A stop-sign vs. a stoplight; when does each make sense? (Part 2, Math modeling)

I know it’s been a while since we visited this, but quick recap:

We want to figure out when it is appropriate to have a stop-sign at an intersection versus a stoplight. In Part 1, we made some assumptions about stop-sign wait time modeling and developed an equation that gave us the total wait time as the dependent variable and the number of cars as the dependent. In this post, we’ll try to model the total wait time based on the number of cars at a stop light.

Now, at a stoplight two of the incoming lanes will be green and the other two red. So any approaching car would have a 50/50 chance of winding up at a red light (ASSUMING NO GAP). Now, if you DO manage to hit a green, you’re golden. The wait time is zero. But if you DO hit a red you’re stuck for 60 seconds (note: could also be 30, or 45, or whatever). And even when you start up again, you’ll essentially be proceeding through like it is a stop sign. Maybe slightly longer if there are more cars. According to our tests in Part 1, it looks like the stop-to-full-speed time is about 15 seconds. Let’s summarize our (my) assumptions:

Assumption #4: This is an equal opportunity intersection. That is, each direction (N,S,E,W) sees 25% of the traffic. So half the traffic will be waiting at any given time, while half will have a wait time of zero.

Assumption #5: Stop-to-full-speed time = 15 seconds per car.

Assumption #6: The stop-light has a stop-time of 60 seconds.

Assumption #7: It’ll take about 2 seconds for each car to get through the intersection. Rather, 2 extra seconds that we wouldn’t have used had there not been any light there.

Total wait time for all cars = 0*1/2 x + (60+15+2)*1/2 x

And if there is in fact a three-second delay in between green lights?

Total wait time for all cars = 0*1/2 x + (60+15+2+3)*1/2 x = 40x

So our equation for the total wait time of all cars at a stop-light (with a 60 second timer) looks like this:

y=40x

Super. So if we compare this model equation against the model equation from Part 1, we get this graph.

So somewhere between 2 and 3 cars being at an intersection is the point at which it becomes beneficial to have a stop-light instead of a stop sign according to our model.

But there’s a bit of a problem: the units (# of cars). “Number of cars” isn’t very descriptive. If we’re evaluating this via traffic, we’ll need something like “number of cars per minute”. And if that “number of cars per minute” exceeds the individual wait time of the cars at the intersection the total wait time will increase exponentially.

So, let’s switch our independent variable in our models from “number of cars” to “number of cars per minute.” So our axes for our graph will look something like this.

Once again, I’m going to give another intermission at this point. Partially because I want to let you ruminate on this quandary, and partially because I need to think about how to model this. Hopefully the intermission between Part 2 and Part 3 of this post won’t be as long as the one between Part 1 and Part 2. Be sure to give feedback/corrections/suggestions in the comments!

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