Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 27K / c

In equilibrium the chain is reversible by detailed balance. Reverse time. Forward departures become reverse accepted arrivals; reversibility identifies the reversed chain with the original one, whose attempted arrivals form a Poisson process. The accepted points are exactly those occurring while the state is below $N$. The standard quasi-reversibility argument therefore makes the equilibrium departure process Poisson, with rate equal to the mean departure rate