Self-Stabilizing Balls and Bins in Batches

AbstractA 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 $$\textsc {Greedy}[{d}]$$GREEDY[d] for $$n=m$$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 $$d=2$$d=2 yields an exponential improvement compared to $$d=1$$d=1. Berenbrink et al. (SIAM J Comput 35(6):1350–1385, 2006. https://doi.org/10.1137/S009753970444435X) extended this to $$m\gg n$$m≫n, showing that for $$d=2$$d=2 the maximal load difference is independent of m (in contrast to the $$d=1$$d=1 case). We propose a new variant of an infinite balls-into-bins process. In each round an expected number of $$\lambda n$$λn 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 $$\textsc {Greedy}[{d}]$$GREEDY[d] distribution scheme in this setting and show a strong self-stabilizing property: for any arrival rate $$\lambda =\lambda (n)<1$$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.) for $$d=1$$d=1 and for $$d=2$$d=2. In particular, $$\textsc {Greedy}[{2}]$$GREEDY[2] has an exponentially smaller system load for high arrival rates.