  We study a fundamental online scheduling problem where jobs with processing
times, weights, and deadlines arrive online over time at their release dates.
The task is to preemptively schedule these jobs on a single or multiple
(possibly unrelated) machines with the objective to maximize the weighted
throughput, the total weight of jobs that complete before their deadline. To
overcome known lower bounds for the competitive analysis, we assume that each
job arrives with some slack $\varepsilon > 0$; that is, the time window for
processing job $j$ on any machine $i$ on which it can be executed has length at
least $(1+\varepsilon)$ times $j$'s processing time on machine $i$. Our
contribution is a best possible online algorithm for weighted throughput
maximization on unrelated machines: Our algorithm is
$O\big(\frac1\varepsilon\big)$-competitive, which matches the lower bound for
unweighted throughput maximization on a single machine. Even for a single
machine, it was not known whether the problem with weighted jobs is "harder"
than the problem with unweighted jobs. Thus, we answer this question and close
weighted throughput maximization on a single machine with a best possible
competitive ratio $\Theta\big(\frac1\varepsilon\big)$. While we focus on
non-migratory schedules, our algorithm achieves the same (up to constants)
performance guarantee when compared to an optimal migratory schedule.
