Proportional Switching in First-in, First-out Networks

Abstract

We consider a family of discrete time multihop switched queueing networks where each packet moves along a fixed route. In this setting, BackPressure is the canonical choice of scheduling policy; this policy has the virtues of possessing a maximal stability region and not requiring explicit knowledge of traffic arrival rates. BackPressure has certain structural weaknesses because implementation requires information about each route, and queueing delays can grow super-linearly with route length. For large networks, where packets over many routes are processed by a queue, or where packets over a route are processed by many queues, these limitations can be prohibitive.In this article, we introduce a scheduling policy for FIFO networks, the Proportional Scheduler, which is based on the proportional fairness criterion. We show that, like BackPressure, the Proportional Scheduler has a maximal stability region and does not require explicit knowledge of traffic arrival rates. The Proportional Scheduler has the advantage that information about the network's route structure is not required for scheduling, which substantially improves the policy's performance for large networks. For instance, packets can be routed with only next-hop information and new nodes can be added to the network with only knowledge of the scheduling constraints.

Stability and Instability of the MaxWeight Policy

“…After some calculation, one can check that (35) is equivalent to the upper bound for a in (5), which completes the proof of the lemma. (With additional work, one can show that κ 2 M ≤ U on G M , by properly quantifying κ, κ 1 , and κ 2 .…”

“…In Section 6, we will briefly compare the stability of the MaxWeight policy for multiclass switched networks with that of the ProportionalScheduler, where the terms Q j σ j in (1) are replaced by Q j log σ j . As shown in Bramson et al [5], the ProportionalScheduler is maximally stable for all multiclass multihop switched networks that are of Kelly type (and hence is automatically maximally stable for all single class multihop switched networks). We will give elementary heuristic reasoning why the term log σ j produces a more stable policy than does σ j .…”

“…where < S > is the convex hull of S. When such a solution is a weighted average of schedules in S, a schedule is chosen at random according to these weights. It was shown in [5] that the ProportionalScheduler is maximally stable for multiclass multi-hop switched queueing networks of Kelly type; it is consequently also maximally stable for single class multi-hop switched queueing networks. Why is maximum stability so much more robust for the ProportionalScheduler than for MaxWeight?…”

Achieving Low-Delay and Fast-Convergence in Stochastic Network Optimization

“…In contrast, without sacrificing any throughput-optimality and without requiring any non-causal statistical knowledge, our Nesterovian approach achieves an O( √ K) delay scaling. We also note that although attempts to get rid of the back-pressure nature of the QCA framework have also been proposed in the liter-ature (see, e.g., [20,21]), convergence performance was not addressed in these works.…”

Modelling complex stochastic systems: approaches to management and stability

“…More recently introduced maximally stable algorithms include the ProportionalScheduler [63,64] and Queue-Proportional Rate Allocation (QPRA) [16] policies. The primary advantage that these policies have over BackPressure is that they do not distinguish between the types of jobs at each station and consequently scale much better with network size.…”