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

The queue length is an M-M-1 queue and hence a birth-death process with birth rate $\lambda$ and death rate $\mu$ away from zero. The detailed-balance equations areWriting $\rho =\lambda /\mu$, they give ${\pi }_n={\pi }_0{\rho }^n$. Since $\rho <1$, this measure is summable and normalization yieldsThe existence of this invariant probability distribution for the irreducible nonexplosive chain proves positive recurrence.