Solution
= Solution
The queue length is a <birth-death process> with birth rate $λr^n$ in state $n$ and death rate $μ\min(n,2)$. Its reversible weights satisfy
$$
π_n=π_0\prod_{j=0}^{n-1}{λr^j\over μ\min(j+1,2)}.
$$
For $0<r<1$, the factor $r^{n(n-1)/2}$ makes their sum finite for every $λ,μ>0$. For $r=1$, the tail is geometric with ratio $λ/(2μ)$, so a <stationary distribution> exists exactly when $λ<2μ$.