= Product-form stationary distribution of an open migration process
{title2=$\pi(n)\propto\prod_j\alpha_j^{n_j}/\prod_{k=1}^{n_j}\phi_j(k)$}
For an <open migration process>, colony $j$ with population $k$ has total departure rate $\phi_j(k)$. Its stationary weight is $b_j(k)=\alpha_j^k/\prod_{h=1}^k\phi_j(h)$. The colony weights factor independently and normalize when $\sum_{k\geq0}b_j(k)<\infty$ for every colony. <Global balance for a continuous-time Markov chain> proves the formula even when routing is not reversible.
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