A general framework for handling commitment in online throughput maximization

We study a fundamental online job admission problem where jobs with deadlines arrive online over time at their release dates, and the task is to determine a preemptive single-server schedule which maximizes the number of jobs that complete on time. To circumvent known impossibility results, we make a standard slackness assumption by which the feasible time window for scheduling a job is at least $1+\varepsilon$ times its processing time, for some $\varepsilon>0$. We quantify the impact that different provider commitment requirements have on the performance of online algorithms. Our main contribution is one universal algorithmic framework for online job admission both with and without commitments. Without commitment, our algorithm with a competitive ratio of $O(1/\varepsilon)$ is the best possible (deterministic) for this problem. For commitment models, we give the first non-trivial performance bounds. If the commitment decisions must be made before a job's slack becomes less than a $δ$-fraction of its size, we prove a competitive ratio of $O(\varepsilon/((\varepsilon-δ)δ^2))$, for $0<δ<\varepsilon$. When a provider must commit upon starting a job, our bound is $O(1/\varepsilon^2)$. Finally, we observe that for scheduling with commitment the restriction to the `unweighted' throughput model is essential; if jobs have individual weights, we rule out competitive deterministic algorithms.