Entropy maximization for loss-network occupancy ($H_{\nu }(x)={\sum }_r[x_r\log {\nu }_r+x_r-x_r\log x_r]$)

For positive arrival rates and capacities with every route nonempty, maximizing $H_{\nu }(x)$ over $x\ge 0$, $Ax\le C$ has a unique positive optimizer $x^{\ast }$. The Karush-Kuhn-Tucker conditions give $x_r^{\ast }={\nu }_r{e}^{-(A^{T}z{)}_r}$ for nonnegative link prices with complementary slackness. Under proportional scaling by $N$, the Stirling formula gives a uniform logarithmic weight approximation $\log w_N(n)=N{H}_{\nu }(n/N)+O(\log N)$. Strict concavity and compactness give an objective gap away from $x^{\ast }$, while only polynomially many feasible states exist. Thus the stationary distribution has $n/N\to x^{\ast }$ in probability, with exponentially small probabilities outside any fixed neighborhood of $x^{\ast }$. In particular $\mathbb{E}[n]=N{x}^{\ast }+o(N)$.

Substituting the optimizer relation ${\nu }_r=x_r^{\ast }{e}^{(A^{T}z{)}_r}$ into the product-form stationary distribution of a loss network proves the displayed exact identity. The independent Poisson random variables give a reference law centered at the optimized occupancies, while the constraint and exponential factor account for saturated resources. An unconstrained Gaussian approximation can miss these boundary effects.