
Wait ages for a bus and then two come along at once - sebgr
http://jasmcole.com/2015/03/02/two-come-along-at-once/
======
cameldrv
The WBEZ article below alludes to the true dynamics problem, but doesn't
explain it. The problem is that say you normally run in 10 minute intervals.
For whatever reason, double parked car, helping a disabled person on the bus,
etc, you get a minute behind schedule. Now suppose at the next stop typically
10 people are waiting. They arrive at the bus stop at one per minute all day.
Since you're a minute late, there will be 11 people, which will take slightly
longer to load. Furthermore, now the buss is a little bit fuller, so it takes
people a little longer to get to and from the doors. Now you get a little more
behind schedule. The bus behind you now has the opposite happening. The extra
guy you picked up should have been his passenger, and so he makes his stops
more quickly, until he catches up with you.

You can try to fix this in a few ways.

1\. You can reduce the perturbations to the system, for example, offboard
payment, dedicated lanes, stricter double parking enforcement, quicker
entry/exit systems for disabled passengers.

2\. You can reduce the feedbacks, or provide a counter-feedback, for example,
frequent scheduled waiting stops to resynchronize, don't collect fares when
the bus is full, traffic light priority when the bus is full

~~~
foobar2020
This is a nice analysis, but the data in the article actually disproves it, at
least to the degree show in the last plot (quite reliably in my opinion). The
distribution of the waiting time is exponential, is a Poisson process, which
means that buses do not influence each other.

~~~
stdbrouw
The existence of one plausible explanation (good model fit) does not
automatically invalidate another plausible explanation (mechanism of action).

~~~
foobar2020
It does if the explanations are contradictory. In this case this means one of
two things:

1) The "first bus collects delay" theory is responsible for a part of the
difference between the data and the model, after all there is some.

2) There is another mechanism at action that statistically reverses the effect
of the "first bus collects delay" theory, not necessarily in the same bus
runs.

Number 1) is coherent with Occam's razor.

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evo_9
Interesting. He should look at data from Japan, along with other countries.
When I was in Tokyo the trains and buses were all spread out and things ran
on-time; you can literally watch the train arrive as the est. time of arrival
clicked down to zero. Buses were similarly on-time and never bunched up. Not
once did I see this pattern while I travelled around Japan for 15 days.

When I arrived back in Denver the first thing I noticed were 2 light-rail
trains that arrived about 5 seconds apart from each other. Buses also get
similar bunched up.

Denver isn't anywhere near London's density; I wonder if this is more related
to the system in place and ethnocentric factors such as the pride that
Japanese operators seemed to have at running everything as precisely as
possible.

~~~
lotharbot
My dad always causes a ~90 second delay for the Denver light rail, because the
operator has to activate the wheelchair ramp for him. How do they handle that
sort of thing in Tokyo?

(Also, I often see multiple light-rail trains bunched up after sporting
events.)

~~~
screwedup
I'd imagine that the platform on most light-rail trains is built so that a
ramp is unnecessary?

~~~
lotharbot
There's a ramp inside the train that needs to be deployed to bridge the gap to
the ramp at the platform.

~~~
reverius42
Sounds like a strange design to me. On most of the train-based public transit
I've seen (and even some of the "rapid" buses), the floor of the car is
aligned with the floor of the boarding area, such that ramps are unnecessary.

~~~
lotharbot
they're aligned, but there's a small gap that needs to be bridged.

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natnat
Also worth reading:
[http://en.wikipedia.org/wiki/Bus_bunching](http://en.wikipedia.org/wiki/Bus_bunching)

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im2w1l
Well seems like he is aggregating data across many lines, stops and times per
day. Not very surprising that the result looks random.

~~~
foobar2020
There are different kinds of randomness. The exponential distribution of the
waiting time is a nice, mathematically pleasing result. Although not a very
nice one to the commuter.

~~~
im2w1l
"Random" was poor wording on my part.

What the exponential distribution means is that bus arrivals are completely
independent. The time since the last bus came doesn't affect the probability
that a bus will come anytime soon.

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quarterto
I've noticed that in the "two come along at once" scenario, the second bus
often leapfrogs the first, skipping a stop while the first bus picks up
passengers. I've no intuitive idea whether this would smooth out the flow more
than waiting behind the first bus though.

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rawnlq
If we accept the exponential waiting time model, then another term for this
phenomenon is Poisson clumping:
[http://en.wikipedia.org/wiki/Poisson_clumping](http://en.wikipedia.org/wiki/Poisson_clumping).

