  We study kill-and-restart and preemptive strategies for the fundamental
scheduling problem of minimizing the sum of weighted completion times on a
single machine in the non-clairvoyant setting. First, we show a lower bound
of~$3$ for any deterministic non-clairvoyant kill-and-restart strategy. Then,
we give for any $b > 1$ a tight analysis for the natural $b$-scaling
kill-and-restart strategy as well as for a randomized variant of it. In
particular, we show a competitive ratio of $(1+3\sqrt{3})\approx 6.197$ for the
deterministic and of $\approx 3.032$ for the randomized strategy, by making use
of the largest eigenvalue of a Toeplitz matrix. In addition, we show that the
preemptive Weighted Shortest Elapsed Time First (WSETF) rule is $2$-competitive
when jobs are released online, matching the lower bound for the unit weight
case with trivial release dates for any non-clairvoyant algorithm. Using this
result as well as the competitiveness of round-robin for multiple machines, we
prove performance guarantees smaller than $10$ for adaptions of the $b$-scaling
strategy to online release dates and unweighted jobs on identical parallel
machines.
