Optimal Bin Packing with Items of Random Sizes III

Abstract

Consider a probability measure $\mu $ on $[0,1]$ and independent random variables $X_1 , \cdots ,X_n $ distributed according to $\mu $. Let $Q_n = Q_n (X_1 , \cdots ,X_n )$ be the minimum number of unit-size bins needed to pack items of size $X_1 , \cdots ,X_n $. Let $c(\mu ) = \lim _{n \to \infty } {{E(Q_n )} / n}$. In this paper it is proved that the random variable ${{(Q_n - nc(\mu ))} / {\sqrt n }}$ converges in distribution. The limit is identified as a distribution of the supremum of a certain Gaussian process canonically attached to $\mu $.