Convex potential for the Erlang fixed point ($F(y)$)

Set $y_j=-\log (1-B_j)$ and let $e_C(a)=E(a,C)$. For positive integer capacities,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.

Update one link blocking probability at a time using the reduced load accepted by all other links, then cycle through the links. In logarithmic acceptance coordinates this is exact coordinate descent for the convex potential for the Erlang fixed point. Potential values decrease within a compact sublevel set. Continuity of a whole-sweep update forces every limit point to minimize every coordinate, hence to equal the unique global minimizer. This proves convergence without requiring a simultaneous substitution scheme.