I have a graph $G=(V,E)$ with $|V|=n$ nodes. Define a markov chain matrix P on G (e.g. Metropolis-Hastings). I have $k$ random walkers which are deployed at time $t=0$ on the vertices of $G$ at random based on the stationary distribution of $P$ (for simplicity). If more than one walkers fall in the same node, they are placed in a buffer. At time $t=1$, the top walker in each non-empty node's buffer leaves that node and makes a random transition to a neighbour.

The problem is how to characterize the statistics of the markovian random vector of buffer sizes. This is equivalent to a repeated experiment of throwing k balls in n bins, with probabilities indicated by the stationary distribution of P (at least if $G$ is complete).

Notice that if two walkers at nodes i and j move to the same node r, then in the next step only one of the two can leave r and the other is buffered. Thus the markov chain describing the evolution of the buffer size vector is non-reversible.