Buses are bosons, and they condensate

By Daniel Holz | March 4, 2010 11:04 pm

I did my graduate work at the University of Chicago, and lived in Hyde Park. On occasion I would take the bus (the #6 Jeffery Express) to downtown. Although the buses were scheduled to run every 15 minutes, I would invariably end up waiting a half hour. Sometimes more. Often in the freezing cold, or the sweltering heat. Most infuriatingly, when the bus finally arrived, there was always another one immediately behind it! The buses inevitably came in pairs. Sometimes even in triples or quads.

Chicago busLet’s assume that the buses are supposed to arrive every 15 minutes. If the buses adhered to their schedule, and I showed up at a random time, I should generally have to wait roughly half the mean bus arrival time: 7.5 minutes. If the buses were totally random, then I would have to wait the average time between bus arrivals: 15 minutes (if you haven’t thought about this before, this statement should sound crazy; perhaps I’ll do a future post on it). So the question is: why did I always end up waiting roughly 30 minutes or more?

I always assumed that the Universe was conspiring against me. This is a common feeling in graduate school. However….

I just stumbled across a blog post of a friend of mine from graduate school, Alex Lobkovsky. In it, he discusses precisely this problem, and presents various reasons for the bunching of buses. I have no doubt that he was inspired from similar suffering. Perhaps at the very same bus stop.

At the end of the day, there’s a fairly straightforward solution. Imagine all of the buses are roughly on time. Now imagine that one bus (call it bus S) happens to fall behind. Because S is running behind, more time has elapsed since the previous bus has passed. This means that more waiting passengers have accumulated, at more bus stops. This in turn means that bus S has to stop more often, and has to pick up more people at each stop. Hence, bus S falls even farther behind. Which means even more people accumulate at each stop. Which means the bus falls even farther behind. And so on. In short: a slow bus gets slower and slower.

Now let us consider the bus behind bus S; we’ll call it bus F. Bus F starts out roughly on schedule. But because bus S is running late, less time than average has elapsed between when bus S last passed and when bus F arrives. This means fewer people have accumulated, at fewer stops. Which means bus F makes fewer stops, and picks up fewer people. Which means that it starts to run faster than average. Which means even fewer people accumulate. Which means it runs even faster. And so on. In short: a fast bus gets faster and faster.

Putting this all together: if a random fluctuation creates a slow bus, then it will get slower and slower, and the bus behind it will get faster and faster, until the two buses meet up. At this point, the buses stick together, and are essentially incapable of separating. Thus, in general, buses will bunch up. This will usually happen in pairs, though on occasion triples and even quads may occur. This argument predicts that the arrival of buses will be random, with pairs of buses arriving more often than not, being separated by on average double the mean bus separation. And this is precisely what I discovered, the hard way, shivering at the corner of 55th St. and Hyde Park Boulevard. (N.B. I spent a year in Berlin. There, the buses are fermions, and always arrive exactly on time. It’s the stereotype, but it turns out to be true.)

After writing this post, I found that wikipedia has already figured it all out. Regardless, it’s nice to know that my suffering was due to statistics, and not because the Universe is out to get me.

CATEGORIZED UNDER: Mathematics, Miscellany, Technology, World
  • max

    I’m pretty sure that the universe is out to get you precisely because of statistics.

  • Albion

    We both know that *Chicago* was out to get us.

  • chris

    samrt bus operators know about the problem and have the solution: once a bus is too close on the heel of its predicessor, it does not stop at intermediate stops any more (or at most stops to let people off the bus, but does not allow anyone to enter). this also works wonders for load balancing.

  • Pieter Kok

    chris, that only further fuels the (in my opinion justified) belief that all bus drivers are fascists…

  • http://rampke.de/ Matthias

    Countermeasures are easy: make the schedule less tight, such that, under “standard” conditions, the bus will arrive slightly ahead of time, or, in the event of a slight delay, will be able to make it up despite the bunching of passengers waiting.

  • Chaz

    I think part of the US/Europe discrepancy is due to the way fares are collected. In the US (and the UK), everyone waits in line at the front of the bus and pays/swipes their card before they can board. In Germany, tickets are checked only occasionally, and most of the time everyone boards and sits immediately (I’ve only ever had my ticket checked once or twice in Europe). The US system can delay the bus by several minutes at each stop during peak times, but it could contribute to bunching even if the delay is only 30 seconds per stop.

  • coryy

    @ #2—


    I heartily agree. Chicago is out to get everybody!

