Mathematics of waiting for the bus
If your bus is late, how long should you wait before starting to walk in the hopes that you'll catch it at the next stop? Mathematicians from Harvard University modeled the problem and determined that the optimal strategy is the "lazy" one: stay put and wait. From New Scientist:
The formula does break down in extreme cases, (Harvard's Scott) Kominers says, when the time interval between buses is longer than an hour, for example, and your destination is only a kilometre away.Link to New Scientist, Link to scientific paper abstract
If you do choose to walk, you should make your decision before you start waiting, he says. You will still reach your destination later than the bus you'd have caught, but it will be much less frustrating than waiting for a while and then watching the bus shoot by.
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TAKUAN: these #1 are truly words of wisdom.
I predict that this man has an Ig Nobel in his future.
I already knew this, without having "done the math." You dont need a harvard egghead to know this, all you need is to have grown up and lived in NYC and had to rely on it's public transportation system.
I think I'm missing something here...
Isn't the bus going to reach its stops at the same time whether you walk to the next one or not? I mean If you start walking and the bus catches up to you at the third stop then the bus is reaching that stop at the same time as it would if you just waited and caught it at the first stop. You aren't saving any time.
Really this is just a matter of whether you get restless or cold and want to walk. Or if you feel lazy and want to wait.
Unless obviously you can just get there faster by walking. If your destination is only a km away maybe you should just be walking there all the time anyways?
The bus stop after mine often fills the bus to capacity. Walking to the next stop would never make sense for me, since I might not actually get on if I wait at a later stop. I suppose I could walk to the previous stop, but then, that's up hill, and who wants to do all that work? :)
When in Brooklyn, and going from the corner of Grand and Union to the corner of Grand and Bushwick, the bus will always pass you if you walk and will never come if you wait, especially if it's cold or raining. Except in those cases where it comes but drives right past you without stopping. No math involved.
This seems to leave out the fact that walking is more pleasant than standing still, particularly on the busy streets where buses tend to run.
From just reading the links, it looks like this optimizes only for speed, not for total expected utility of the journey.
I guess that would be tricky, as it would introduce all sorts of unpredictable constraints - negative utility from missing appointments, relative utility of time spent at the destination vs. time spent walking there, etc.
Unfortunately, the MBTA (which is what runs by/through Harvard) is rather often in the allegedly "extreme" case where the interval between buses is longer than an hour, even when the schedule says twenty minutes.
Sounds to me like their model is too simple to be helpful in the real world, even assuming a value of "Cambridge" for "real world".
#5, I don't get it either. Assuming there's only one bus, there's only a single time at which you can possibly arrive at your destination no matter where you catch the bus, unless you walk the whole way. So if the goal is to maximize the time spent on the bus vs. outside in the cold, then obviously it's better to wait since you can't catch the bus earlier at a later stop.
Why do we need to pretend there's a math problem here?
i ride the bus all the time, and this is all very true. however, there's also the "fuck it, i'm just walking" factor. this tends to play out when cold, restless, or drunk.
#5 and #11: Yeah, it's usually pointless to walk to the next stop, and the walk generally isn't very enjoyable because you spend the entire time looking over your shoulder to see if your bus has shown up and whether you should sprint back to the old stop, or try to sprint to the new stop.
The only time I've ever had any point (time wise) in just giving up and walking was when I was particularly close as the pedestrian walks, my bus was hourly, and the route the bus is on was particularly squiggly. I was admittedly a little lost and my feet hurt like blazes, but I arrived ... two whole minutes before the bus showed up. :P
At least I got exercise, and had vowed to buy new shoes :P
I seem to remember that situations like this can be described accurately by stochastic process models, particularly 'hazard' functions, but I can't for the life of me remember how. Some hardcore stats BBer help me out!
# 13
But that's the fun part! Constantly analyzing the risk that any particular segment of the walk will leave you stranded between two bus stops at the precise time the bus shows up makes the walk so much more exciting. Bridges are especially good that way - they're riskiest, but they're also the most fun to walk across.
Also re #11 - I find that, unless it's very windy and the bus stop has good shelter, I'm much warmer walking than standing still. In summer, I'll more likely stay put or walk to a bus stop in the shade, but in winter I tend to walk between stops, just to stay warm.
