 Stationary law of a two-resource linear flow network

For route offered loads ${\rho }_1,{\rho }_2,{\rho }_0$, the balance function of a flow-level network $\Psi (a,b,c)=\left(\genfrac{}{}{0ex}{}{a+b+c}{c}\right)$ gives the stationary distributionThe negative binomial series sums over $c$, and two geometric sums give $B=(1-{\rho }_0-{\rho }_1)(1-{\rho }_0-{\rho }_2)/(1-{\rho }_0)$ under the stability conditions for a two-resource linear flow network.