= Entropy maximization for loss-network occupancy
{title2=$H_\nu(x)=\sum_r[x_r\log\nu_r+x_r-x_r\log x_r]$}
For positive arrival rates and capacities with every route nonempty, maximizing $H_\nu(x)$ over $x\geq0$, $Ax\leq C$ has a unique positive optimizer $x^*$. The <Karush-Kuhn-Tucker conditions> give $x_r^*=\nu_r e^{-(A^Tz)_r}$ for nonnegative link prices with <complementary slackness>. Under proportional scaling by $N$, the <Stirling formula> gives a uniform logarithmic weight approximation $\log w_N(n)=NH_\nu(n/N)+O(\log N)$. Strict <concavity> and compactness give an objective gap away from $x^*$, while only polynomially many feasible states exist. Thus the <stationary distribution> has $n/N\to x^*$ in probability, with exponentially small probabilities outside any fixed neighborhood of $x^*$. In particular $\mathbb E[n]=Nx^*+o(N)$.
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