  We investigate online scheduling with commitment for parallel identical
machines. Our objective is to maximize the total processing time of accepted
jobs. As soon as a job has been submitted, the commitment constraint forces us
to decide immediately whether we accept or reject the job. Upon acceptance of a
job, we must complete it before its deadline $d$ that satisfies $d \geq
(1+\epsilon)\cdot p + r$, with $p$ and $r$ being the processing time and the
submission time of the job, respectively while $\epsilon>0$ is the slack of the
system. Since the hard case typically arises for near-tight deadlines, we
consider $\varepsilon\leq 1$. We use competitive analysis to evaluate our
algorithms. Our first main contribution is a deterministic preemptive online
algorithm with an almost tight competitive ratio on any number of machines. For
a single machine, the competitive factor matches the optimal bound
$\frac{1+\epsilon}{\epsilon}$ of the greedy acceptance policy. Then the
competitive ratio improves with an increasing number of machines and approaches
$(1+\epsilon)\cdot\ln \frac{1+\epsilon}{\epsilon}$ as the number of machines
converges to infinity. This is an exponential improvement over the greedy
acceptance policy for small $\epsilon$. In the non-preemptive case, we present
a deterministic algorithm on $m$ machines with a competitive ratio of $1+m\cdot
\left(\frac{1+\epsilon}{\epsilon}\right)^{\frac{1}{m}}$. This matches the
optimal bound of $2+\frac{1}{\epsilon}$ of the greedy acceptance policy for a
single machine while it again guarantees an exponential improvement over the
greedy acceptance policy for small $\epsilon$ and large $m$. In addition, we
determine an almost tight lower bound that approaches $m\cdot
\left(\frac{1}{\epsilon}\right)^{\frac{1}{m}}$ for large $m$ and small
$\epsilon$.
