I propose to develop a very simple deterministic model of traffic flow. We will begin with a pattern in which cars flow downward on a Y-shaped freeway system, so that two streams of cars are merging together. Each car will have position and velocity. Time will occur in discrete increments. In each update of time, all the cars attain a new position and velocity, and the rules for this update will be set up so that collisions are not possible. There are a variety of ways to do this, but we will begin with a simple set of rules:
First, each car always moves forward in each time increment by the amount that its velocity at the start of that increment requires. In other words, there is no acceleration or deceleration during a time increment.
Second, each car is required to reset its velocity at the end of a time increment in such a way that, even if the car in front of it stops on a dime, this will not lead to a collision. In other words, the velocity of a car multiplied by the time increment can never exceed the distance to the car in front of it.
Third, there will be a maximum velocity, i.e., a speed limit. The minimum velocity will be zero; cars are not allowed to reverse on the freeway.
Fourth, every car always travels as fast as it can, keeping to the preceding rules.
Although it is unrealistic, we will assume for now that every car is a point, i.e., has length 0, and two cars may occupy the same point at the same time, though one of them must always be thought of as being infinitesimally behind the other. In this situation, the following car would have to set its velocity to zero, in keeping with the rules about velocity and following distance. Minimum velocity is zero, which, of course, can occur in a severe traffic jam. One of the branches of the Y will have to be given right of way, so that, if two cars reach the merge point at the same time, we will know which one should be placed behind the other. To begin, there will be no upper upper or lower bound on acceleration, only on velocity. We will have to decide how cars enter into the top of the Y -- how many, how often and at what velocity. It is typical in this situation to move away from our deterministic model, using an "exponential" distribution, but we would have to modify this to account for velocities and our rules for avoiding collisions. We will probably begin by simply sending cars into the top of the Y at a constant rate each with the maximum velocity and minimum following distance, possibly staggered.
Such a model should not take too much time to develop, and then the goal will be add in realism. For example, one should assume a maximum acceleration which possibly varies with velocity. We could implement an exponential distribution to control the entry of cars into the freeway system. Putting in a maximum deceleration is a bit more tricky, given the requirement that there be no collisions, but it could be done. We could give each car a length greater than 0.
In the Twin Cities area, there is a system of "car metering", in which traffic lights are placed on the entrance ramps to the freeway system. Cars are allowed through at a regulated pace, one per green light. The delays experienced by drivers awaiting their turn has stirred a significant political controversy leading to a multimillion dollar study to determine the effectiveness of metering. We could add into our model a metering system on one or both branches of the Y, so that the cars will be forced to spread out before merging. We could compare the efficiency of metered and unmetered traffic flow.
Finally, we could try to make our road system more complicated than a simple Y. The ideal situation would be one where we could drag various standard objects (a line segment, a Y with traffic going down, a Y with traffic going up, a curve, a metering system, etc.) around on a screen to construct any desired road system. Perhaps, however, such a thing is too ambitious for an eight week project.
In any case, I hope that this is all reasonably clear. The inital model is, in fact, fairly simple, and any point of confusion should be quickly clarified in conversation.