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

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,α a], where a>0 and α>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) 1+α/2 in the case where 1<α<2; (2) 3/2 in the case where 2≥α<5; and (3) 4+α/6 in the case where 5≥α<6. We further propose an algorithm, called B-ONLINE, and prove that in the case where 25/14 ≤ α 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 ∑ in advance. We first show a lower bound 1+α/2 in the case where 1<α<2, then we propose an algorithm B-SUM-ONLINE which is optimal in the case where Σ ≥ 2α/α-1 a and 1<α<2.