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Improved queue-size scaling for input-queued switches via graph factorization

Part of: Operations research and management science Graph theory Special processes Computer system organization

Published online by Cambridge University Press:  24 September 2020

Jiaming Xu Open the ORCID record for Jiaming Xu [Opens in a new window]  and
Yuan Zhong
Jiaming Xu*
Affiliation:
Fuqua School of Business, Duke University
Yuan Zhong*
Affiliation:
Booth School of Business, University of Chicago
*
*Postal address: 100 Fuqua Drive, Durham, NC 27708, USA. Email: jx77@duke.edu
**Postal address: 5807 South Woodlawn Ave, Chicago, IL 60637, USA. Email: yuan.zhong@chicagobooth.edu
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Abstract

This paper studies the scaling of the expected total queue size in an $n\times n$ input-queued switch, as a function of both the load $\rho$ and the system scale n. We provide a new class of scheduling policies under which the expected total queue size scales as $O\big( n(1-\rho)^{-4/3} \log \big(\!\max\big\{\frac{1}{1-\rho}, n\big\}\big)\big)$, over all n and $\rho<1$, when the arrival rates are uniform. This improves on the best previously known scalings in two regimes: $O\big(n^{1.5}(1-\rho)^{-1} \log \frac{1}{1-\rho}\big)$ when $\Omega\big(n^{-1.5}\big) \le 1-\rho \le O\big(n^{-1}\big)$ and $O\big(\frac{n\log n}{(1-\rho)^2}\big)$ when $1-\rho \geq \Omega(n^{-1})$. 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.

Keywords

Input-queued switchqueue-size scalinggraph factorization

MSC classification

Primary: 60K25: Queueing theory
Secondary: 05C70: Factorization, matching, partitioning, covering and packing 68M20: Performance evaluation; queueing; scheduling 90B36: Scheduling theory, stochastic
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 3 , September 2020 , pp. 798 - 824
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Copyright
© Applied Probability Trust 2020

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