I was doing some simulations on societal structures. For example, given a population, with a heirachy of N levels. e.g. one prime-minister, 10 cabinet members, 100 MPs, and so on. Let $P$ be the number of people that a person on a higher level has under their control. (In this case P=10).
And assuming that everyone starts their carreer on the bottom of the heirachy. And that everyone retires at a specified age. When someone retires, one person on the level below randomly gets promoted (and someone moves up to replace them and so on) and one new person starts at the bottom level. (We can assume some steady birth rate so the working population stays constant.)
Then I asked questions such as how long will the person at the top stay in power?
Let $L$ be the length of anyone's career before retirement. (e.g. 50 years).
I found that the average length of time the top person will stay in their job will be $L/N$ years.
In other words for a flatter heirachy the ruler will stay in basically half their life. Whereas for a tall heirachy the ruler will usually be very old by the time they get to the top and only stay there for a short time. (Think of the case where $N\approx 1$ and the case where $P\approx 1$).
In fact I compared this with the UK heirachical structure (I set $P=30$ and $N\approx 5.3$) and I get the length of time for a prime minister to be roughly the right 10 years. Which is kind of right.
But I also wanted to calculate the probability that someone in the lowest heirachichy will never get promoted and stay on the bottom rung all their life? Or to be more general the probability that someone will reach level $M$ before retirement.