= Product-form stationary distribution of a closed migration process
{title2=$\pi(n)\propto\prod_j\theta_j^{n_j}/\prod_{k=1}^{n_j}\phi_j(k)$}
For an irreducible <closed migration process> with finite population $N$, positive departure rates and routing matrix $P$, choose positive traffic weights satisfying $\theta=\theta P$. On $\sum_jn_j=N$, its <stationary distribution> has the displayed form. For a predecessor $m=n+e_j-e_k$, its weighted transition into $n$ is $\pi(n)(\theta_j/\theta_k)P_{jk}\phi_k(n_k)$. Summing over $j$ gives $\pi(n)\phi_k(n_k)$ by the traffic equation, and summing over $k$ proves <global balance for a continuous-time Markov chain>. The distribution need not satisfy <detailed balance>.
// Target: probability-and-statistics.bigb
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