Buffer Management for Packets with Processing Times

Abstract

We discuss the well known job scheduling problem with release times and deadlines, alongside an extended model - buffer management for packets with processing requirements. For job scheduling, an $\Omega(\sqrt{\frac{\log{\kappa}}{\log{\log{\kappa}}}})$ lower bound for any randomized preemptive algorithm was shown by Irani and Canetti (1995), where κ is the the maximum job duration or the maximum job value (the minimum is assumed to be 1). The proof of this well-known result is fairly elaborate and involved. In contrast, we show a significantly improved lower bound of Ω(logκ) using a simple proof. Our result matches the easy upper bound and closes a gap which was supposedly open for 20 years.

We also discuss an interesting extension of job scheduling (for tight jobs). We discuss the problem of handling a FIFO buffer of a limited capacity, where packets arrive over time and may be preempted. Most of the work in buffer management considers the case where each packet has unit processing requirement. We consider a model where packets require some number of processing cycles before they can be transmitted. We aim to maximize the value of transmitted packets. We show an $\Omega(\frac{\log{\kappa}}{\log{\log{\kappa}}})$ lower bound on the competitive ratio of randomized algorithms in this setting. We also present bounds for several special cases. For packets with unit values we also show a ϕ ≈ 1.618 lower bound on the competitive ratio of deterministic algorithms, and a 2-competitive algorithm for this problem. For the case of packets with constant densities we present a 4-competitive algorithm.