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

Abstract

Power-of-d-choices 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-choices 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-choices achieves asymptotic zero delay like the join-the-shortest-queue (JSQ) algorithm, which is a special case of power-of-d-choices 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 and $0\leq\alpha . This paper establishes that when $d=\omega-\left(\frac{1}{1-\lambda}\right)$ , the probability that an incoming job is routed to a busy server is asymptotically zero, i.e. a job experiences zero queueing delay with probability one asymptotically; and when $d=O\left(\frac{1}{1-\lambda}\right)$ ’ the probability that a job is routed to a busy server is lower bounded by a positive constant independent of $N$ . Therefore, our results show that $d=\omega(\frac{1}{1-\lambda})$ is sufficient and almost necessary for achieving zero delay with the power-of-d-choices load balancing policy.