Self-stabilizing balls & bins in batches: The power of leaky bins [Extended Abstract]
conference contributionposted on 06.02.2018, 13:44 by Petra Berenbrink, Tom Friedetzky, Peter Kling, Frederik Mallmann-Trenn, Lars NagelLars Nagel, Chris Wastell
© 2016 ACM. 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 to n bins (servers). In a seminal work, Azar et al.  proposed the sequential strategy Greedy[d] for n = m. When thrown, a ball queries the load of d random bins and is allocated to a least loaded of these. Azar et al. showed that d = 2 yields an exponential improvement compared to d = 1. Berenbrink et al.  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. In each round an expected number of n 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 superexponential) 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 has an exponentially smaller system load for high arrival rates.
Petra Berenbrink is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). Peter Kling is partly supported by the Natural Sciences and Engi- neering Research Council of Canada (NSERC) and the Pacific Institute for the Mathematical Sciences (PIMS). Lars Nagel is supported by the German Ministry of Education and Research under Grant 01IH13004. Christopher Wastell is supported by EPSRC.
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