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Part I: Are Traffic Lights Feasible?

Prerequisites: Basic Algebra.
Mathematical Reasoning/Computer Experimentation balance: primarily MR.

How can you use the information you have (the thirteen numbers d, ..., d, c, l, and L) to decide whether traffic lights are even feasible, or whether a more drastic solution is required, such as widening the roads or building overpasses? The following steps should help you to answer this question.

Step 1. Traffic lights are feasible only if they're able to let through as many cars per minute as are trying to get through; otherwise there will be an ever-increasing backlog.

When a light is red it lets through no cars at all. When it's green or yellow it lets through cars up to a certain maximum number M per minute (which depends on the speed limit and the way that cars try to space themselves out).

If g is the fraction of time that the light is green or yellow, then the time-averaged maximum number of cars per minute that can pass through the light is Mg (see if you can convince yourself of this). In order to avoid an ever-increasing backlog, this must not be less than the average number of cars per minute that are trying to use the lane.

Think about a light where an east-west lane with traffic density d intersects a north-south lane with density d'. If g is the fraction of time it lets through east-west traffic, then 1-g is the largest fraction of time for which it can let through north-south traffic. To accommodate all east-west traffic, one must have Mg >= d. To accommodate all north-south traffic, one must also have M(1-g) >= d'. The question is, is there any number g that will make both of those inequalities work? See if you can come up with a condition that M, d, and d' have to satisfy in order for such a g to exist.

If you do this for all six lights, you should find a set of equations or inequalities that the numbers M, d, ..., d have to satisfy in order for traffic lights to be feasible.

Step 2. Find a formula for M in terms of the numbers c, l, and L that describe traffic behaviour. To do this you will need to know the details of the mathematical model we are assuming the cars follow.

Hint: Suppose cars are travelling at constant speed v km/h (which is 50v/3 metres per minute) with spacing x metres between the cars. Then the number of cars per minute that will pass a given spot is N = 50v/(3x).

In the section Details on the Mathematical Model, we describe how v is related to x, c, l, and L. Use this to eliminate x from the formula for N (still under the assumption that you have a stream of traffic moving at constant speed v). This gives you a value for N that depends of v, c, l, and L.

Ask youself, "thinking of c, l, and L as constants, but allowing v to change, what is the largest value that N can be as different speeds v are considered?". Answering that question tells you what M is (still under the assumption that the traffic is moving at constant speed). You should have a formula now for M in terms of c, l, and L.

Finally, discuss as best you can what, if any, modifications need to be made to the formula if the stream of traffic is accelerating or decelerating.

Step 3. Combine your answers to questions 1 and 2 into a unified procedure for how to decide, if someone tells you the values of d, ..., d, c, l, and L, whether or not traffic lights are feasible. You may find it helpful to pick some values for these variables, plug them into your formulas, and predict whether or not it's possible to time the lights so as to avoid an ever-increasing backlog. Then enter them into the computer simulation and actually see if can or cannot. Do the results of your experiments confirm your predictions?