  • anonymous


    I’m just happy that I understand the difference between bosons and fermions enough to appreciate this analogy. :)

  • Georg

    When You
    observe bubble chains in champagne or gazified water starting
    at some point in the glas, You sometimes will find such strings
    made up from pairs of bubbles moving closely connected.

  • Joseph

    Their operators may also commute. :)

  • JD

    Quick question: How did the schedule work? Was it e.g.: bus at 8:07, 8:22, 8:37, 8:52 …, or was it “from 7 to 10 AM: bus every 15 minutes”?

    I’m asking because I currently live in Montreal, where buses use the arrival time schedule system, and I used to live in Quebec City, where they used the interval time schedule. In Montreal, buses rarely bunch (except when there are frequent buses (<8 minutes intervals) on crowded streets. Bus 165 is a good example: buses every 6 minutes or less , traffic can get heavy, so sometimes you get two buses bunched together), but in Quebec City, they always bunched. Buses 800 and 801 come to mind: trips were daily occurences, quads happened multiple times a week, and I once saw 6 buses bunched together.

    Lobkovsky's explanation makes a lot of sense, but I always had the (perhaps irrational and unfounded) suspicion that the interval time schedule made buses more prone to bunch up. It would be nice to see if there's any correlation.

  • Paul Schofield
  • onymous

    Interesting. The model may need further refinement to explain why some buses bunch much more than others; the 6 never caused me the same degree of problems that the 55 did, for instance.

  • Paul

    I too suffered at that same bus stop, waiting for that same bus. Somehow, though, it seemed that the Jeffery was more prone to this than other lines.

  • NoJoy

    I’m not a physicist, and am wondering whether “condensate”, used as a verb, is a term of art, distinct from “condense”?

    Joseph – That was hilarious.

  • yj

    I agree with NoJoy. Condensate is not a verb!!!

  • James D

    @6. Chaz: Agreed. Also, in Europe, bus stop spacing tends to be wider, so each bus tends to make exactly the same stops, rather like light rail, rather than having a flag stop on every street corner.

  • Eugene

    I agree with Albion : Chicago is always out to get us.

    MTA Trains are bosons too. And I have learned to realize that if I have to wait slightly longer than the usual for the train, *and* the train is packed when it arrives, there is usually one empty one just right behind. Ah, I feel like a New Yawker.

  • Steve

    The Jeffrey!

    *raises fist and shakes it at the sky*

    I spent plenty of time on the bus and waiting for that bus during my undergrad time at UofC. Thanks for explaining the waiting. Gotta love sitting in the accordian though.

  • Rohan Mehra

    Why give the buses letters (S for slow and F for fast) if no maths is to be done with these?
    It reminds me a little of the trick some pop-philosophy writers use to sound scientific!
    Not that you’re doing that of course 😉

    Bunching buses always reminds me of watching droplets of rain on a side window when in a car or on a train.

  • David

    “If the buses were totally random, then I would have to wait the average time between bus arrivals: 15 minutes (if you haven’t thought about this before, this statement should sound crazy; perhaps I’ll do a future post on it).”

    I would enjoy a post on this. I’ve been trying to figure this out, but I’m not exactly sure how to correctly interpret the buses being “totally random.”

  • Iolaum

    I ve seen it happen many times in route 58 in Thessaloniki (when I lived there years ago) and I always thought it was because the drivers were drinking coffee and left for work at the same time 😮

  • Oana

    We now have CTA bus tracker here in Chicago, so no need to wait more than a couple of minutes. It’s purely amazing, especially in winter.

  • steeleweed

    Once saw the math explaining why elevators are never on your floor or headed in your direction. I vote for ‘the universe is conspiring’ theory.

  • Jdhuey

    Well, I have only a limited experience riding the public transit light rail/ bus system here in San Diego County but I noticed that, at least on the routes that I took, the timing was very punctual. I also noticed that the bus drivers were very careful not to get off of their schedule – but mostly they made sure they didn’t get *ahead* of schedule. On many occasions, the driver would just sit at the bus stop until the official clock on the dash hit the right time. I suspect that buses that get ahead of schedule can cause just as much of a disruption as buses that get behind.

  • Travis Garrett

    Comment 9:

    When You
    observe bubble chains in champagne or gazified water starting
    at some point in the glas, You sometimes will find such strings
    made up from pairs of bubbles moving closely connected.

    pretty much sums it all up doesn’t it?

