Detailed balance for a birth-death process
= Detailed balance for a birth-death process
For birth rates $b_i$ and death rates $d_i$, a reversible measure satisfies
$$
\pi_i b_i=\pi_{i+1}d_{i+1}.
$$
Thus $\pi_{i+1}/\pi_i=b_i/d_{i+1}$, and it is an invariant distribution when the resulting measure is summable and the chain is nonexplosive.