Poisson exponential tilting for a loss network (${\pi }_N(n)\propto 1_{\{An\le NC\}}{e}^{z^T(An-NC)}{\prod }_r\mathrm{Pr}\{\mathrm{Poisson}(N{x}_r^{\ast })=n_r\}$)

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.