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 modeled 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. (SIAM J Comput 29(1): 180-200, 1999.https://doi.org/10.1137/S0097539795288490) 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. (1999) showed that d = 2 yields an exponential improvement compared to d = 1. Berenbrink et al. (SIAM J Comput 35(6): 1350-1385, 2006.https://doi.org/10.1137/S009753970444435X) extended this to m n, showing 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 lambda n new balls arrive and are distributed (in parallel) to the bins. Subsequently, 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 lambda = lambda(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-lambda . log n/1-lambda) for d = 1 and O(log n/1-lambda) for d = 2. In particular, Greedy[2] has an exponentially smaller system load for high arrival rates.
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