The waiting time paradox: why is my bus always late? (2018)

This seems more to say that the average time of all passengers waiting will be close to the interval, but that the average time for any individual in a given stop will be closer to half? (Similarly, if you are discussing the longest time you will wait throughout the day and you have multiple stops you have to wait at, it will drift up to the the interval time.)

That is, they sound like similar questions, but they are not. How long can one random person expect to wait at a stop is different from how long a population will wait at a given spot. In large because a person can only arrive at a single time in the waiting interval, but more passengers become less likely the closer to departure time.

(I realize I didn't word all of this as a question, but I am not asserting I'm correct here. Genuinely curious if I understand correctly.)

It's not about population vs. individual. The underlying principle is the assumption that we (the people taking the bus) arrive at a random time. And we're more likely to arrive in a shit interval (because they're longer) than in a lucky interval (because they're shorter).

Here's a trivial example.

Buses are supposed to arrive at a 10 minute interval: 12:00, 12:10 and 12:20. But today the second bus arrives a bit early, at 12:07. So they arrive at 12:00, 12:07 and 12:20. We arrive at the bus stop at a random moment >12:00 and <= 12:20.

If we arrive in the interval 12:00-12:07, our average waiting time will be 3.5 minutes. What's the chance that we do arrive in this interval? 7 mins/20 mins.

If we arrive in the interval 12:07-12:20, our average waiting time will be 6.5 minutes. What's the chance that we do arrive in this interval? 13 mins/20 mins.

So our expected waiting time is not 5 minutes but 3.5 * 7/20 + 6.5 * 13/20 = 5.45.

Basically "the shit intervals are longer so we're more likely to arrive in them. the lucky intervals are shorter so we're less likely to arrive in them.". If we arrive at a random time, which is the core assumption here.

Now you might say "but 5.45 doesn't feel close to 2N". And that's where the other assumption that probably does not reflect reality comes in - the bus arrival times are simulated as uniform random numbers. I mean, it depends on where you live, haha. But it's pretty much a worst case scenario, so in reality it's not as bad. Which the writer shows using the real-world data.

Nevertheless, unless it's the Japanese subway which always arrives exactly on time, it's always going to be bigger than 2N.

And what if we don't arrive at a random time, but arrive according to some pattern guided by the bus schedule? That change everything.

Still, it's actually pretty common to arrive at a random time, and buses (and some subways, or other things in life) do tend to arrive not exactly on time, in which case it holds. To some extent.

This seems to be attacking from a different perspective, though? You are requiring the change from a bus that is not to schedule. This article pointed out that that was not necessary. And, indeed, you can presume perfectly scheduled busses and still see a distribution quirk where the average wait time of the population is at the interval level. Right?

If the buses are perfectly at schedule and people arrive at random (uniform), the mean (and the median) waiting time will be half the scheduled interval.

If both are completely uniformly random, the mean waiting time will be the mean interval between buses. (In fact the distribution of the passenger arrival doesn't matter anymore.)

The real world is somewhere between those two.

Isn't that at odds with the article? Quoting, "The reason is that there are (of course) more students in the larger classes, and so you oversample large classes when computing the average experience of students." This leans on the experiences of the students, which necessarily leans on the population. (Granted, this is more clearly a different question/scenario.)

I'll try and play with the simulation some. And to be clear, I don't disagree with your statements. I just feel these are still different questions/statements. How long I expect to wait at a given trip/stop is not the same as how long I expect I have waited at that stop over the days.