= Solution
The objective approximates the logarithm of the <product-form stationary distribution of a loss network>. To make the approximation precise, multiply arrival rates and capacities by $N$ and write $x=n/N$. The <Stirling formula> gives, uniformly on the feasible compact set,
$$
\log\prod_r\frac{(N\nu_r)^{n_r}}{n_r!}=N H_\nu(n/N)+O(\log N),\qquad H_\nu(x)=\sum_r\bigl[x_r\log\nu_r+x_r-x_r\log x_r\bigr],
$$
with $0\log0=0$. The feasible set is $K=\{x\geq0:Ax\leq C\}$. Positive capacities and a nonempty route for each class make $K$ compact with a strictly positive feasible point. Since $H_\nu$ is a <strictly concave function>, its maximizer $x^*$ is unique; its infinite inward derivative at $x_r=0$ makes every component positive.
The <Karush-Kuhn-Tucker conditions> give nonnegative link <Lagrange multipliers> $z_j$ satisfying
$$
\boxed{x_r^*=\nu_r e^{-(A^Tz)_r},\qquad z_j\bigl(C_j-(Ax^*)_j\bigr)=0.}
$$
For any neighborhood of $x^*$, continuity and uniqueness give a strictly positive objective gap outside it. There are only polynomially many feasible lattice states, while their weights differ exponentially in $N$. The feasible point $\lfloor Nx^*\rfloor$ has objective tending to $H_\nu(x^*)$. It follows that the stationary law concentrates at $x^*$ on the scale $n/N$; in particular,
$$
\boxed{\frac nN\longrightarrow x^*\text{ in probability},\qquad \mathbb E[n]=Nx^*+o(N).}
$$
Thus NETWORK predicts the dominant occupancy and mean occupancy in a large <loss network>. For unscaled inputs it gives a continuous approximation to the modal integer state.
It also gives an exact useful change of variables. Substituting $\nu_r=x_r^*e^{(A^Tz)_r}$ into the equilibrium weights yields
$$
\pi_N(n)\propto\mathbf1_{\{An\leq NC\}}e^{z^T(An-NC)}\prod_r\Pr\{\operatorname{Poisson}(Nx_r^*)=n_r\}.
$$
This is <Poisson exponential tilting for a loss network>: independent <Poisson random variables> centered at the predicted occupancies, restricted to the feasible region and penalized for slack in positively priced resources. It explains why an unconstrained <Gaussian approximation> should not automatically be applied at a saturated capacity boundary.
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