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