Self-stabilizing Balls & Bins in Batches

A fundamental problem in distributed computing is the distribution of requests to a set of uniformservers without a centralized controller. Classically, such problems are modelled as static balls intobins processes, where m balls (tasks) are to be distributed to n bins (servers). In a seminal work,Azar et al. [4] proposed the sequential strategy Greedy[d] for n = m. When thrown, a ball queriesthe load of d random bins and is allocated to a least loaded of these. Azar et al. showed that d = 2yields an exponential improvement compared to d = 1. Berenbrink et al. [7] extended this to m n,showing that the maximal load difference is independent of m for d = 2 (in contrast to d = 1).We propose a new variant of an infinite balls into bins process. Each round an expected numberof λn new balls arrive and are distributed (in parallel) to the bins. Each non-empty bin deletes oneof its balls. This setting models a set of servers processing incoming requests, where clients can querya server’s current load but receive no information about parallel requests. We study the Greedy[d]distribution scheme in this setting and show a strong self-stabilizing property: For any arrival rateλ = λ(n) < 1, the system load is time-invariant. Moreover, for any (even super-exponential) roundt, the maximum system load is (w.h.p.) O(11−λ · log n1−λ)for d = 1 and O(log n1−λ)for d = 2. Inparticular, Greedy[2] has an exponentially smaller system load for high arrival rates. ∗[email protected].†[email protected].‡[email protected]. Supported in part by the Pacific Institute for the Mathematical Sciences.§[email protected]¶[email protected]. Supported by the German Ministry of Education and Research under Grant 01IH13004.‖[email protected]. Supported in part by EPSRC. 1arXiv:1603.02188v1[cs.DC]7Mar2016