$$ \Gamma(G)= \max\{\frac{|E(W)|}{\lfloor |W|/2\rfloor}: W\subseteq V, 3\leq |W|\equiv 1\pmod{2}\}, $$
where $E(W)$ is the set of edges having both ends in $W$. We show that the chromatic index $\chi'(G)$ is asymptotically $\max\{D(G),\Gamma(G)\}$. The latter is, by Edmonds' Matching Polytope Theorem, the fractional chromatic index of $G$. The proof uses a probabilistic approach, based on entropy-maximizing (also called ``normal'') distributions on matchings, to go from fractional to ordinary (integer) colorings.