Queues are a fact of everyday life and understanding what makes the length of lines vary and how to minimize wait times is of course an important factor for any establishment that has to deal with customers. One of the things that fascinated me in my first statistics course was queuing theory that dealt with this very question. I remember one particular insight. If you look at any train or subway station that has three escalators that carry people between the platform and the street, you will find that it is almost always the case that two of them are going from the platform to the street level and only one the other way. This does not matter if the street level is above or below the platform, nor does it depend on the time of day. The reason is quite simple. People coming to catch a train arrive at random times and thus can be accommodated with a single escalator but when a train arrives at a station, it disgorges a whole bunch of people at once and this requires more exit capability. The 2-1 arrangement is designed to take care of that.

This article by Vikash V. Gayah and S. Ilgin Guler at Pennsylvania State University, explains a commonly experienced annoying phenomenon of why you sometimes wait a long time for a bus that is late only to find a couple of them, or even more, arriving together.

[O]nce a bus gets behind schedule, it’s nearly impossible for it to get back on track. It will continue to get further and further behind schedule until the next bus on the route catches up.

The same thing happens to buses that are early: They continue to get earlier and earlier as they travel through their route, until they catch up to the bus just ahead.

Equations that describe how buses move along a route identify why this happens. The time buses spend serving passengers at a stop is related to the amount of time between consecutive bus arrivals, commonly known as bus headway. When a bus runs late, its headway increases and more passengers arrive that need to be served at its next stop. But the more passengers waiting at a stop, the longer a bus needs to spend there. So late buses need to spend more time at each subsequent stop, causing them to run even later. The opposite happens for a bus that’s early. This cycle continues until multiple buses eventually catch up to each other and bunch.

So what can be done about this?

One strategy is to instruct late buses to skip stops where passengers don’t need to get off or to limit the number of people allowed to board late buses at each stop. Both of these allows the late bus to spend less time at each stop, which allows it the opportunity to catch up. Of course, doing so can leave potential users stranded.

Another common strategy is to build more time than needed into a bus’s schedule. This additional time – called slack – helps accommodate the variability in bus travel time. Buses that are early are instructed to hold at selected stops until the scheduled time to depart. However, this strategy does not help late buses recover.

The catch is that passengers waiting at a stop get infuriated when they see a bus zipping by without stopping. You are also not going to win any friends by limiting the number of people who can get on a bus when they can see that there is space inside. Those already on a bus will be annoyed at simply sitting in one place for no apparent reason.

The development of real-time technology that tells drivers and dispatches exactly how far they are away from the next bus in front or behind can help nudge them faster and slower and keep them within a given range of the adjacent buses. This will be important if we are to encourage the use of mass transit.