We consider two flow control schemes for Best Effort traffic in on-chip architectures, which can be deemed as the solutions to the boundary extremes of a class of utility maximization problem. At one extreme, we consider the so-called Rate-Sum flow control scheme, which aims at improving the performance of the underlying system by roughly maximizing throughput while satisfying capacity constraints. At the other extreme, we deem the Max-Min flow control, whose concern is to maintain Max-Min fairness in rate allocation by fairly sacrificing the throughput. We then elaborate our argument through a weighting mechanism in order to achieve a balance between the orthogonal goals of performance and fairness. Moreover, we investigate the implementation facets of the presented flow control schemes in on-chip architectures. Finally, we validate the proposed flow control schemes and the subsequent arguments through extensive simulation experiments.