= Four-cycle partition function
{title2=$B(a,b,c,d)$}
For offered loads on consecutive routes $a,b,c,d$ of the reversible <proportionally fair allocation on a four-cycle>, the normalizing sum of the <stationary distribution> is
$$
B=\frac{(1-a)(1-b)(1-c)(1-d)-abcd}{(1-a-b)(1-a-d)(1-c-b)(1-c-d)}.
$$
This formula applies in the <stability region for the reversible four-cycle flow network>. Grouping populations on opposite routes gives $\sum_{s,t}\binom{s+t}{s}h_s(a,c)h_t(b,d)$, where $h_s(a,c)=\sum_{i=0}^sa^ic^{s-i}$. For distinct opposite loads, substituting $h_s=(a^{s+1}-c^{s+1})/(a-c)$ and summing the four geometric generating series proves the formula; <continuity> covers coincident loads. If all four loads equal $z<1/2$, it reduces to $(1-2z+2z^2)/(1-2z)^3$.
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