Danger! Only for math buffs!
As we all know, Elo is Gaussian. In theory. Except it isn't. But my question is not about that distribution. Your Elo might also list a second number: the number of tournaments rated. To my best knowledge, FIDE Elo only has the "K number". My German national Elo ("DWZ") lists this number, though. Can you say something about the statistical distribution of that variable, making some natural assumptions on the process (players take up and leave chess with a constant rate; the number of tournaments per time of individual players might be Gaussian)? I also would be interested in a numerical simulation of that process.
Legend: The lower curve was compiled by me and shows the number of rated tournaments of all rated Hamburg players, just pre-Corona. (I'm a chess maniac and show-off, I'm #2 or so :-) The upper curve - which determines the axis numbers! - is from my CS bachelor thesis and completely unrelated, but I found it always funny how similar it looks, except for the cutoff. Unfortunately I don't know a formula for that one, either. In first approximation, both are Poisson or Zipf or so.)
EDIT! As announced, here is the output of a very primitive simulation. We have an array A of length 1000 (players), with p=0.01 a chessplayer drops out and at the same time a new one starts at 0 (which is of course not realistic). With q=0.1, a player gets a new tournament rating (again, as stated in a comment, more than one should be updated). Essentially the only free variable is p/q (assuming both are smallish). Red: 50000 runs, Blue: 100000. (A player with 92 rated tournaments leads by far.) Evidently, a dynamic steady state was reached.