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

Xin Liu, Lei Ying

Research output: Contribution to journalArticle

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>$0&lt;\gamma&lt;1.$</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>$0\leq \alpha&lt;1/6,$</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) IEEE Transactions on Network Science and Engineering https://doi.org/10.1109/TNSE.2018.2878403 Accepted/In press - Jan 1 2018

### Keywords

• Data centers
• Delays
• Heavy traffic
• Mean-field model
• Power-of-d-Choices
• Probes
• Servers