 Solution

On states $0,\dots ,N$, births have rate $1$ below $N$ and deaths have rate $\mu$ above $0$. For $\mu >0$ this finite continuous-time Markov chain is irreducible, so its unique stationary law isFor $\mu =1$ this is uniform. If the parameter value $\mu =0$ is admitted, state $N$ is absorbing and the unique stationary law is instead the point mass at $N$.