^{*}/v$ is called headway.

Hence, $q(v)=\frac {v}{l(1+v/10)}$

Hence, $q(v)$ is a monotonically increasing function of v. Now, $\lim_{v \rightarrow +\infty}q/v=10/l$

This implies that there should be no speed limit!

For a typical Vancouver car such as a 1983 Ford Fairmont, $l \approx 1/325$ miles. Hence the optimal flow rate is $q_{opt}=\lim_{v \rightarrow +\infty}=3250$ cars per hour.

Note that when $v=15$ mph, q is 1950 cars per hour.

Note that when $v=30$ 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, $q=q(\rho)=\frac {10}{l}-10\rho$

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

For the love of Maths: the Queen of all Sciences

Elementary Calculus and traffic flow problem

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