Self-stabilizing balls and bins in batches : The power of leaky bins

A fundamental problem in distributed computing is the distribution of requests to a set of uniform servers without a centralized controller. Classically, such problems are modelled as static balls into bins processes, where m balls (tasks) are to be distributed among n bins (servers). In a seminal work, Azar et al. [4] proposed the sequential strategy Greedy[d] for n = m. Each ball queries the load of d random bins and is allocated to a least loaded of them. Azar et al. showed that d = 2 yields an exponential improvement compared to d = 1. Berenbrink et al. [7] extended this to m n, showing A preliminary version of this paper was published in the proceedings of PODC’16 Petra Berenbrink Universität Hamburg, 22527 Hamburg, Germany The author did some of the research while affiliated with Simon Fraser University. E-mail: berenbrink@informatik.uni-hamburg.de Tom Friedetzky Durham University, Durham DH1 3LE, U.K. E-mail: tom.friedetzky@dur.ac.uk Peter Kling Universität Hamburg, 22527 Hamburg, Germany The author did some of the research while affiliated with Simon Fraser University. E-mail: peter.kling@uni-hamburg.de Frederik Mallmann-Trenn École normale supérieure, 75005 Paris, France & Simon Fraser University, Burnaby, B.C., V5A 1S6, Canada E-mail: frederik.mallmann-trenn@ens.fr Lars Nagel Johannes Gutenberg-Universität Mainz E-mail: nagell@uni-mainz.de Chris Wastell Durham University, Durham DH1 3LE, U.K. E-mail: christopher.wastell@dur.ac.uk 2 P. Berenbrink et al. that for d = 2 the maximal load difference is independent of m (in contrast to the d = 1 case). We propose a new variant of an infinite balls-into-bins process. In each round an expected number of λn new balls arrive and are distributed (in parallel) to the bins, and each non-empty bin deletes one of its balls. This setting models a set of servers processing incoming requests, where clients can query a 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 rate λ = λ(n) < 1, the system load is time-invariant. Moreover, for any (even super-exponential) round t, the maximum system load is (w.h.p.) O ( 1 1−λ · log n 1−λ ) for d = 1 and O ( log n 1−λ ) for d = 2. In particular, Greedy[2] has an exponentially smaller system load for high arrival rates.