= Solution
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
$$
\boxed{μ\mathbb P(X>0)=1-π_N.}
$$
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