The Waiting Time Paradox, Or, Why Is My Bus Always Late?
jakevdp.github.io
The Waiting Time Paradox, Or, Why Is My Bus Always Late?
1–10 of 98 posts
Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?
#2Would not be easier to actually time the actual waiting times as he waited for the bus every day?
Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?
#3Would not be easier to actually time the actual waiting times as he waited for the bus every day?
That would probably make it seem, though, that the buses actually don't arrive (on average) every 10 minutes, since you'd oversample the buses that take longer than 10 minutes.
Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?
#4Would not be easier to actually time the actual waiting times as he waited for the bus every day?
That would probably make it seem, though, that the buses actually don't arrive (on average) every 10 minutes, since you'd oversample the buses that take longer than 10 minutes.
That's exactly the phenomenon the article explores.
Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?
#5OneBusAway 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 return route has traffic or other delays. Or maybe a drunk or drug user got in a fight with the driver and the police were needed. Or someone in a wheelchair had a problem getting onto the lift.
Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?
#6The memorylessness of the Poisson process makes the statistical aspect a bit trivial. But here's an interesting variant: how should you update your beliefs while waiting if there is a certain probability that the bus won't come at all? "The Ups and the Downs of the Hope Function in a Fruitless Search", Falk et al 1994: https://www.gwern.net/docs/statistics/bayes/1994-falk
Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?
#7Highly recommend reading this to any folks that are just sitting the discussions.
The simulations were worth the article on their own. The real world analysis was a great bonus.
Anecdotally, i was expecting confirmation bias to be the main culprit. Pleasantly surprised to seei was wrong.
Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?
#8Nice article. It reminds me of my year living in London, and taking the bus everyday to Imperial College from West End Lane in West Hampstead. There was a stop on both sides of the road - one for the outbound bus, and one for the inbound (the bus went from central London to a terminus and then returned mostly on the same route). Now we did not use schedules - way too inaccurate at rush hour, and the busses there were pretty frequent anyway. But we did expect an even chance of the inbound bus arriving before an outbound one did. My daughter and I became convinced after a while that this was not happening, so we invented a game (which we called "The Game of Life".) When our bus (inbound) arrived first, we added 1 to our score. We subtracted 1 for every outbound bus that passed before ours arrived (there were often more than 1). We realized that the result would be slightly skewed to the negative, but we expected the outcome to be close to 0 over time. Of course it was not. Anyway we extended the game to many statistical situations. For example, you go to the checkout line at the supermarket, and there are N people in front of you. When you get to the front of the line, you count the people behind you - call that M. If M is bigger than N, you scored life points. If it is smaller, you lost some. So you add M-N to your running score, and you get an idea of how lucky you are in life. However, I never followed up with any real analysis, so I enjoyed this article.
Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?
#9Would not be easier to actually time the actual waiting times as he waited for the bus every day?
That would probably make it seem, though, that the buses actually don't arrive (on average) every 10 minutes, since you'd oversample the buses that take longer than 10 minutes.
I'd argue that it's not oversampling 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 measure empirically, or even model, than just average wait time for that one person, since it requires counting passengers boarding, not just bus arrival times.
Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?
#10Hah great analysis. One factor with bus' is the schedule is likely planned to minimize early arrivals at the risk of being late more often. Usually when a bus is early it has to sit and wait until its departure time. A late running bus can be more efficient, and if kept until departure time might not ever get a chance to average down the bursts of lateness.