Product-form stationary distribution of an open migration process ($\pi (n)\propto {\prod }_j{\alpha }_j^{n_j}/{\prod }_{k=1}^{n_j}{\varphi }_j(k)$)

For an open migration process, colony $j$ with population $k$ has total departure rate ${\varphi }_j(k)$. Its stationary weight is $b_j(k)={\alpha }_j^k/{\prod }_{h=1}^k{\varphi }_j(h)$. The colony weights factor independently and normalize when ${\sum }_{k\ge 0}{b}_j(k)<\infty$ for every colony. Global balance for a continuous-time Markov chain proves the formula even when routing is not reversible.