We consider resource sharing networks of the form introduced in the work of Massoulié and Roberts(2000) as models for Internet flows. The goal is to study the open problem, formulated in Harrison et al. (2014), of constructing simple form rate allocation policies for broad families of resource sharing networks with associated costs converging to the Hierarchical Greedy Ideal performance in the heavy traffic limit. We consider two types of cost criteria, an infinite horizon discounted cost, and a long time average cost per unit time. We introduce a sequence of rate allocation control policies that are determined in terms of certain thresholds for the scaled queue length processes and prove that, under conditions, both type of costs associated with these policies converge in the heavy traffic limit to the corresponding HGI performance. The conditions needed for these results are satisfied by all the examples considered in Harrison et al. (2014).