I have a set of marbles and a number line from 0 to infinity. Every step I either put a new marble on the number 0 or I move one existing marble (chosen uniformly) to the next number. The ratio between new marbles and propagating existing marbles is $r$. That is, for every new marble I put down, $r$ marbles have been propagated since the last new marble.
At any given time I want to know the distribution of my marbles. How would I model this? I don't really know where to start.
Also, eventually I would want to change how I select the marble to propagate. Maybe I want to favor the marbles closer to 0, or maybe I want to guarantee that each number on the line has at most $k$ marbles. This second suggestion sounds quite a bit more complex though.
Any thoughts?
After $n$ marbles are on the line, the sum of the positions is $(n-1)r$. The average position of the first marble to be placed is $r(1+\frac 12 + \frac 13 + \dots \frac 1{n-1})=rH_{n-1} \approx r (\log (n-1)+ \gamma)$ because it is certain to be moved the first $r$ moves, has $\frac 12$ chance for the next $r$ and so on. $H_k$ is the $k$th harmonic number. The average position of the second marble is $r(H_{n-1}-1)$ and so on. There will be substantial dispersion.
The other way to model it is to write a program that throws random numbers, using whatever distribution you want, and collect the statistics. Do a bunch of tries and see what comes out. If the choice of which marble to move is not uniform, you will need this.