Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 27K / c

For state $n\ge 1$, births occur at rate $\lambda$ and departures at rateThe birth--death detailed-balance weights areTheir ratios tend to zero, so the series is summable and normalization gives an invariant probability distribution.

The chain is irreducible and nonexplosive, and it has an invariant probability distribution. Therefore every state is positive recurrent. The increasingly strong abandonment rate is what stabilizes the system even when $\lambda \ge \mu$.