Simple form control policies for resource sharing networks with HGI performance

Abstract

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 Massoulié and Roberts (Telecommun. Syst. 15 (2000) 185–201) 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. (Stoch. Syst. 4 (2014) 524–555), 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.