 Solution

The queue length is a birth-death process with birth rate $\lambda r^n$ in state $n$ and death rate $\mu \min (n,2)$. Its reversible weights satisfyFor $0<r<1$, the factor $r^{n(n-1)/2}$ makes their sum finite for every $\lambda ,\mu >0$. For $r=1$, the tail is geometric with ratio $\lambda /(2\mu )$, so a stationary distribution exists exactly when $\lambda <2\mu$.