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

With state-dependent admission, the effective birth rate in state $n$ iswhile the death rate is $\mu$ for $n\ge 1$. The stationary ratios for an M-M-1 queue with state-dependent admission are thereforeInduction gives ${\pi }_n={\pi }_0{\rho }^n/n!$, and normalization identifies a Poisson distribution of mean $\rho$: