  We give the first polynomial-time approximation scheme (PTAS) for the
stochastic load balancing problem when the job sizes follow Poisson
distributions. This improves upon the 2-approximation algorithm due to Goel and
Indyk (FOCS'99). Moreover, our approximation scheme is an efficient PTAS that
has a running time double exponential in $1/\epsilon$ but nearly-linear in $n$,
where $n$ is the number of jobs and $\epsilon$ is the target error. Previously,
a PTAS (not efficient) was only known for jobs that obey exponential
distributions (Goel and Indyk, FOCS'99).
  Our algorithm relies on several probabilistic ingredients including some
(seemingly) new results on scaling and the so-called "focusing effect" of
maximum of Poisson random variables which might be of independent interest.
