On queue-size scaling for input-queued switches

Abstract

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 $\rho$, which has been conjectured to be $\Theta(n/{(}1-\rho{)})$ (cf. [15]). In a recent work [16], the validity of this conjecture has been established for the regime where $1-\rho=O(1/n^{2})$. 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 $O\big(n^{1.5}(1-\rho)^{-1}\log\big(1/(1-\rho)\big)\big)$ when $1-\rho=O(1/n)$. This is an improvement over the state of the art; for example, for $\rho=1-1/n$ the best known bound was $O(n^{3})$, while ours is $O(n^{2.5}\log n)$.