# Distribution of marbles on number line

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.
• By `and so on' do you mean that the third marble would have average position $r (H_{n-1} - (1 + \frac{1}{2}))$ and the fourth would have average position $r (H_{n-1} - (1 + \frac{1}{2} + \frac{1}{3}))$? Commented Jul 3, 2014 at 20:13
• I don't see anything easy. Of course if $r$ is small, you know the last few marbles are below $20$, and the next to last few are very likely below $20$ Commented Jul 5, 2014 at 4:26
• It seems to follow an exponential distribution perfectly. I'm using $\lambda = 1/r$. plot image Is there any way to prove this? Commented Jul 5, 2014 at 15:03