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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,with $0\log 0=0$. The feasible set is $K=\{x\ge 0:Ax\le C\}$. 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 $x^{\ast }$ is unique; its infinite inward derivative at $x_r=0$ makes every component positive.

The Karush-Kuhn-Tucker conditions give nonnegative link Lagrange multipliers $z_j$ satisfyingFor any neighborhood of $x^{\ast }$, 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 $\lfloor N{x}^{\ast }\rfloor$ has objective tending to $H_{\nu }(x^{\ast })$. It follows that the stationary law concentrates at $x^{\ast }$ on the scale $n/N$; in particular,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 ${\nu }_r=x_r^{\ast }{e}^{(A^{T}z{)}_r}$ into the equilibrium weights yieldsThis 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.