  Online load balancing for heterogeneous machines aims to minimize the
makespan (maximum machine workload) by scheduling arriving jobs with varying
sizes on different machines. In the adversarial setting, where an adversary
chooses not only the collection of job sizes but also their arrival order, the
problem is well-understood and the optimal competitive ratio is known to be
$\Theta(\log m)$ where $m$ is the number of machines. In the more realistic
random arrival order model, the understanding is limited. Previously, the best
lower bound on the competitive ratio was only $\Omega(\log \log m)$.
  We significantly improve this bound by showing an $\Omega( \sqrt {\log m})$
lower bound, even for the restricted case where each job has a unit size on two
machines and infinite size on the others. On the positive side, we propose an
$O(\log m/\log \log m)$-competitive algorithm, demonstrating that better
performance is possible in the random arrival model.
