  Minimizing the sum of weighted completion times on $m$ identical parallel
machines is one of the most important and classical scheduling problems. For
the stochastic variant where processing times of jobs are random variables,
M\"ohring, Schulz, and Uetz (1999) presented the first and still best known
approximation result achieving, for arbitrarily many machines, performance
ratio $1+\frac12(1+\Delta)$, where $\Delta$ is an upper bound on the squared
coefficient of variation of the processing times. We prove performance ratio
$1+\frac12(\sqrt{2}-1)(1+\Delta)$ for the same underlying algorithm---the
Weighted Shortest Expected Processing Time (WSEPT) rule. For the special case
of deterministic scheduling (i.e., $\Delta=0$), our bound matches the tight
performance ratio $\frac12(1+\sqrt{2})$ of this algorithm (WSPT rule), derived
by Kawaguchi and Kyan in a 1986 landmark paper. We present several further
improvements for WSEPT's performance ratio, one of them relying on a carefully
refined analysis of WSPT yielding, for every fixed number of machines $m$,
WSPT's exact performance ratio of order $\frac12(1+\sqrt{2})-O(1/m^2)$.
