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

Abstract

Abstract-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 subcritical 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.

Maximum weight matching with hysteresis in overloaded queues with setups

“…One way to reduce the complexity of evaluating MWM is utilizing Local MWM, which only considers the nearest neighbor service vectors as alternatives to switch to. Ross and Bambos [33] demonstrate that such local optimization results in a stabilizing policy (in a setting without setups) that mimics the standard MWM with some delay. Such local optimization could also be applied to the MWM-H policy.…”

“…First, one might consider whether introducing hysteresis to other policies, such as Projective Cone Scheduling from [33], would also ensure identical stability regions in the presence of setup times. Next, it may be possible to extend this work to networks of parallel queueing systems, by relying on results from [34].…”

“…In the absence of setup times for service reconfigurations, a well-investigated class of scheduling policies is maximum weight matching (MWM), which has been studied by several authors, including [2,33,35,37], and more recently, in overloaded systems, by [34] (discussed below).…”

A general class of throughput optimal routing policies in multi-hop wireless networks

“…and downlinks) and input-queued switches have also been of great interest [2], [7], [12]- [15]. Next, we show that throughput optimal scheduling for input-queued switches is a special case of our framework and discuss the f -scheduling.…”

“…In this subsection, we compare f -scheduling with two known throughput optimal scheduling policies: 1) Longest Queue First (LQF) [2]; and 2) Projective Cone Scheduling (PCS) [15]. Before we proceed, a brief overview of LQF and PCS is provided.…”