Lazy and Impractical Mathematicians
Tagesschau.de ran a piece about a mathematical solution to the question if a person waiting for a bus should keep waiting or walk the distance to the next bus stop. The authors of the paper called “Walk versus Wait: The Lazy Mathematician Wins” came to the obvious conclusion that it is - in most cases - more practical to wait for the next bus, that is if one is not able to walk the whole distance up to the destination in a shorter time than would the bus need to arrive and then drive the whole distance.
The authors Justin G. Chen, Scott D. Kominers and Robert W. Sinnott argue that if you’re lucky and you reach the next bus stop before the next bus arrives, you would still not have saved any time, since the bus would’ve arrived at that time anyway, even if you boarded it at the first stop. Moreover if you didn’t arrive on time, you would miss the next bus! Either way, unless you wanted to walk the whole distance to your destination it’s much easier and cheaper (from an energetic point of view) to wait for the next bus.
The only problem with the paper is its very shallow analysis of the problem at hand. There are some things the authors should have better considered before coming to that rather rash conclusion:
- Pedestrians have the ability to walk different routes than buses (or any motorized vehicles). Especially bus routes are rarely planned to be very effective for someone who wants to get from one random point to another. Rather they tend to make every turn possible to collect as many passengers as possible. Often a pedestrian is able to take a shortcut to another bus stop (not necessarily the next one).
- Buses don’t drive with a constant velocity! Buses like any other motorized vehicle have to accelerate, decelerate and sometimes yield right of way for some other vehicle. They are bound by traffic lights, road works and other such obstacles. Pedestrians too are bound by at least some of those obstacles, but it’s a time-proven fact that over small distances a human being can travel faster than a vehicle.
- Further more buses have to stop at bus stops, they have to wait for passengers to board, sometimes tickets have to be paid, printed or at least looked at by the driver. This all takes time!
So, if a bus just started driving away from the first bus stop and the next bus stop is only 100 meter away with two traffic lights, a construction sight and many many vehicles in between, it is probably the best thing to run to the next bus stop, because you can get there before the bus does!
In general, it is possible to save time by walking to one of the next bus stops as long as the time of departure of the previous bus is reasonably near and the distance to the next bus stop is reasonably small. The only hint one can give is: be smart! It’s also possible to take another bus route - which drives from a different bus stop but crosses the same destination point or comes near to it - if the system allows. At any rate, what we can say to be very probably true is that you will loose time, if you start pondering your chances of catching the next bus at the next stop. Every other possibility for saving time has got to be considered at the specific moment.
So, in conclusion, it has to be left to the individual to decide weather he wants to wait or walk a bit to get his lazy bottom moving once in a while..
By the way: with this paper mathematicians strive further to prove once more, that they are in fact very disconnected from the real world. They seem to think that reality really bends to fulfill their private approximations.. In this case the bus drives with constant velocity by some bus stops which are evenly distributed along it’s trajectory with no obstacles in the between. Actually, that’s not the case!
Update (3. Februar 2008):
Over at BoingBoing one Chillar suggests that the stochastic algorithm in the paper might be buggy:
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).
I’d have to wrap my head around the formulas to give my opinion, i only dealt with the model itself, and that is already very flawed..
Update #2
Over at the google-group “sci.math” “FCS” gives an example of what is wrong about the system used by the three scientists:
For example a 7 minute walk from a bus stop with a route frequency of 1 per hour can end up 4 stops away at a far better served bus stop with a route frequency of at least one every ten minutes. A further 3 stops from this can still be got to by bus in about 10 minutes, total.
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— 4. Februar 2008 @ 13:02
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By , 4. Februar 2008 @ 17:10
As I see it, the authors actually did not need to take your three points from above into account, because the variables v_b and v_w implicitly contain whatever velcity distribution over the route. Consider a bus arriving at time t. Waiting passengers don’t care whether the bus had to stop at a red sign or where it drove slower and where it drove faster, they only care about the arrival time t. The same applies to the possibility for a pedestrian to take short cuts: in this scenario it is only important, how much time elapses until you reach your destination nad not what path you take (ok, there is only one variable d, but one could adjust v_w to reflect the advantage of short cuts).
So all in all I don’t think there are any seriouse mistakes made. The irritating thing is that the result is so trivial: Get a bus schedule and use that transportation that gets you to your destination first.
By , 5. Februar 2008 @ 09:51
@Munir: actually it makes a big difference if a vehicle drives with a constant velocity and does not take into account people getting inside. Especially if the distance between two bus stops is small, the velocity distribution becomes important.
The model used by the three doesn’t take shortcuts into account and therefor doesn’t consider catching up with the previous bus as a possibility, since pedestrians travel slower than buses in general. The whole thing is generally not so linear as the scientists try to show.
Life is much more complicated.