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 On queue-size scaling for input-queued switches
 Devavrat Shah, Massachusetts Institute of TechnologyJohn N. Tsitsiklis, Massachusetts Institute of TechnologyYuan Zhong, Columbia University

 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 $$O(n^{2.5} \log n)$$. AMS 2000 subject classifications: Primary 60K20, 68M12; secondary 68M20.Keywords: Input-queued switches, queue-size scaling.
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Shah, Devavrat, Tsitsiklis, John N., Zhong, Yuan, On queue-size scaling for input-queued switches, Stochastic Systems, 6, (2016), 1-25 (electronic). DOI: 10.1214/14-SSY151.

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