Semi-online scheduling on 2 machines under a grade of service provision with bounded processing times

Abstract

We study the problem of semi-online scheduling on 2 machines under a grade of service (GoS). GoS means that some jobs have to be processed by some machines to be guaranteed a high quality. The problem is online in the sense that jobs are presented one by one, and each job shall be assigned to a time slot on its arrival. Assume that the processing time p i of every job J i is bounded by an interval [a,? a], where a>0 and ?>1 are two constant numbers. By knowing the bound of jobs' processing times, we denote it by semi-online problem. We deal with two semi-online problems. The first one concerns about bounded processing time constraint. First, we show that a lower bound of competitive ratio is: (1) $\frac{1+\alpha}{2}$ in the case where 1<?<2; (2) $\frac{3}{2}$ in the case where 2??<5; and (3) $\frac{4+\alpha}{6}$ in the case where 5??<6. We further propose an algorithm, called B-ONLINE, and prove that in the case where $\frac{25}{14}\leq \alpha$ and the optimal makespan C OPT ?20a, B-ONLINE algorithm is optimal. For the second problem, we further know the sum of jobs' processing times Σ in advance. We first show a lower bound $\frac{1+\alpha}{2}$ in the case where 1<?<2, then we propose an algorithm B-SUM-ONLINE which is optimal in the case where $\Sigma \geq \frac{2\alpha}{\alpha-1}a$ and 1<?<2.