I think one intuitive way to explain it is if you're flipping some biased coin
continuously, then if someone walks in, you are more likely to currently be in
a long streak and have stuff clump afterwards. For exampling flipping a coin
with probability 1/60 of heads each minute for an hour (so average wait time
should be 10 mins):

    
    
        > ''.join('H' if random.randint(1, 10) == 10 else 'T' for i in xrange(60))
        'TTTHTTTTTTTTTTTTTTTTTTTTTTTTTTHTHTTTTTTTTTTTHTTTTTTTTTTHTTTH'

~~~
mungoman2
This sounds interesting, but I don't understand. "if someone walks in, you are
more likely to currently be in a long streak and have stuff clump afterwards"
What does that mean?

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dylansal
Great article. And here's some academic research that models the dynamics of
bus bunching and proposes a decentralized control scheme to minimize a
combination of passenger wait time and travel time:

[http://www.its.berkeley.edu/publications/UCB/2011/VWP/UCB-
IT...](http://www.its.berkeley.edu/publications/UCB/2011/VWP/UCB-ITS-
VWP-2011-1.pdf)

...And here's a product we created to commercialize this research:

[https://www.youtube.com/watch?v=YQXZybHO52A](https://www.youtube.com/watch?v=YQXZybHO52A)

Full disclosure: I am a co-founder along with the authors of the above paper
of Via Analytics:

v-a.io

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erik_p
At least with SF Muni the bunching of busses issue is more
human/contractual/process related.

When the bus runs late (because stuff happens) and thereby completes it's
route late and turns around, and is basically right up against the "next"
bus... they send them both out BECAUSE the thought of paying a muni worker to
wait for 10 minutes to create a sane space between busses is unthinkable in a
penny wise, results foolish budgeting process.

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warfangle
Interestingly enough, I ran into an extreme of this on Saturday; while I
wasn't inconvenienced by it (the bus showed up 2 minutes after I arrived at
the stop), I overheard the bus driver talking to riders he seemed to know as
regulars.

This bus was on time, but there hadn't been a bus for over a 45 minutes; the
schedule defined the interval as 1 bus / 7 minutes.

There were seven busses on the same route directly behind us.

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FLGMwt
I thought this was going to be a rehash of this
[http://www.wbez.org/series/curious-city/why-buses-arrive-
bun...](http://www.wbez.org/series/curious-city/why-buses-arrive-
bunches-110941) but then I saw maths. I'd love to see more articles/analyses
like this.

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dnautics
it's worth noting that this measures the distribution of bus intervals. The
distribution of wait times will be more extreme because there will be more
passengers waiting at a bunched stop, so the 'bunched' experience will be more
common than the 'unbunched' experience. Moreover, the hours when the bunching
phenomenon occurs are probably biased to high commute volume times.

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simonster
Another, very different pattern of bus arrival times evidently arises when
buses slow down to prevent bunching:
[https://www.quantamagazine.org/20130205-in-mysterious-
patter...](https://www.quantamagazine.org/20130205-in-mysterious-pattern-math-
and-nature-converge/)

------
dragon88
Bunching happens a lot when one delay causes one bus/streetcar/train to hold
up all the ones behind it.

Best solution is letting transit vehicles going the opposite direction to turn
ahead ahead of the bottleneck and serve the passengers waiting at the stations
in front of the delay area.

~~~
bentcorner
Another solution would be to have the lead bus skip pickup at a stop (or two).

You'd need clear and solid signage to assuage people waiting though. Something
like "Behind schedule, dropoff only. Next bus arrives in 2 minutes".

~~~
omegant
Or set a dropoff only stop 50 yards from the designated stop. That way the
people from the stop don't try to board.

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cozzyd
Does London use pay as you board or proof of payment for buses?

Here in Chicago, I think boarding delays significantly contribute to bus
bunching (or at least make it worse).

~~~
quarterto
London buses are cashless. Either you use an Oyster or contactless payment
card, or buy a ticket or pass in advance.

~~~
thaumasiotes
So, pay as you board.

~~~
lmm
Not in a way that delays you - the driver doesn't count change or anything.

~~~
cozzyd
Almost nobody in Chicago uses cash to board, but if there are 20-30 people
boarding a bus, it can still take over a minute as they all have to go through
the front door and tap their cards (and of course, it doesn't always work the
first time for some reason).

