Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 2 / 18H / b / i / Solution

The chain is a birth-death chain. Detailed balance between $0$ and $1$ givesFor $i\ge 1$, detailed balance between $i$ and $i+1$ givesConsequentlyThe expected occupation time of the positive even states during a return cycle to state $1$ is therefore