One of the assumptions used in the actual article is suspect to me, as a resident of Cambridge. The assumption that the speed of a bus is strictly faster than the speed of walking is just not realistic for the area around Harvard.
This is all wrong because it assumes that you know the bus is late by the time you arrive at the bus stop when in fact, you most likely won't know the bus is late until the time that the bus should have arrived has passed (and never mind the fact that your local MTA may not consider a bus to be late unless it is at least 10 mins late). The real question is how late is the bus?
Weather considerations aside, I generally walk along the bus route. Get a little exercise, once in a while get to feel superior when a bus has broken down and I get where I need to be on time anyway.
But I've only done this on urban routes which mostly travel a long distance in a single direction, without tall hills to block the view back down the line. Generally these buses stop every 2-3 blocks anyway, so at most I might need to sprint forward or back a block and a half when I spy it coming.
Or you could just move to Portland. TriMet uses a GPS-based "TransitTracker" - call the phone number or check their website for real time bus and train arrivals. I get to stay in my office reading BoingBoing until the last possible minute.
@17 ursonate:
Agreed you don't know if the bus is late, but you can usually see by the number of people in the stop, if one just passed by, and you will have to wait the full time beetewn passes to catch the next. In this case, the shall I wait, or shall I walk question comes to mind. In my personal experience, walking does not pay timewise, but i sometimes walk, because its better to keep walking than to just stand waiting. If its not raining that is!
Oh the buses of Cambridge! I've waited for the 86 countless times, only to surmise after waiting 40 minutes that A) the previous bus was 10 mins early, and B) the subsequent bus is 15 mins late. All told, that means I'll get to work about 25 mins tardy.
The saving grace is that my bus stop is right outside the Sherman Cafe, which means I'm enjoying tasty breakfast treats the entire time (and prob getting to watch a funeral across the street). I can't say Scott enjoys the same benefits.
I usually ride my bike, but I got doored last week. Hmph.
Oh the buses of Cambridge! I've waited for the 86 countless times, only to surmise after waiting 40 minutes that A) the previous bus was 10 mins early, and B) the subsequent bus is 15 mins late. All told, that means I'll get to work about 25 mins tardy.
The saving grace is that my bus stop is right outside the Sherman Cafe, which means I'm enjoying tasty breakfast treats the entire time (and prob getting to watch a funeral across the street). I can't say Scott enjoys the same benefits.
I usually ride my bike, but I got doored last week. Hmph.
Regardless of the conclusion the paper draws, the 'mathematics' of this 'paper' are all wrong. There are some real mathematicians working on these kind of problems, and the authors should really examine what's out there.
For instance (from page one of their 'paper'), what does it mean for the probability of the bus arriving at time t to be p(t)? Unless you have only a finite number of places where p(t) is nonzero, then this makes no sense (and since they are later integrating p(t), they are assuming the opposite).
Think of it this way: what is the probability that the bus arrives at time t = pi, at EXACTLY t = pi? Of course, it is 0. That is why people use things called distributions: One must integrate a function over a PERIOD of time to get a probability that the event will occur in that interval.
This is Stat/Probability 101. When such an error occurs in the first few lines of such a 'paper', the rest must be rubbish (and indeed is).
Hey, I found his website, http://www.scottkom.com/ .
It looks like he actually does some real math, rather than just inciting British blokes to be lazy.
#5 & #10
My understanding is that the model allows Justin to make a seperate decision to either wait or walk at EACH stop along the way. He can continue to walk or change his mind in light of new information (presumably in the form of a catchable bus, missed bus or continued absence of either)at each decision point. If the bus never comes he's making slow progress, if he can catch a bus at a later stop he's no worse off. BUT, there's the risk of getting caught in between and missing the bus.
i believe that tracks. though i should point out that my personal expertise is in type "A" behavior rather than mathematics per se.
Actually, the model used by the scientists is very flawed and therefor their conclusions are quite system-inherit.
If we consider the non-linearity of most bus tracks and how buses have to stop (they don't drive with a constant velocity!!), wait for traffic and traffic lights, then the equation and the solution to the question get quite complicated and there are more possibilities in which one would save time by walking a bit. It doesn't have to be on the track of the bus, but pedestrians can take shortcuts..
I've written my comment here:
http://tech.toomuchcookies.net/46/lazy-and-impractical-mathematicians/