Online scheduling of jobs with kind release times and deadlines on a single machine

Abstract

This paper studies online scheduling of jobs with kind release times on a single machine. Here “kind release time” means that in online setting, no jobs can be released when the machine is busy. Each job J has a kind release time r(J) ≥ 0, a processing time p(J) > 0 and a deadline d(J) > 0. The goal is to determine a schedule which maximizes total processing time (Σp(J)E(J)) or total number (ΣE(J)) of the accepted jobs. For the first objective function Σp(J)E(J), we first present a lower bound $\sqrt 2$, and then provide an online algorithm LEJ with a competitive ratio of 3. This is the first deterministic algorithm for the problem with a constant competitive ratio. When p(J) ∈ {1, k}, k > 1 is a real number, we first present a lower bound min{(1+k)/k, 2k/(1+k)}, and then we show that LEJ has a competitive ratio of 1+⌈k⌉/k. In particular, when all the k length jobs have tight deadlines, we first present a lower bound max{4/(2 + k), 1} (for Σp(J)E(J)) and 4/3 (for ΣE(J)). Then we prove that LEJ is ⌈k⌉/k-competitive for Σp(J)E(J) and we provide an online algorithm H with a competitive ratio of 2⌈k⌉/(⌈k⌉ + 1) for the second objective function ΣE(J).