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Only routes with $n_r>0$ contribute to proportional fairness or to the weighted logarithmic utility; omit inactive routes from expressions with a rate in the denominator. Active routes have positive rates whenever a positive feasible allocation exists. For a feasible competitor $y$, the elementary logarithm inequality $\log u\le u-1$ givesA zero competitor rate on an active route has utility $-\infty$ and also cannot improve the objective. Therefore a proportionally fair allocation maximizes the logarithmic objective. Conversely, the feasible set is a convex set, and differentiating the objective along $x+t(y-x)$ at $t=0+$ shows that an optimizer satisfies the proportional fairness inequality. Strict concavity gives uniqueness on active coordinates; rates assigned to nonexistent flows are immaterial.

In the linear flow network, put $z_0=n_0{x}_0$ and $z_i=n_i{x}_i$ for active routes. A local route $i$ uses only resource $i$. If $n_i>0$ and $z_0+z_i<1$, increasing $x_i$ increases the objective without violating any other constraint. Thus each active local route saturates its resource:Write $L={\sum }_{i=1}^I{n}_i$ and $N=n_0+L$. For $n_0>0$ and $L>0$, substitute $z_i=1-z_0$ into the objective. Apart from constants its dependence on $z_0$ isIts derivative vanishes when $n_0/z_0=L/(1-z_0)$, and its second derivative is negative. Hence the per-flow through-route rate isIf $L=0<n_0$, the optimal through-route aggregate rate is one, so $x_0=1/n_0=1/N$ still holds. If $n_0=0<L$, each active local route has $x_i=1/n_i$, also given by the displayed local formula. When $N=0$ there are no flows and no departures. Inactive coordinates need not be assigned these undefined per-flow formulas.

Independent arrivals and exponential document sizes make this a flow-level network model. A document of type $r$ has residual-size hazard ${\mu }_r$ per unit of data transmitted; when its transmission speed is $x_r(n)$, its completion hazard per unit time is ${\mu }_r{x}_r(n)$. Thus the continuous-time Markov chain of flow counts has transition intensitiesFor $N>0$ these departure intensities are ${\mu }_0{n}_0/N$ for the through route and ${\mu }_{i}L/N$ for an active local route. Total arrival intensity is constant and total departure intensity is bounded by ${\sum }_r{\mu }_r$, ensuring a nonexplosive Markov chain.

Set ${\rho }_r={\nu }_r/{\mu }_r$, the offered traffic, and define the balance function of a flow-level networkThe binomial coefficients satisfyThese identities also cover states with only one kind of active route. They show that the weight $a(n)=\Phi (n){\prod }_{r=0}^I{\rho }_r^{n_r}$ satisfiesThus detailed balance for a continuous-time Markov chain proves that this is a reversible Markov chain and reduces the stationary distribution calculation to normalization.

Sum over $n_0$ first. The negative binomial series gives, for fixed local counts and $L={\sum }_i{n}_i$,All summands are nonnegative, so the Tonelli theorem permits exchanging the sums. The partition sum is thereforeIt is finite exactly when ${\rho }_0+{\rho }_i<1$ for every $i$, for $I\ge 1$. These are the resource-load conditions: resource $i$ receives the through-route offered traffic plus its local offered traffic. The normalized stationary distribution isWith positive arrival rates this is the unique stationary distribution of the irreducible feasible flow-count chain. Zero arrival rates restrict its closed class to the corresponding zero coordinates.

Summing out $n_0$ also shows that the local counts are independent, with geometric distributions on $0,1,\dots$:Hence their mean flow counts areThe binomial balance factor is essential: independent geometric counts for all routes would not give these departure rates or the correct resource-load normalization.