  We consider a fundamental online scheduling problem in which jobs with
processing times and deadlines arrive online over time at their release dates.
The task is to determine a feasible preemptive schedule on a single or multiple
possibly unrelated machines that maximizes the number of jobs that complete
before their deadline. Due to strong impossibility results for competitive
analysis on a single machine, we require that jobs contain some slack
$\varepsilon>0$, which means that the feasible time window for scheduling a job
is at least $1+\varepsilon$ times its processing time on each eligible machine.
Our contribution is two-fold: (i) We give the first non-trivial online
algorithms for throughput maximization on unrelated machines, and (ii), this is
the main focus of our paper, we answer the question on how to handle commitment
requirements which enforce that a scheduler has to guarantee at a certain point
in time the completion of admitted jobs. This is very relevant, e.g., in
providing cloud-computing services, and disallows last-minute rejections of
critical tasks. We present an algorithm for unrelated machines that is
$\Theta\big(\frac{1}\varepsilon\big )$-competitive when the scheduler must
commit upon starting a job. Somewhat surprisingly, this is the same optimal
performance bound (up to constants) as for scheduling without commitment on a
single machine. If commitment decisions must be made before a job's slack
becomes less than a $\delta$-fraction of its size, we prove a competitive ratio
of $\mathcal{O}\big(\frac{1}{\varepsilon - \delta}\big)$ for $0 < \delta <
\varepsilon$. This result nicely interpolates between commitment upon starting
a job and commitment upon arrival. For the latter commitment model, it is known
that no (randomized) online algorithm admits any bounded competitive ratio.
