### 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 | |

State | Accepted/In press - Jan 1 2018 |

### Fingerprint

### 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

### Cite this

**On Achieving Zero Delay with Power-of-d-Choices Load Balancing.** / Liu, Xin; Ying, Lei.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - On Achieving Zero Delay with Power-of-d-Choices Load Balancing

AU - Liu, Xin

AU - Ying, Lei

PY - 2018/1/1

Y1 - 2018/1/1

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.

AB - 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.

KW - Data centers

KW - Delays

KW - Heavy traffic

KW - Load management

KW - Load modeling

KW - Mean-field model

KW - Power-of-d-Choices

KW - Probes

KW - Servers

KW - Steady-state

UR - http://www.scopus.com/inward/record.url?scp=85055692582&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85055692582&partnerID=8YFLogxK

U2 - 10.1109/TNSE.2018.2878403

DO - 10.1109/TNSE.2018.2878403

M3 - Article

JO - IEEE Transactions on Network Science and Engineering

JF - IEEE Transactions on Network Science and Engineering

SN - 2327-4697

ER -