M-M-1 queue with state-dependent admission

If potential arrivals form a Poisson process of rate $\lambda$ and an arrival seeing $n$ customers joins with probability $p(n)$, the queue length is a birth-death process with birth rate $b_n=\lambda p(n)$ and death rate $d_n=\mu$ for $n\ge 1$. Its stationary ratios, when normalizable, are