= Convex potential for the Erlang fixed point
{title2=$F(y)$}
Set $y_j=-\log(1-B_j)$ and let $e_C(a)=E(a,C)$. For positive integer capacities,
$$
U(z,C)=e^{-z}e_C^{-1}(1-e^{-z}),\qquad
F(y)=\sum_r\nu_r e^{-(A^Ty)_r}+\sum_j\int_0^{y_j}U(z,C_j)\,dz.
$$
The function $U$ is the <carried load of an Erlang loss resource> evaluated at its inverse blocking parametrization. It is continuous and strictly increasing from zero to $C$. Hence $F$ is a <coercive function> and a <strictly convex function> on the nonnegative orthant. Its first-order conditions are the fixed-routing <Erlang fixed point> equations, including zero prices at unused links.
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