= Solution
For the <link-route incidence matrix> $A$, the <reduced-load approximation> and <Erlang loss formula> give
$$
\boxed{B_j=E(a_j,C_j),\qquad a_j=\sum_r A_{jr}\nu_r\prod_{k\ne j}(1-B_k)^{A_{kr}}.}
$$
Introduce $y_j=-\log(1-B_j)$ and accepted route flows $x_r=\nu_r e^{-(A^Ty)_r}$. Multiplying the reduced load by the link's own acceptance probability gives
$$
a_j(1-B_j)=\sum_r A_{jr}x_r=U(y_j,C_j).
$$
These are precisely the <first-order optimality conditions> for the <convex potential for the Erlang fixed point>, since
$$
\frac{\partial F}{\partial y_j}=-\sum_r A_{jr}\nu_r e^{-(A^Ty)_r}+U(y_j,C_j).
$$
Each exponential term is a <convex function>, and the integral of the strictly increasing function $U(\cdot,C_j)$ is a <strictly convex function>. Hence $F$ is a <strictly convex function>. Moreover $U(z,C_j)\to C_j>0$ as $z\to\infty$, by the saturation of the <carried load of an Erlang loss resource>. Its integrals tend to infinity at least linearly, so $F$ is a <coercive function> on the nonnegative orthant and has a unique minimizer.
At $y_j=0$, $U(0,C_j)=0$. If link $j$ is used by any positive-rate route, then $\partial_jF<0$, excluding a boundary minimizer there. An unused link instead minimizes at $y_j=0$ and has zero derivative. Thus the minimizer satisfies the displayed flow equations for every link. Conversely, those equations imply $\nabla F=0$ and therefore minimize the <convex potential for the Erlang fixed point>. Finally $U(y,C)=a[1-E(a,C)]$ with $y=-\log(1-E(a,C))$ converts the flow equations back to $B_j=E(a_j,C_j)$. \b[The Erlang fixed point approximation has exactly one solution under fixed routing.]
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