  We consider a family of resource sharing networks, known as bandwidth sharing
models, in heavy traffic with general service and interarrival times. These
networks, introduced in Massoulie and Roberts (2000) as models for internet
flows, have the feature that a typical job may require simultaneous processing
by multiple resources in the network. We construct simple form threshold
policies that asymptotically achieve the Hierarchical Greedy Ideal (HGI)
performance. This performance benchmark, which was introduced in Harrison et
al. (2014), is characterized by the following two features: every resource
works at full capacity whenever there is work for that resource in the system;
total holding cost of jobs of each type at any instant is the minimum cost
possible for the associated vector of workloads. The control policy we provide
is explicit in terms of a finite collection of vectors which can be computed
offline by solving a system of linear inequalities. Proof of convergence is
based on path large deviation estimates for renewal processes, Lyapunov
function constructions, and analyses of suitable sample path excursions.
