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Asymptotically Optimal Load Balancing Topologies

Authors:

Debankur Mukherjee,

Sem C. Borst,

Johan S.H. van LeeuwaardenAuthors Info & Claims

Proceedings of the ACM on Measurement and Analysis of Computing Systems, Volume 2, Issue 1

Article No.: 14, Pages 1 - 29

https://doi.org/10.1145/3179417

Published: 03 April 2018 Publication History

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Abstract

We consider a system of N servers inter-connected by some underlying graph topology G_N. Tasks with unit-mean exponential processing times arrive at the various servers as independent Poisson processes of rate λ. Each incoming task is irrevocably assigned to whichever server has the smallest number of tasks among the one where it appears and its neighbors in G_N.

The above model arises in the context of load balancing in large-scale cloud networks and data centers, and has been extensively investigated in the case G_N is a clique. Since the servers are exchangeable in that case, mean-field limits apply, and in particular it has been proved that for any λ < 1, the fraction of servers with two or more tasks vanishes in the limit as N → ∞. For an arbitrary graph G_N, mean-field techniques break down, complicating the analysis, and the queue length process tends to be worse than for a clique. Accordingly, a graph G_N is said to be N-optimal or √N-optimal when the queue length process on G_N is equivalent to that on a clique on an N-scale or √N-scale, respectively.

We prove that if G_N is an Erdöo s-Rényi random graph with average degree d(N), then with high probability it is N-optimal and √N-optimal if d(N) → ∞ and d(N)/√Nlog(N)) → ∞ as N → ∞, respectively. This demonstrates that optimality can be maintained at N-scale and √N-scale while reducing the number of connections by nearly a factor N and √N/log(N) compared to a clique, provided the topology is suitably random. It is further shown that if G_N contains Θ(N) bounded-degree nodes, then it cannot be N-optimal. In addition, we establish that an arbitrary graph G_N is N-optimal when its minimum degree is N - o(N), and may not be N-optimal even when its minimum degree is cN + o(N) for any 0 < c < 1/2. Simulation experiments are conducted for various scenarios to corroborate the asymptotic results.

References

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M. Adler, S. Chakrabarti, M. Mitzenmacher, and L. Rasmussen. 1995. Parallel randomized load balancing. In Proc. STOC '95. 238--247.

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[2]

S. Albers, M. Charikar, and M. Mitzenmacher. 2001. Delayed information and action in on-line algorithms. Inform. Comput. Vol. 170, 2 (2001), 135--152.

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Goldsztajn DBorst Svan Leeuwaarden J(2024)Server Saturation in Skewed NetworksACM SIGMETRICS Performance Evaluation Review10.1145/3673660.365508052:1(97-98)Online publication date: 13-Jun-2024
https://dl.acm.org/doi/10.1145/3673660.3655080
Goldsztajn DBorst Svan Leeuwaarden J(2024)Server Saturation in Skewed NetworksProceedings of the ACM on Measurement and Analysis of Computing Systems10.1145/36560188:2(1-37)Online publication date: 29-May-2024
https://dl.acm.org/doi/10.1145/3656018
Goldsztajn DBorst Svan Leeuwaarden JGaretto MMarin ACiucu FFanti GRighter R(2024)Server Saturation in Skewed NetworksAbstracts of the 2024 ACM SIGMETRICS/IFIP PERFORMANCE Joint International Conference on Measurement and Modeling of Computer Systems10.1145/3652963.3655080(97-98)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3652963.3655080
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Asymptotically Optimal Load Balancing Topologies

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Asymptotically Optimal Load Balancing Topologies
SIGMETRICS '18

We consider a system of N ~servers inter-connected by some underlying graph topology~G N . Tasks with unit-mean exponential processing times arrive at the various servers as independent Poisson processes of rate lambda. Each incoming task is irrevocably ...
Asymptotically Optimal Load Balancing Topologies
SIGMETRICS '18: Abstracts of the 2018 ACM International Conference on Measurement and Modeling of Computer Systems

We consider a system of N ~servers inter-connected by some underlying graph topology~G N . Tasks with unit-mean exponential processing times arrive at the various servers as independent Poisson processes of rate lambda. Each incoming task is irrevocably ...
Asymptotically optimal control of N-systems with $$H_2^*$$H2ź service times under many-server heavy traffic

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cover image Proceedings of the ACM on Measurement and Analysis of Computing Systems

Proceedings of the ACM on Measurement and Analysis of Computing Systems Volume 2, Issue 1

March 2018

603 pages

EISSN:2476-1249

DOI:10.1145/3203302

Editors:
Augustin Chaintreau
Columbia University
,
Aditya Akella
University of Wisconsin-Madison
,
Adam Wierman
California Institute of Technology

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Publication History

Published: 03 April 2018

Published in POMACS Volume 2, Issue 1

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Netherlands Organization for Scientific Research (NWO) Gravitation Networks
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Goldsztajn DBorst Svan Leeuwaarden J(2024)Server Saturation in Skewed NetworksACM SIGMETRICS Performance Evaluation Review10.1145/3673660.365508052:1(97-98)Online publication date: 13-Jun-2024
https://dl.acm.org/doi/10.1145/3673660.3655080
Goldsztajn DBorst Svan Leeuwaarden J(2024)Server Saturation in Skewed NetworksProceedings of the ACM on Measurement and Analysis of Computing Systems10.1145/36560188:2(1-37)Online publication date: 29-May-2024
https://dl.acm.org/doi/10.1145/3656018
Goldsztajn DBorst Svan Leeuwaarden JGaretto MMarin ACiucu FFanti GRighter R(2024)Server Saturation in Skewed NetworksAbstracts of the 2024 ACM SIGMETRICS/IFIP PERFORMANCE Joint International Conference on Measurement and Modeling of Computer Systems10.1145/3652963.3655080(97-98)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3652963.3655080
Rutten DMukherjee D(2023)Mean-field Analysis for Load Balancing on Spatial GraphsACM SIGMETRICS Performance Evaluation Review10.1145/3606376.359355251:1(27-28)Online publication date: 27-Jun-2023
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Rutten DMukherjee DSmirni EAvrachenkov KGill PUrgaonkar B(2023)Mean-field Analysis for Load Balancing on Spatial GraphsAbstract Proceedings of the 2023 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems10.1145/3578338.3593552(27-28)Online publication date: 19-Jun-2023
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Cardinaels EBorst Svan Leeuwaarden J(2022)Power-of-two sampling in redundancy systems: The impact of assignment constraintsOperations Research Letters10.1016/j.orl.2022.10.00650:6(699-706)Online publication date: Nov-2022
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Borst S(2022)Load balancing in large-scale heterogeneous systemsQueueing Systems: Theory and Applications10.1007/s11134-022-09819-x100:3-4(397-399)Online publication date: 1-Apr-2022
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Weng WZhou XSrikant R(2021)Optimal Load Balancing with Locality ConstraintsProceedings of the ACM on Measurement and Analysis of Computing Systems10.1145/34283304:3(1-37)Online publication date: 15-Jun-2021
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Banerjee SMukherjee D(2020)Join-the-Shortest Queue diffusion limit in Halfin–Whitt regime: Sensitivity on the heavy-traffic parameterThe Annals of Applied Probability10.1214/19-AAP149630:1Online publication date: 1-Feb-2020
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