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

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 (N is the number of servers in the system), 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 λ, approaches to one as N increases, and assume λ =1-γ N-α for 0<γ <1. This paper establishes that when d=Ω log N 1-λ, finite buffer size b and 0 α 1/6, 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) and infinite buffer size b=∞, 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=Ω \left(\frac{\log N}{1-λ }\right) is sufficient and almost necessary for achieving zero delay with the power-of-d-choices policy.