Stationary law of a proportionally fair four-cycle ($\pi (n)\propto \left(\genfrac{}{}{0ex}{}{{\sum }_r{n}_r}{n_1+n_3}\right){\prod }_r{\rho }_r^{n_r}$)

The proportionally fair allocation on a four-cycle has aggregate service $P/(P+Q)$ on each active odd route and $Q/(P+Q)$ on each active even route. With Poisson arrival rates ${\nu }_r$ and exponential document-size parameters ${\mu }_r$, the departure intensity is ${\mu }_r{n}_r{x}_r$. The adjacent ratios of the displayed binomial coefficient weight cancel those intensities in detailed balance, with ${\rho }_r={\nu }_r/{\mu }_r$.

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 isThis formula applies in the stability region for the reversible four-cycle flow network. Grouping populations on opposite routes gives ${\sum }_{s,t}\left(\genfrac{}{}{0ex}{}{s+t}{s}\right)h_s(a,c)h_t(b,d)$, where $h_s(a,c)={\sum }_{i=0}^s{a}^i{c}^{s-i}$. 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 $(1-2z+2z^2)/(1-2z{)}^3$.

Group the normalizing sum by its odd and even flow counts. The coefficient sums are $h_p({\rho }_1,{\rho }_3)={\sum }_{i=0}^p{\rho }_1^i{\rho }_3^{p-i}$ and the analogous $h_q$ for even routes. If $a$ and $b$ are the two maximum loads, then $a^p\le h_p\le (p+1)a^p$ and similarly for $b$. The generating identity ${\sum }_{p,q}\left(\genfrac{}{}{0ex}{}{p+q}{p}\right)a^p{b}^q=(1-a-b{)}^{-1}$ proves necessity and sufficiency of the displayed strict inequality. It is equivalent to strictly subunit offered load on each of the four resources.