On Queue-Size Scaling for Input-Queued Switches

We study the optimal scaling of the expected total queue size in an $n\times n$ input-queued switch, as a function of the number of ports $n$ and the load factor $ρ$, which has been conjectured to be $Θ(n/(1-ρ))$. In a recent work, the validity of this conjecture has been established for the regime where $1-ρ= O(1/n^2)$. In this paper, we make further progress in the direction of this conjecture. We provide a new class of scheduling policies under which the expected total queue size scales as $O(n^{1.5}(1-ρ)^{-1}\log(1/(1-ρ)))$ when $1-ρ= O(1/n)$. This is an improvement over the state of the art; for example, for $ρ= 1 - 1/n$ the best known bound was $O(n^3)$, while ours is $O(n^{2.5}\log n)$.

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