Improved Queue-Size Scaling for Input-Queued Switches via Graph Factorization

This paper studies the scaling of the expected total queue size in an n × n input-queued switch, as a function of both the load - and the system scale n. We provide a new class of scheduling policies under which the expected total queue size scales as O(n(1 - p)-4/3 log (max 1 1-p ,n) , over all n and p < 1, when the arrival rates are uniform. This improves over the previously bestknown scalings in two regimes: O (n1.5 (1 - p)-1 log 1 1-p) when Ω(n-1.5) ≤ 1 - p ≤ O(n-1) and O (n log n (1-p)2) when 1 - p ≥ Ω(n-1). A key ingredient in our method is a tight characterization of the largest k-factor of a random bipartite multigraph, which may be of independent interest. 1