Bandwidth-sharing networks as considered by Massoulie & Roberts provide a natural modeling framework for describing the dynamic flow-level interaction among elastic data transfers. Although valuable stability results have been obtained, crucial performance metrics such as flow-level delays and throughputs in these models have remained intractable in all but a few special cases. In particular, it is not well understood to what extent flow-level delays and throughputs achieved by standard bandwidth-sharing mechanisms such as alpha-fair strategies leave potential room for improvement. In order to gain a better understanding of the latter issue, we set out to determine the scheduling policies that minimize the mean delay in some simple linear bandwidth-sharing networks. We compare the performance of the optimal policy with that of various alpha-fair strategies so as to assess the efficacy of the latter and gauge the potential room for improvement. The results indicate that the optimal policy achieves only modest improvements, even when the value of a is simply fixed, provided it is not too small.