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 \lambda n, where \lambda >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\geq 2, there is an integer-valued function m_d(n)=\ln \ln n/\ln d+O(1) such that, in the equilibrium distribution, the maximum load of a bin is concentrated on the two values m_d(n) and m_d(n)-1, with probability tending to 1, as n\to \infty. We show also that the maximum load usually does not vary by more than a constant amount from \ln \ln n/\ln d, even over quite long periods of time.