We will be exploring a generalization of real time scheduling problem sometimes called the real time throughput maximization problem. Our input is a sequence of jobs specified by their release time, deadline and processing time. We assume that jobs are announced before or at their release time. At each time step, the algorithm must decide whether to schedule a job based on the information so far. The goal is to maximize the value of the sum of the processing times of jobs that finish before their deadline, this is often called real time throughput with proportional weights. We extend this problem by defining a notion of $t$-advance-notice, a measure of how far in advance each job is announced relative to their processing time. We show that there exists a $\frac{t}{2t+1}$-competitive algorithm when all jobs have $t$-advance-notice for $t\in [0,1]$, this gives us a competitive ratio of $\frac{1}{3}$ when $t$ is greater than or equal to $1$. We also show that this ratio is optimal for all algorithms with $t$-advance-notice and that the upper bound of $\frac{t}{2t+1}$-competitiveness holds for all $t$, in particular that regardless of how much advance-notice is given, no algorithm can reach $\frac{1}{2}$-competitiveness.

翻译：本文探讨实时调度问题的一种推广形式，常被称为实时吞吐量最大化问题。输入为一组作业序列，每个作业由其释放时间、截止期限和处理时间定义。我们假设作业在其释放时间或之前被通知。在每个时间步，算法必须基于当前已知信息决定是否调度某个作业。目标是在截止期限前完成的作业的处理时间之和最大化，这通常被称为带比例权重的实时吞吐量。我们通过定义$t$提前通知的概念扩展了该问题，该概念度量每个作业相对于其处理时间的提前通知程度。我们证明，当所有作业具有$t$提前通知且$t\in [0,1]$时，存在一个$\frac{t}{2t+1}$竞争比算法；当$t$大于等于$1$时，该算法竞争比为$\frac{1}{3}$。我们还证明该竞争比对于所有具有$t$提前通知的算法是最优的，且$\frac{t}{2t+1}$竞争比的上界对所有$t$均成立，特别地，无论提前通知量多少，任何算法均无法达到$\frac{1}{2}$竞争比。