Suppose I have $n \in \mathbb{N}$ bins, each having unit volume. Let $a_1,a_2,\ldots,a_{n-1} \in \mathbb{R},\;a_k \geq 1\;\forall\;k$.
An integer sequence is generated as follows:
A man shows up with a random quantity of water $x$. He marks bin $1$ with chalk and pours water into it until he either runs out of water or the bin is full. If he runs out of water, he adds "$1$'' to the sequence and leaves. If the bin is full, he pours out all but $\frac{1}{a_1}$ units of water in it onto the ground. Then he marks bin $2$ with chalk and begins pouring the remainder of bin $1$ into bin $2$. He does this until either bin $2$ is full or he runs out of water.
If he runs out of water, he replaces (the now empty) bin $1$ and resumes pouring the remainder of $x$ into it. If it fills up again, he repeats the process.
If instead he fills up bin $2$, he pours out all but $\frac{1}{a_2}$ of it, marks bin $3$ and starts pouring the remainder of bin $2$ into it. And so on and so forth. If the $n$'th bin is filled up, he just dumps it out entirely and goes back to the $n-1$'th bin.
Eventually he must run out of water. The number he adds to the sequence is the highest bin he's marked with chalk. Then he erases all the chalk marks and leaves.
He repeats this process $m$ times, with random amounts of water each time.
As $m \to \infty$, the frequencies of the integers in the sequence will tend to a distribution based on $\{a_k\}_k$ and the distribution of $x$. The process is vaguely reminiscent of dithering or error diffusion in image processing, although one-dimensional and distinctly different.
My question is: Is there a formal name for processes of this type?
Examples
If $x = \frac{1}{2}$ and $a_k = 2 \;\forall\;k$, the sequence produced is $(1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,\ldots)$.
If $x = \frac{1}{3}$ and $a_k = 2 \;\forall\;k$, the sequence produced is $(1,1,2,1,1,3,1,1,2,1,1,4,1,1,2,1,1,\ldots)$.
If $x = \frac{1}{2}$ and $a_k = 3 \;\forall\;k$, the sequence produced is $(1,2,1,2,1,3,1,2,1,2,1,4,1,2,1,2,1,\ldots)$.
If $x = \frac{1}{3}$ and $a_k = 3 \;\forall\;k$, the sequence produced is $(1,1,2,1,1,2,1,1,3,1,1,2,1,1,2,1,1,\ldots)$.