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The Waiting Time Paradox, Or, Why Is My Bus Always Late?

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Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?

#41

Post by bonoboTP »

twtw wrote: post #13

> a Poisson process is a memoryless process that assumes the probability of an arrival is entirely independent of the time since the previous arrival. In reality, a well-run bus system will have schedules deliberately structured to avoid this kind of behavior: buses don't begin their routes at random times throughout the day, but rather begin their routes on a schedule chosen to best serve the transit-riding public.…

Phone calls. Say, an office worker gets 8*6 phone calls in an average shift of 8 hours, so one every 10 minutes on average. It doesn't matter how long ago the last call was, since the customers don't coordinate.

This real world example still doesn't perfectly match the theory. For example, if there was no call for a long time, it may indicate that it's some special day or the phone line is malfunctioning or whatever and it could mean that the next call is probably further in the future than the model would say.

Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?

#42

Post by theoh »

LolWolf wrote: post #32
theoh wrote: post #29

Earlier quoted context omitted.

It is impossible to catch more than one inbound bus on any given occasion, whereas any number of outbound buses might pass. BTW, on a slightly unrelated point, if there's no timetable, but the interval between buses is maintained reliably, the expected waiting time is uniformly distributed over that interval. If you have to get a second bus, you need to convolve two of those two uniform distributions to find out the…

Also, gaussians are great approximations for large n, too, since the convolution of any distribution with itself n times (for n "large enough") is close to gaussian (by the CLT. More generally, there are very nice error estimates for many distributions). I suspect this analysis can be carried out and yield quite good results in the gaussian case (a careful analysis might even yield error bounds on the result).

Yes. If you spend your whole life on one long multi-transfer bus journey, you'll end up with a gaussian.

It's a bit less clear that gaussians should be used when e.g. fitting a coordinate to an astronomical feature, which might not actually be symmetrical.

The other useful property that the gaussian has is its separability, in the 2D case. That is unique to the gaussian and counts for a lot.

Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?

#43

Post by howard941 »

twtw wrote: post #13

> a Poisson process is a memoryless process that assumes the probability of an arrival is entirely independent of the time since the previous arrival. In reality, a well-run bus system will have schedules deliberately structured to avoid this kind of behavior: buses don't begin their routes at random times throughout the day, but rather begin their routes on a schedule chosen to best serve the transit-riding public.…

I think your question is deeper than it first appears, at least it was for me cause I forgot to distinguish gaussian from poisson. One phenomenon is retinal response (per card 8 at this nicely designed site https://quizlet.com/124228940/optometry-stevenson-lectures-f...)

Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?

#44

Post by wiml »

QML wrote: post #24
stephengillie wrote: post #5

OneBusAway is surprisingly accurate, at least in my experience. Google Maps has very good transit support too. One reason buses are late is because a bus must travel a circuit. Cars provide linear transportation, so the delay can only happen in the direction of your travel. Since buses run a circuit, they are impacted by delays in the direction opposite of your travel as well. Your bus might be late because the retur…

Why is modeling required? Can’t we just put a sensor on every bus, and just return the empirical expected time it takes for the next bus to drive to your station given the time of day and day of week?

That's how it works (at least the OneBusAway feed in the Seattle area). They started in the 90s with an RFID transponder on bus stops, read by the bus as it went past; more recently they use things like odometry or GPS to feed information into the system.

But:

> just return the empirical expected time it takes for the next bus

There is a world of complexity in "the empirical expected time", there... expected according to what models?

Anecdotally, I think it's especially hard to model because any given delay is probably attributable to one or a few specific incidents. This isn't a situation where everything averages out and we can use a nice tractable AWGN model; we're down in the muck and the shot-noise.

Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?

#45

Post by agumonkey »

Slightly related, my ghost town had few buses and sparses. I could never rely on printed hours. If I got there 10 min earlier to be sure, I'd still never be sure I'd wait 20 min for nothing because it was 11 min early. Of course half the time if I decide to walk to the next town where buses are many, I'd see all my town buses (both ways) pass me

I think it made me completely careless about time, I would just go between stops and take the first one, go with the flow. By experience I'd know the range it would take for me to reach big places around the area.

I had a friend who was completely foreign to this mode of thinking, she was very dilligent and fully trusting (although she mostly used trains so a lot less divergence).

It reminds me of kid studies about intelligence / wealth ratios. When you're environment is random, you think random. When it's predictable you planify.

Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?

#46

Post by hammock »

hammock wrote: post #23
mmt wrote: post #9

Earlier quoted context omitted.

I'd argue that it's not over sampling at all, but, rather, that the measure of "average bus arrival time" is what's invalid or misleading. After all, the point of the bus arrivals isn't in service of the bus (or driver) but of the passengers. Observed average wait time at each bus stop is a better measure. The even better measure would be average wait time weighted by number of passengers [1]. [1] which is tougher to…

That's a nice idea but ignores all the people sitting in their offices or homes, choosing to go or not go out of their places down to the bus stop. Better to consider each bus stop as an asset to invest in, the more valuable it is, the more people you can serve.

@mmt to clarify, you seem to be treating bus stops independent of alternative means of transportation. Measuring the average wait time of people at the bus stop is not enough: there are people who chose to ride a bike today instead of waiting at the bus stop, because of what happened to them yesterday at the bus stop.

Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?

#47

Post by ChrisFoster »

It strikes me that even with a perfectly regular starting schedule, buses might clump together in time because the schedule is probably dynamically unstable. To explain, picking up passengers from a stop costs time and a long time between buses implies a high probability that passengers will be waiting at a given stop. This further adding to the delay and shortens the time to the next bus in the schedule.

I'm sure drivers try to actively manage this, but if they didn't I suspect the system would naturally evolve toward pairs of buses leapfrogging each other on long routes.

Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?

#48

Post by icebraining »

hammock wrote: post #46
hammock wrote: post #23

Earlier quoted context omitted.

That's a nice idea but ignores all the people sitting in their offices or homes, choosing to go or not go out of their places down to the bus stop. Better to consider each bus stop as an asset to invest in, the more valuable it is, the more people you can serve.

@mmt to clarify, you seem to be treating bus stops independent of alternative means of transportation. Measuring the average wait time of people at the bus stop is not enough: there are people who chose to ride a bike today instead of waiting at the bus stop, because of what happened to them yesterday at the bus stop.

mmt can't downvote you; HN doesn't allow people to downvote direct replies to their posts.

Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?

#50

Post by stornetn »

Reminds me of a similar article that measured a similar kind of question about the wait times for NYC subways conditional on how long you've been waiting (https://erikbern.com/2016/04/04/nyc-subway-math.html). I think it's a pretty safe bet that people who like this post will like this article as well.
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