Projective Cone, Scheduling (PCS) Algorithms for Packet Switches of Maximal Throughput

We study the (generalized) packet switch scheduling problem, where service configurations are dynamically chosen in response to queue backlogs, so as to maximize the throughput without any knowledge of the long term traffic load. Service configurations and traffic traces are arbitrary. First, we identify a rich class of throughput-optimal linear controls, which choose the service configuration S maximizing the projection < S, BX > when the backlog is X. The matrix B is arbitrarily fixed in the class of positive-definite, symmetric matrices with negative or zero off-diagonal elements. In contrast, positive off-diagonal elements may drive the system unstable, even for sub-critical loads. The associated rich Euclidian geometry of projective cones is explored (hence the name projective cone scheduling PCS). The maximum-weight-matching (MWM) rule is seen to be a special case, where B is the identity matrix. Second, we extend the class of throughput maximizing controls by identifying a tracking condition which allows applying PCS with any bounded time-lag without compromising throughput. It enables asynchronous or delayed PCS implementations and various examples are discussed.