Solution
= Solution
On states $0,\ldots,N$, births have rate $1$ below $N$ and deaths have rate $μ$ above $0$. For $μ>0$ this finite <continuous-time Markov chain> is irreducible, so its unique stationary law is
$$
π_n={μ^{-n}\over\sum_{j=0}^N μ^{-j}},\qquad0\leq n\leq N.
$$
For $μ=1$ this is uniform. If the parameter value $μ=0$ is admitted, state $N$ is absorbing and the unique stationary law is instead the point mass at $N$.