Concave switching in single-hop and multihop networks

Abstract

Switched queueing networks model wireless networks, input-queued switches, and numerous other networked communications systems. We consider an ($$\alpha ,g$$?,g)-switch policy; these policies provide a generalization of the MaxWeight policies of Tassiulas and Ephremides (IEEE Trans Autom Control 37(12):4936---1948, 1992) and the weighted $$\alpha $$?-fair with allocations of Mo and Walrand (IEEE/ACM Trans Netw 8(5):556---567, 2000) which are typically applied to Bandwidth Sharing Networks (Massoulie and Roberts in IEEE/ACM Trans Netw 10(3):320---328, 2002). For single-hop switch networks, we prove the maximum stability property for this class of randomized policies. Thus these policies have the same first-order behavior as the MaxWeight policies. However, for multihop networks some of these generalized polices address a number of critical weakness of the MaxWeight/BackPressure policies. For multihop networks with fixed routing, we consider a policy called the Proportional Scheduler (or (1, log)-policy). In this setting, the BackPressure policy is maximum stable, but must maintain a queue at each node for every route destination, which typically grows rapidly with a network's size. However, the Proportional Scheduler only needs to maintain a queue for each outgoing link, which is typically bounded in number. As is common with Internet routing, by maintaining per-link queueing, each node only needs to know the next hop for each packet and not its entire route. Further, in contrast to BackPressure, the Proportional Scheduler does not compare downstream queue lengths to determine weights; only local link information is required. This leads to greater potential for decomposed implementations of the policy. Through a reduction argument and an entropy argument, we demonstrate that, while maintaining substantially less queueing overhead, the Proportional Scheduler achieves maximum throughput stability.