  We consider the stochastic extensible bin packing problem (SEBP) in which $n$
items of stochastic size are packed into $m$ bins of unit capacity. In contrast
to the classical bin packing problem, the number of bins is fixed and they can
be extended at extra cost. This problem plays an important role in stochastic
environments such as in surgery scheduling: Patients must be assigned to
operating rooms beforehand, such that the regular capacity is fully utilized
while the amount of overtime is as small as possible.
  This paper focuses on essential ratios between different classes of policies:
First, we consider the price of non-splittability, in which we compare the
optimal non-anticipatory policy against the optimal fractional assignment
policy. We show that this ratio has a tight upper bound of $2$. Moreover, we
develop an analysis of a fixed assignment variant of the LEPT rule yielding a
tight approximation ratio of $(1+e^{-1}) \approx 1.368$.
  Furthermore, we prove that the price of fixed assignments, related to the
benefit of adaptivity, which describes the loss when restricting to fixed
assignment policies, is within the same factor. This shows that in some sense,
LEPT is the best fixed assignment policy we can hope for. We also provide a
lower bound on the performance of this policy comparing against an optimal
fixed assignment policy. Finally, we obtain improved bounds for the case where
the processing times are drawn from a particular family of distributions, with
either a bounded Pietra index or when the familly is stochastically dominated
at the second order.
