Imagine a hierarchy with $N$ layers (maybe $N=6$, say). And under each person except the bottom layer there are $M$ people ($M=30$, say). A person works for a length of time $T$ ($T=50$ years, say). After this time they retire. If they are a boss, one of the $M$ random employees gets promoted. Also, a new person joins the hierarchy at the bottom. Seed this simulation with random aged people and let this simulation run for a while. After it has run for a while a new person joins the hierarchy at the bottom.
What is the probability $P(X)$ that within a lifetime this person reaches level $X$? What is the probability $P(N)$ that (s)he reaches all the way to the top? i.e. becomes president?
Bonus questions: What is the average age of the president over time? And average age of people on each level of the hierarchy? How long on average does someone stay president until they retire?
My instincts are that the majority of people don't get promoted very far. And also that the president is always quite old.
Is there a name for this type of problem?
Edit: Running some simulations I seem to get that for $N=6$, and $T=50$ a president stays on average for about 10 years it seems to not depend too much on M which is odd. In fact I get an empirical estimate of roughly $T/N$ years for an average presidential term from simulations. Can this be explained? Only that it would take longer for a president to make it up more layers and so have a shorter time in office until retirement. I guess this makes sense.