Probability in single-lane traffic flow: What are the odds of "choke points" being encountered?

Let's say you have a single-lane road (single in this case meaning single lane in each direction, or what you could also call two-lane). Let's say you have a random number of vehicles on a given 10-mile stretch of road. The speed limit is 45 mph, but some people go faster and some go slower.

Is there any math that describes the likelihood that faster vehicles will overtake slower vehicles at "choke points" (points where oncoming traffic makes passing impossible)? (Assume a straight, flat road where safe passing is always allowed.) EDIT: Choke points are points where oncoming cars make it impossible to pass without the overtaking vehicle having to slow down and wait for the oncoming car (or cars) to pass.

Obviously, the answer has to be some kind of curve, because if there is only one vehicle on the road the probability is 0 that a choke point will be encountered, and if the road is completely filled with vehicles in both directions (or even one) the probability is 1 that a choke point will be encountered at any given time.

Sorry if this is a stupid question. But it seems like there should be a way to express this mathematically.

EDIT 2: Limiting the scope of the problem

Thanks to Mike Spivey, and I hope this helps narrow the problem so that may be in some way answerable. If not, I'll just declare defeat and retire from the field.

Let's say that we have two cars going in one direction: a slow vehicle (Vehicle A) traveling at 30mph and a faster vehicle (Vehicle B) traveling at 60mph. At the start of the problem the A is somewhere between mile 2 and mile 4 on the 10-mile road, and B is at mile 0. Additionally, there will be four oncoming vehicles randomly distributed, and these vehicles are randomly traveling at 30 or 60mph. The overtaking vehicle requires .5 miles of "clear" space to pass safely if the oncoming vehicle is traveling at 60mph, and .25 if it is traveling at 30mph.

Assume further that the car lengths do not matter, but that a "choke point" (or bottleneck, if you prefer) is not reached until B gets within .05 miles of the rear bumper of A, at which point B has to pass A or decelerate. We will call the latter condition a "deceleration event," (DE) and I would like to be able to calculate the probability that at least one DE will occur for Vehicle A along the 10-mile route.

Perhaps this clarification is still insufficient to render the problem solvable, but I appreciate anyone taking the time to consider it.

Solution 1:

Your question is an interesting one, but I think the reason you haven't received an answer to it yet is that it's ill-constrained. That's not a criticism but a technical term. Basically it means that the problem isn't well-defined enough for there to be a clear, distinct answer. (This is what the second part of Rahul Narain's comment is getting at.) To borrow from the Math Overflow FAQ: "The site works best for well-defined questions: math questions that actually have a specific answer."

Instead, your question would be great in my mathematical modeling course. I would pose it and then ask the students what assumptions would need to be made in order to get a well-defined mathematical model, or what aspects of the problem would be parameters to the model that we could vary and then see how the solution changes (such as those Rahul Narain mentions), or what existing mathematical tools we could use to help understand traffic behavior (such as queuing theory, as Yuval Filmus mentions). An answer to the question would then require some student taking this on as an extended assignment. He or she would have to make and justify assumptions, create a model (or two or three, which might require some programming), testing it (or them), refine the model(s), vary parameters, and then interpret the output from the model(s). There probably wouldn't be a single answer but a range of answers along the lines of "Based on these assumptions our model says this."

(I've actually just described the process of mathematical modeling.)

Another aspect of your question is that traffic problems are hard. They frequently show up as problems in the Mathematical Contest in Modeling (see, for example, the 2009 and 2005 contests).

Traffic problems are an active area of research, too, because there's a lot we don't understand about the way traffic behaves. There's not even a consensus about the best way to go about modeling traffic. Fluid flow models seem to be the most popular, but some people use queuing, and I've also seen discrete dynamical system and even cellular automata models.

If you're interested in learning more about research on traffic, you could also check out this article ("Traffic jam mystery solved by mathematicians") or the book Mathematical Theories of Traffic Flow.