Poker and Random Bunching
In June 1966 the British group the Hollies topped the charts with their song Bus Stop.
Bus stop, wet day, she's there, I say
Please share my umbrella
Bus stop, bus goes, she stays, love grows
Under my umbrella
The love song did not capture the experience of most bus riders on Boston’s number 1 bus going from Dudley Square, passing the MIT campus on its way to Harvard Square. Most passengers waiting for the bus have to endure long waits – usually in cold, rainy, or snowy weather. Furthermore, after a miserable wait they are typically frustrated to see two #1 buses arriving together, the first one fully loaded and the second virtually empty. This being MIT, it may be time for a good theory or an underlying model of the phenomenon.
There are two possible explanations for this observable fact. The first is known as the “poker table” theory. It surmises that the drivers on the number 1 bus are a friendly bunch. They like each other’s company and prefer the thrills of Texas-hold-em to leaving the station one at a time, driving alone along Mass Avenue, collecting umbrella-holding couples on time. The result is a convoy of two and three buses coming together to pick up and deliver hurried students and visitors to MIT.
The other theory is known as “random bunching” (aka “bus pairing”). It assumes that drivers leave the first station exactly on schedule, in, say, ten-minute intervals. But due to some random event – such as missing a green light or waiting for an old passenger who takes a long time to alight – one bus falls slightly behind schedule.
The chances are, then, that when the bus arrives a little late at the next station, and the number of people waiting for it will be larger than average. The bus will therefore spend more time than average picking those passengers up, and fall further behind its schedule, finding an even larger number of passengers at the next station, and falling still further behind.
To add insult to injury, the bus behind it now finds fewer and fewer people at the stations and gets further and further ahead of its schedule. Pretty soon the two buses start moving in a kind of convoy with the first bus full and the second one empty. Furthermore, one of the results of this phenomenon is that most of the customers are waiting a long time and only a few experience short wait times. So unless an attractive partner is waiting in the station and sharing an umbrella – in which case it may lead to positive outcomes as the Hollies suggest – most customers experience bad service.
Observations support the contention that random bunching is always at play, but the first theory is hard to rule out since poker games of drivers while on shift are difficult to observe without resorting to NSA methods.
Supporting evidence for the random bunching theory (aside from its scientific-sounding name) can be found by observing automatic systems that exhibit similar characteristics. For example, elevator systems tend to bunch just like buses .You can wait a long time for an elevator in a high rise with an elevator bank, then two or more will arrive together. This phenomenon is even stronger in Boston than in New York.
The random bunching explanation in the elevator case is similar to the one pertaining to buses. When polite people hold the elevator door for a latecomer, the elevator falls behind and there is a higher chance that it will be stopped at the next floor. Meanwhile, the other elevators in the group speed up and arrive, in many cases, simultaneously. The problem is less severe in New York, since Yankee fans are less inclined than genteel Red Sox fans to hold the door for latecomers.
The moral of the story is that holding the elevator door is actually anti-social behavior. The considerate driver who waits for a late passenger at a bus station is actually harming the system, while the driver of an early bus who waits at a station while you urge him to leave (when you are already on board, of course) is actually doing the right thing from a system point of view.
Another lesson is that you’ve got to know when to hold them and when to fold them.
Photo: Shutterstock.com
Vice President, Supply Chain Product Strategy at Oracle
10 年So what's the best operations management strategy to recover from bunching? A) Empty bus overtakes full bus. (In the case of subway bunching, this isn't possible.) B) Driver asks all passengers for the next few local stops to disembark, and then drives directly to the more distant stop while the empty bus takes the others C) Passengers are only allowed to disembark at each stop (only rear doors open) until loads are balanced D) Buses go to alternative stops (odd/even) so they can load in parallel. Passengers on an even bus wanting an odd stop walk forward/back or take a bus in the opposite direction to arrive. E) The empty bus waits until it can collect more passengers, but not so long that it repeats the process with the next following bus. F) The full bus becomes so full that loading time is actually reduced, because no one can get on. Eventually the empty bus fills up with passengers who can't get on the first one. G) Passengers save time by climbing in and out of windows (I saw this in Egypt.) It's clear that some approaches are more effective than others in theory, but in practice, they all have costs. For example, it takes yet more time to tell passengers to change buses, and to actually get the right ones to obey the command. The bus drivers' and passengers' incentives also need to be aligned around the best overall plan, rather than the 50 best individual plans. They usually aren't, so the result is typically F), at least in Cambridge.
Risk Management, Cybersecurity, & Privacy Specialist at Law Office of Charles J McCarthy
10 年Interesting post, Yossi. On the subway, which has an e-sign with estimated arrival times of the next trains, I will not pack into a subway car if the next train is only 3 minutes away. Instead I will wait for the next train which is practically empty since most just pack themselves into whatever train is in front of them. So, if busses had posted real-time arrival times, would this help eliminate the bunch-up from self-loading cargo?
Wonderful theory with explicative power. I will never look at bus stops the same. That said, I would still rather live in a city where the bus driver did wait a few extra seconds and I would like it even better if he said aloha when the malfeasant passenger boarded. There is something admirable about a more efficient transportation system, as in say Tokyo. But nothing beats a city full of interactive and friendly human beings.
Stealth Startup
10 年Interestingly, very similar phenomena can be regularly observed on many Chicago routs. I wonder if similar can be observed in Europe, in particular in Germany. If not, how do the Germans do it?
Director of Manufacturing at Catalyst Pharmaceuticals, Inc.
10 年Great post, Yossi! I'm old enough to say I love the Hollies and that song, which made it even more enjoyable. Whether logistics or execution in a plant, there is a great deal of truth to your example. Thanks for sharing - I intend to do the same with your article!