  We study the online bin packing problem under two stochastic settings. In the
bin packing problem, we are given n items with sizes in (0,1] and the goal is
to pack them into the minimum number of unit-sized bins. First, we study bin
packing under the i.i.d. model, where item sizes are sampled independently and
identically from a distribution in (0,1]. Both the distribution and the total
number of items are unknown. The items arrive one by one and their sizes are
revealed upon their arrival and they must be packed immediately and irrevocably
in bins of size 1. We provide a simple meta-algorithm that takes an offline
$\alpha$-asymptotic approximation algorithm and provides a polynomial-time
$(\alpha + \varepsilon)$-competitive algorithm for online bin packing under the
i.i.d. model, where $\varepsilon$>0 is a small constant. Using the AFPTAS for
offline bin packing, we thus provide a linear time
$(1+\varepsilon)$-competitive algorithm for online bin packing under i.i.d.
model, thus settling the problem.
  We then study the random-order model, where an adversary specifies the items,
but the order of arrival of items is drawn uniformly at random from the set of
all permutations of the items. Kenyon's seminal result [SODA'96] showed that
the Best-Fit algorithm has a competitive ratio of at most 3/2 in the
random-order model, and conjectured the ratio to be around 1.15. However, it
has been a long-standing open problem to break the barrier of 3/2 even for
special cases. Recently, Albers et al. [Algorithmica'21] showed an improvement
to 5/4 competitive ratio in the special case when all the item sizes are
greater than 1/3. For this special case, we settle the analysis by showing that
Best-Fit has a competitive ratio of 1. We make further progress by breaking the
barrier of 3/2 for the 3-Partition problem, a notoriously hard special case of
bin packing, where all item sizes lie in (1/4,1/2].
