Optimal Rate-Matrix Pruning For Heterogeneous Systems

We consider large-scale load balancing systems where processing time distribution of tasks depend on both task and server types. We analyze the system in the asymptotic regime where the number of task and server types tend to infinity proportionally to each other. In such heterogeneous setting, popular policies like Join Fastest Idle Queue (JFIQ), Join Fastest Shortest Queue (JFSQ) are known to perform poorly and they even shrink the stability region. Moreover, to the best of our knowledge, in this setup, finding a scalable policy with provable performance guarantee has been an open question prior to this work. In this paper, we propose and analyze two asymptotically delay-optimal dynamic load balancing approaches: (a) one that efficiently reserves the processing capacity of each server for "good" tasks and route tasks under the Join Idle Queue policy; and (b) a speed-priority policy that increases the probability of servers processing tasks at a high speed. Introducing a novel analytical framework and using the mean-field method and stochastic coupling arguments, we prove that both policies above achieve asymptotic zero queueing, whereby the probability that a typical task is assigned to an idle server tends to 1 as the system scales.