Elevator Paradox - Modeling The Elevator Problem

Modeling The Elevator Problem

Several attempts (beginning with Gamow and Stern) were made to analyze the reason for this phenomenon: the basic analysis is simple, while detailed analysis is more difficult than it would at first appear.

Simply, if one is on the top floor of a building, all elevators will come from below (none can come from above), and then depart going down, while if one is on the second from top floor, an elevator going to the top floor will pass first on the way up, and then shortly afterward on the way down – thus, while an equal number will pass going up as going down, downwards elevators will generally shortly follow upwards elevators (unless the elevator idles on the top floor), and thus the first elevator observed will usually be going up. The first elevator observed will be going down only if one begins observing in the short interval after an elevator has passed going up, while the rest of the time the first elevator observed will be going up.

In more detail, the explanation is as follows: a single elevator spends most of its time in the larger section of the building, and thus is more likely to approach from that direction when the prospective elevator user arrives. An observer who remains by the elevator doors for hours or days, observing every elevator arrival, rather than only observing the first elevator to arrive, would note an equal number of elevators traveling in each direction. This then becomes a sampling problem — the observer is sampling stochastically a non uniform interval.

To help visualize this, consider a thirty-story building, plus lobby, with only one slow elevator. The elevator is so slow because it stops at every floor on the way up, and then on every floor on the way down. It takes a minute to travel between floors and wait for passengers. Here is the arrival schedule for people unlucky enough to work in this building; as depicted above, it forms a triangle wave:

If you were on the first floor and walked up randomly to the elevator, chances are the next elevator would be heading down. The next elevator would be heading up only during the first two minutes at each hour, e.g., at 9:00 and 9:01. The number of elevator stops going upwards and downwards are the same, but the odds that the next elevator is going up is only 2 in 60.

A similar effect can be observed in railway stations where a station near the end of the line will likely have the next train headed for the end of the line. Another visualization is to imagine sitting in bleachers near one end of an oval racetrack: if you are waiting for a single car to pass in front of you, it will be more likely to pass on the straight-away before entering the turn.