Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 27J / b / i / Solution

{c}

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 $n\geq1$. Its stationary ratios, when normalizable, are
$$
\frac{\pi_{n+1}}{\pi_n}=\frac{\lambda p(n)}\mu.
$$

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 Incoming links: M-M-1 queue with state-dependent admission