Is it enough to drain the heaviest bottlenecks?

This paper takes a philosophically new approach to throughput-optimal scheduling queueing systems with interference. All existing popular approaches (e.g. max-weight, greedy, ”pick-and-compare” etc.) focus on the weights of individual queues. We take an alternative approach, by focusing instead on the aggregate queues of bottlenecks. A bottleneck is a set of mutually-interfering queues; a schedule drains a bottleneck if it removes a packet from any one of its queues. We consider (the standard) switch scheduling problem, where the bottlenecks are the nodes. We establish the following phasetransition (1) ensuring only that the very heaviest nodes are drained is not enough for throughput optimality, but (2) ensuring scheduling for all nodes with weight within (1 − α) of the heaviest is enough for throughput optimality, for any α>0. The proof uses a new Lyapunov function: the weight of the critical bottleneck. Our alternate node-focused view also enables the development of new algorithms for scheduling. We show (a) how any policy can be made throughput-optimal by doing a small number of extra operations, (b) a new algorithm — Maximum Vertex-weighted Matching (MVM) — has (empirical) delay performance better than the current state of the art, and lower complexity than Max-(edge)weighted Matching, and (c) a class o f throughput-optimal policies that trade off between complexity and delay.