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

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 > 2, there is an integer-valued function m(d)(n) = 1n1nn/1nd + 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 -> infinity. We show also that the maximum load usually does not vary by more than a constant amount from 1n1nn/1nd, even over quite long periods of time.