Asymptotically optimal load balancing topologies

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Abstract

We consider a system of N servers inter-connected by some underlying graph topology GN. 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 GN. 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 GN 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 GN, mean-field techniques break down, complicating the analysis, and the queue length process tends to be worse than for a clique. Accordingly, a graph GN is said to be N-optimal or √N-optimal when the queue length process on GN is equivalent to that on a clique on an N-scale or √N-scale, respectively. We prove that if GN 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 GN contains Θ(N) bounded-degree nodes, then it cannot be N-optimal. In addition, we establish that an arbitrary graph GN 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.
Original languageEnglish
Article number14
Number of pages29
JournalProceedings of the ACM on Measurement and Analysis of Computing Systems
Volume2
Issue number1
DOIs
Publication statusPublished - Apr 2018

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Resource allocation
Servers
Topology
Processing
Experiments

Keywords

  • asymptotic optimality, cloud networking, data centers, delay performance, load balancing, load balancing on graphs, power-of-d scheme, scaling limits

Cite this

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title = "Asymptotically optimal load balancing topologies",
abstract = "We consider a system of N servers inter-connected by some underlying graph topology GN. 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 GN. 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 GN 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 GN, mean-field techniques break down, complicating the analysis, and the queue length process tends to be worse than for a clique. Accordingly, a graph GN is said to be N-optimal or √N-optimal when the queue length process on GN is equivalent to that on a clique on an N-scale or √N-scale, respectively. We prove that if GN is an Erd{\"o}o s-R{\'e}nyi 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 GN contains Θ(N) bounded-degree nodes, then it cannot be N-optimal. In addition, we establish that an arbitrary graph GN 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.",
keywords = "asymptotic optimality, cloud networking, data centers, delay performance, load balancing, load balancing on graphs, power-of-d scheme, scaling limits",
author = "Debankur Mukherjee and Borst, {Sem C.} and {van Leeuwaarden}, {Johan S.H.}",
year = "2018",
month = "4",
doi = "10.1145/3179417",
language = "English",
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journal = "Proceedings of the ACM on Measurement and Analysis of Computing Systems",
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Asymptotically optimal load balancing topologies. / Mukherjee, Debankur; Borst, Sem C.; van Leeuwaarden, Johan S.H.

In: Proceedings of the ACM on Measurement and Analysis of Computing Systems, Vol. 2, No. 1, 14, 04.2018.

Research output: Contribution to journalArticleAcademicpeer-review

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T1 - Asymptotically optimal load balancing topologies

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AU - Borst, Sem C.

AU - van Leeuwaarden, Johan S.H.

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N2 - We consider a system of N servers inter-connected by some underlying graph topology GN. 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 GN. 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 GN 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 GN, mean-field techniques break down, complicating the analysis, and the queue length process tends to be worse than for a clique. Accordingly, a graph GN is said to be N-optimal or √N-optimal when the queue length process on GN is equivalent to that on a clique on an N-scale or √N-scale, respectively. We prove that if GN 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 GN contains Θ(N) bounded-degree nodes, then it cannot be N-optimal. In addition, we establish that an arbitrary graph GN 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.

AB - We consider a system of N servers inter-connected by some underlying graph topology GN. 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 GN. 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 GN 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 GN, mean-field techniques break down, complicating the analysis, and the queue length process tends to be worse than for a clique. Accordingly, a graph GN is said to be N-optimal or √N-optimal when the queue length process on GN is equivalent to that on a clique on an N-scale or √N-scale, respectively. We prove that if GN 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 GN contains Θ(N) bounded-degree nodes, then it cannot be N-optimal. In addition, we establish that an arbitrary graph GN 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.

KW - asymptotic optimality, cloud networking, data centers, delay performance, load balancing, load balancing on graphs, power-of-d scheme, scaling limits

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