On the power of two choices: Balls and bins in continuous time

Abstract

Suppose that there are n bins, and balls arrive in a Poisson process at rate λn, where λ > 0 is a constant. Upon arrival, each ball chooses a fixed number d of random bins, and is placed into one with least load. Balls have independent exponential lifetimes with unit mean. We show that the system converges rapidly to its equilibrium distribution; and when d ≥ 2, there is an integer-valued function md(n) = In ln n/ln d + O(1) such that, in the equilibrium distribution, the maximum load of a bin is concentrated on the two values md(n) and md(n) - 1, with probability tending to 1, as n → ∞. We show also that the maximum load usually does not vary by more than a constant amount from In In n /In d, even over quite long periods of time. © Institute of Mathematical Statistics, 2005.