Solution
= Solution
With state-dependent admission, the effective birth rate in state $n$ is
$$
b_n=\lambda p(n)=\frac{\lambda}{n+1},
$$
while the death rate is $\mu$ for $n\geq1$. The stationary ratios for an <M-M-1 queue with state-dependent admission> are therefore
$$
\frac{\pi_{n+1}}{\pi_n}=\frac{\rho}{n+1}.
$$
Induction gives $\pi_n=\pi_0\rho^n/n!$, and normalization identifies a <Poisson distribution> of mean $\rho$:
$$
\boxed{\pi_n=e^{-\rho}\frac{\rho^n}{n!}}.
$$