Real Time Proportional Throughput Maximization: How much advance notice should you give your scheduler?

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.