Maximizing Online Utilization with Commitment

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 $d \geq (1+ε)\cdot p + r$, with $p$ and $r$ being the processing time and the submission time of the job, respectively while $ε>0$ 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 $\frac{1+ε}ε$ of the greedy acceptance policy. Then the competitive ratio improves with an increasing number of machines and approaches $(1+ε)\cdot\ln \frac{1+ε}ε$ as the number of machines converges to infinity. This is an exponential improvement over the greedy acceptance policy for small $ε$. In the non-preemptive case, we present a deterministic algorithm on $m$ machines with a competitive ratio of $1+m\cdot \left(\frac{1+ε}ε\right)^{\frac{1}{m}}$. This matches the optimal bound of $2+\frac{1}ε$ of the greedy acceptance policy for a single machine while it again guarantees an exponential improvement over the greedy acceptance policy for small $ε$ and large $m$. In addition, we determine an almost tight lower bound that approaches $m\cdot \left(\frac{1}ε\right)^{\frac{1}{m}}$ for large $m$ and small $ε$.