On Achieving Zero Delay with Power-of-d-Choices Load Balancing

Xin Liu; Lei Ying

doi:10.1109/TNSE.2018.2878403

On Achieving Zero Delay with Power-of-d-Choices Load Balancing

Xin Liu, Lei Ying

Research output: Contribution to journal › Article

1 Scopus citations

Abstract

Power-of-<formula><tex>$d$</tex></formula>-choices (Pof$d$) is a popular load balancing algorithm for many-server systems such as large-scale data centers. For each incoming job, the algorithm probes <formula><tex>$d$</tex></formula> servers, chosen uniformly at random from a total of <formula><tex>$N$</tex></formula> servers, and routes the job to the least loaded one. It is well known that Power-of-<formula><tex>$d$</tex></formula>-chocies reduces queueing delays by orders of magnitude compared to the policy that routes each incoming job to a randomly selected server. The question to be addressed in this paper is how large <formula><tex>$d$</tex></formula> needs to be so that Power-of-<formula><tex>$d$</tex></formula>-chocies achieves asymptotic zero delay like the join-the-shortest-queue (JSQ) algorithm, which is a special case of Power-of-<formula><tex>$d$</tex></formula>-chocies with <formula><tex>$d=N$</tex></formula>. We are interested in the heavy-traffic regime where the load of the system, denoted by <formula><tex>$\lambda$</tex></formula>, approaches to one as <formula><tex>$N$</tex></formula> increases, and assume <formula><tex>$\lambda=1-\gamma N^{-\alpha}$</tex></formula> for <formula><tex>$0<\gamma<1.$</tex></formula> This paper establishes that when <formula><tex>$d=\Omega\left(\frac{\log N}{1-\lambda}\right)$</tex></formula>, finite buffer and <formula><tex>$0\leq \alpha<1/6,$</tex></formula> the probability that a job is routed to a busy server is asymptotically zero, and when <formula><tex>$d=O\left(\frac{1}{1-\lambda}\right)$</tex></formula> and infinite buffer, the probability that a job is routed to busy servers is a positive constant. Therefore, our results show <formula><tex>$d=\Omega\left(\frac{\log N}{1-\lambda}\right)$</tex></formula> is sufficient and almost necessary for achieving zero delay with Power-of-<formula><tex>$d$</tex></formula>-chocies policy.

Original language	English (US)
Journal	IEEE Transactions on Network Science and Engineering
DOIs	https://doi.org/10.1109/TNSE.2018.2878403
State	Accepted/In press - Jan 1 2018

Keywords

Data centers
Delays
Heavy traffic
Load management
Load modeling
Mean-field model
Power-of-d-Choices
Probes
Servers
Steady-state

ASJC Scopus subject areas

Control and Systems Engineering
Computer Science Applications
Computer Networks and Communications

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Liu, X., & Ying, L. (Accepted/In press). On Achieving Zero Delay with Power-of-d-Choices Load Balancing. IEEE Transactions on Network Science and Engineering. https://doi.org/10.1109/TNSE.2018.2878403

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title = "On Achieving Zero Delay with Power-of-d-Choices Load Balancing",

abstract = "Power-of-$d$-choices (Pof$d$) is a popular load balancing algorithm for many-server systems such as large-scale data centers. For each incoming job, the algorithm probes $d$ servers, chosen uniformly at random from a total of $N$ servers, and routes the job to the least loaded one. It is well known that Power-of-$d$-chocies reduces queueing delays by orders of magnitude compared to the policy that routes each incoming job to a randomly selected server. The question to be addressed in this paper is how large $d$ needs to be so that Power-of-$d$-chocies achieves asymptotic zero delay like the join-the-shortest-queue (JSQ) algorithm, which is a special case of Power-of-$d$-chocies with $d=N$. We are interested in the heavy-traffic regime where the load of the system, denoted by $\lambda$, approaches to one as $N$ increases, and assume $\lambda=1-\gamma N^{-\alpha}$ for $0<\gamma<1.$ This paper establishes that when $d=\Omega\left(\frac{\log N}{1-\lambda}\right)$, finite buffer and $0\leq \alpha<1/6,$ the probability that a job is routed to a busy server is asymptotically zero, and when $d=O\left(\frac{1}{1-\lambda}\right)$ and infinite buffer, the probability that a job is routed to busy servers is a positive constant. Therefore, our results show $d=\Omega\left(\frac{\log N}{1-\lambda}\right)$ is sufficient and almost necessary for achieving zero delay with Power-of-$d$-chocies policy.",

keywords = "Data centers, Delays, Heavy traffic, Load management, Load modeling, Mean-field model, Power-of-d-Choices, Probes, Servers, Steady-state",

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N2 - Power-of-$d$-choices (Pof$d$) is a popular load balancing algorithm for many-server systems such as large-scale data centers. For each incoming job, the algorithm probes $d$ servers, chosen uniformly at random from a total of $N$ servers, and routes the job to the least loaded one. It is well known that Power-of-$d$-chocies reduces queueing delays by orders of magnitude compared to the policy that routes each incoming job to a randomly selected server. The question to be addressed in this paper is how large $d$ needs to be so that Power-of-$d$-chocies achieves asymptotic zero delay like the join-the-shortest-queue (JSQ) algorithm, which is a special case of Power-of-$d$-chocies with $d=N$. We are interested in the heavy-traffic regime where the load of the system, denoted by $\lambda$, approaches to one as $N$ increases, and assume $\lambda=1-\gamma N^{-\alpha}$ for $0<\gamma<1.$ This paper establishes that when $d=\Omega\left(\frac{\log N}{1-\lambda}\right)$, finite buffer and $0\leq \alpha<1/6,$ the probability that a job is routed to a busy server is asymptotically zero, and when $d=O\left(\frac{1}{1-\lambda}\right)$ and infinite buffer, the probability that a job is routed to busy servers is a positive constant. Therefore, our results show $d=\Omega\left(\frac{\log N}{1-\lambda}\right)$ is sufficient and almost necessary for achieving zero delay with Power-of-$d$-chocies policy.

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