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

Abstract

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 class of algorithms \(\tau-\textsc{Persist}\) parametrized by some value \(\tau\in [1,\infty)\). If an input sequence has \(t\)-advance-notice, \(\tau-\textsc{Persist}\) is \(\frac{\tau - 1}{\tau^2 +\tau - 1}\)-competitive. In particular, we show that for any \(t \leq \frac{1}{2}\), there is an algorithm that achieves \(\frac{t-t^2}{1+t-t^2}\)-competitiveness and for any \(t \geq \frac{1}{2}\), there is an algorithm that achieves \(\frac{1}{5}\)-competitiveness. We also give an upper bound of any algorithm that relies on input sequences having \(t\)-advance-notice. We show that the competitive ratio of any algorithm can be at most \(\frac{t}{2t+1}\) against input sequences that have \(t\)-advance-notice. In particular, we show that regardless of how much advance-notice is given, no algorithm can reach \(\frac{1}{2}\)-competitiveness.