= Poisson exponential tilting for a loss network
{c}
{title2=$\pi_N(n)\propto\mathbf1_{\{An\leq NC\}}e^{z^T(An-NC)}\prod_r\Pr\{\operatorname{Poisson}(Nx_r^*)=n_r\}$}
Substituting the optimizer relation $\nu_r=x_r^*e^{(A^Tz)_r}$ 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.
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