  We consider the stochastic scheduling problem of minimizing the expected
makespan on $m$ parallel identical machines. While the (adaptive) list
scheduling policy achieves an approximation ratio of $2$, any (non-adaptive)
fixed assignment policy has performance guarantee $\Omega\left(\frac{\log
m}{\log \log m}\right)$. Although the performance of the latter class of
policies are worse, there are applications in which non-adaptive policies are
desired. In this work, we introduce the two classes of $\delta$-delay and
$\tau$-shift policies whose degree of adaptivity can be controlled by a
parameter. We present a policy - belonging to both classes - which is an
$\mathcal{O}(\log \log m)$-approximation for reasonably bounded parameters. In
other words, an exponential improvement on the performance of any fixed
assignment policy can be achieved when allowing a small degree of adaptivity.
Moreover, we provide a matching lower bound for any $\delta$-delay and
$\tau$-shift policy when both parameters, respectively, are in the order of the
expected makespan of an optimal non-anticipatory policy.
