We investigate balls-and-bins processes where *m* weighted balls are
placed into *n* bins using the “power of two choices” paradigm, whereby a
ball is inserted into the less loaded of two randomly chosen bins. The case where
each of the *m* balls has unit weight had been studied extensively. In a
seminal paper Azar ηl showed that when *m=n* the most loaded bin has
*Θ(log log n)* balls with high probability. Surprisingly, the gap in
load between the heaviest bin and the average bin does not increase with *m*
and was shown by Berenbrink ηl to be *Θ(log log n)* with high
probability for arbitrarily large *m*. We generalize this result to the
weighted case where balls have weights drawn from an arbitrary weight
distribution. We show that as long as the weight distribution has finite second
moment and satisfies a mild technical condition, the gap between the weight of
the heaviest bin and the weight of the average bin is independent of the number
balls thrown. This is especially striking when considering heavy tailed
distributions such as Power-Law and Log-Normal distributions. In these cases, as
more balls are thrown, heavier and heavier weights are encountered. Nevertheless
with high probability, the imbalance in the load distribution does not increase.
Furthermore, if the fourth moment of the weight distribution is finite, the
expected value of the gap is shown to be independent of the number of balls.