  • spyder

    I guess this is a Chicago, NYC sort of problem, not experienced by me in any of the regions i have lived over the last sixty years. Santa Monica Transit ran just fine in the 60s and 70s, Alameda County worked wonderfully with SamTrans in the late 70’s and early 80s, Spokane is just perfect in the 21st century.

  • gopher65

    Jdhuey: I live in Saskatoon, Canada, and I’ve made the same observation. In Daniel’s model the fast buses cause much of the bunching. Here, when a bus is running fast, they’ll just sit at a stop for a couple minutes. This prevents bunching from ever happening (except near the university, where there are several buses a minute pulling in and leaving during peak times. But that’s different.).

  • Jeb

    What is most annoying to me about this phenomenon is that half the people don’t just wait two seconds for the other bus

  • emeris

    I would just point out that the 6 is now called the jackson park express (http://www.transitchicago.com/riding_cta/busroute.aspx?RouteId=165)

  • Shaun

    In other words, buses really do arrive according to a Poisson process :)

  • Kaleberg

    I remember this from probability class. There are two Poisson processes involved. The one for the buses and the one for passengers. Even without clumping effects, there were always more passengers stuck waiting longer periods than one might expect.

    If you throw in the delay feedback problem, it just gets worse.

    If we had reliable information about bus arrivals, more people might wait for the next, less full bus, but for most passengers, “There’s a bus right behind this one” translates as “The check is in the mail.”

  • ChicagoMolly

    It’s more complicated than that. What with street repairs, wheelchair passengers, and nannies with those enormous SUVStrollers, buses can start to bunch up. The drivers will sometimes leapfrog each other if they don’t have passengers who need to exit at a certain stop, but they’re not supposed to. The route schedules tell the drivers what time they’re supposed to make each stop, and CTA sends supervisors out every day running spot checks to ensure that drivers are sticking to their schedules. If the drivers are leapfrogging, and Bus #2 meets up with a supervisor while he’s ahead of Bus #1, the driver will have to wait for the other bus to get ahead of him, and if he’s significantly off-sched may have to wait a few minutes more to get back to the proper spacing. Over the long haul it does really average out, but when you happen to be on the bus that gets held back (and you’re running late for work) you want to scream something impolite.

  • http://cae.homepages.wisc.edu/~khuff Katy Huff

    Sean, perhaps it’s been too long since you were in HP! The Jefferey is the #14. The #6, which goes to downtown, is the ‘Jackson Park’ Express.

    Anyway, the rest of this post is right on target! I was there this weekend and, I kid you not, witnessed (from the third bus) three #6 buses converge on 55th street and Hyde Park Blvd.

  • Neal J. King

    So it seems like there are a few driving-policy changes that would eliminate this bunching effect.

    Any possibility of communicating them to the management of the bus system?

  • Charon


    “the buses were scheduled to run every 15 minutes”

    No, they weren’t. My recollection was every ~6 minutes at peak (I’m remembering this from 8-10 years ago, when the #6 was the Jeffrey, probably about when you were there). Currently the CTA site says it arrives every ~8 minutes (5-12, depending on the time of day).

  • Jimmy James

    I remember this problem appearing in my continuum physics class when we were discussing shock waves- I think the professor found it in Tritton’s Physical Fluid Dynamics.

    Speaking of things condensing like bosons, there was a good joke during a statistical mechanics midterm- the TA thinks everyone’s too bunched together, so he says, “No, no! You guys should be like fermions! Spread out a bit!” Some wiseguy says, “If we’re fermions, should we not put our names on the test?”

  • Paul Kaplan

    I’m an undergraduate physics major at the University, and just so you know the 6 is the “Jackson Park Express” not the Jeffrey express!


  • Anne

    In my memory, Hyde Park Blvd. and 55th were parallel, so for a simpler mathematical account of why you waited so long for the Jeffery to arrive at the corner of those two streets, look to Euclid’s Definition 23.

  • Mike Maxwell

    Anne, this is Riemann geometry. Not only do (South) Hyde Park Blvd and 55th intersect, it appears that (South) Hyde Park Blvd and (East) Hyde Park Blvd intersect.

    I’m sure bus routes were much simpler in Euclid’s time.

  • Anne

    My memory worked a lot better back then, too.

  • Mike Maxwell

    And in Euclid’s time, the Earth was flat, so of course Riemann geometry was not applicable; ergo no way Ύδε πάρκ ΒόυλεΒαρδ could have intersected Ύδε πάρκ ΒόυλεΒαρδ!


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