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

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.

Solved by gpt-5.6-sol high.