An f-coloring of a multigraph \(G=(V,E)\) is a coloring of edges E such that each color appears at each vertex \(v\in V\) at most f(v) times. The minimum number of colors needed to f-color G is called the f-chromatic index \(q^*_ f(G)\) of G. Various scheduling problems on networks are reduced to finding an f-coloring of a multigraph. This paper gives an upper bound on the f-chromatic index. Denote by d(v) the degree of vertex \(v\in V\), and define \(d_ f(G)=\max_{v\in V}\lceil d(v)/f(v)\rceil\). Denote by E(H) and V(H) the edge and vertex sets of a subgraph H of G respectively, and define \[ r_ f(G)=\max_{H\subset G}\lceil | E(H)| /\lfloor \sum_{v\in V(H)}f(v)/2\rfloor \rceil \] where H runs over all subgraphs of G having at least three vertices. Then our bound is \[ q^*_ f(G)\leq \max \{r_ f(G),\lfloor (9d_ f(G)+6)/8\rfloor \}. \] The proof is constructive, and yields a polynomial-time algorithm to f-color G with at most \(\lfloor (9q^*_ f(G)+6)/8\rfloor\) colors.