My examples understate the real life problem though.
This missing factor is that it also costs more shaft time to serve a higher floor with a longer travel distance than it does a lower floor with a shorter travel distance. So in the simple elevator case, people from floor one hundred are going to be spending more time in the shaft than people from floor ten.
Still not exponential, but cubic-ish growth puts a hard limit on things.
Due to the passengers the rate of travel for an elevator is not constant:
* There's a maximum rate of acceleration and deceleration that people are comfortable with.
* A longer travel distance means more time to speed up / slow down.
* Thus an elevator or two which are dedicated to operate at the longer distance travel that distance at a more effective speed and provide more rapid service.
* Even more importantly, it also cuts down on the number of slow elevator shafts necessary to service floors by allowing them to be stacked.
There's one in Leicester University (not the only one in the UK, I'm sure).
Really not that dangerous, with a cable near the top of each aperture that triggers a stop if it's touched.
I'm glad they still have it, I was nervous when I searched that I would find out it had been ripped out since I was last there.
There are effectively two vertical shafts, and the elevator car goes up one and down the other. If the car flipped over as it goes over the top to the other shaft, then what was the floor would become the ceiling. So, tape something to the floor to identify it as it's going up, then wait for that car to come down the other shaft and look at where your identifying paper is.
It does seem pretty obvious that it wouldn't flip over - that seems like an insanely dangerous design.
There must be more spiral escalators now. I know there's one in Nordstrom's too.
EDIT: Unless there's some dependence between m and n, like if taller buildings always need more shafts or something.
The dependence is on keeping wait times reasonable. Taller buildings mean each shaft's elevator spends more time travelling, i.e. less time being available.
It's really rough in convention center hotels when a huge convention comes to town a lot of the time it's practically impossible to get an elevator down from the middle floors because it's already full from the floors above you.
Most people are going between the ground floor and whatever floor their destination is on, so a shaft with two cars that go 0-50 and 51-100 wouldn't be very useful.
Say you design a 50 story building and it has one bank of elevators that all run from 1 to 50. What if you wanted to make it twice as high, and have the same elevator capacity for the top half?
You can add another identically sized bank that starts at the ground floor, skips up to 51 and then covers from there to 100.
In the top half of the building you used the same fraction of space for your elevator shafts as you'd used originally. Great! Except the elevator shafts to get there had to go through floors 2-50, and they ate up as much space there as the original elevator bank did. The building has 2x the original volume, but you're spending 3x the space on elevator shafts so you didnt double the useful floor area.
As you try to design taller and taller buildings, the increasing cost of vertical circulation gets worse and worse.
Say you have a 5 story building that needs a one shaft elevator. 5 floors tiles 100 sqft per floor is 500 sqft of wasted shaft space. Then say you have a 10 story building that needs two shafts. Now you are wasting 10 floors times two shafts or 2,000 sqft. And so on.
An old and terrifing solution to this was to use two shafts and run a continuous loop of open door elevators.
https://m.youtube.com/watch?v=Ro3Fc_yG3p0