Online scheduling on two uniform machines to minimize the makespan

Abstract

a b s t r a c tWe consider two problems of online scheduling on two uniform machines: online scheduling under a grade of service (GoS) and online scheduling with reassignment. These problems are online in the sense that when a job presents, we have to irrevocably assign it to one of the machines before the next job is seen. The objective is to minimize the makespan.In the first problem, GoS means that some jobs have to be processed by some machine so that they can be guaranteed a higher quality. Assume that the speed of the higher GoS machine is normalized to 1, while the speed of the other one is s. We show that a lower bound of competitive ratio is 1 + 2s s+2 in the case 0 < s ≤ 1 and 1 + s+1 s(2s+1) in the case s > 1. Then we propose and analyze two online algorithms: HSF algorithm and EX-ONLINE algorithm. HSF is optimal in the case where s > 1 and Σ 1 ≥ Σ 2 s , where Σ 1 and Σ 2 denote the total processing time of jobs which request higher GoS machine and the total processing time of jobs which request the normal one, respectively. EX-ONLINE is optimal in the case 2(In the second problem, we study two subproblems P L and P A proposed in [Z. Tan, S. Yu, Online scheduling with reassignment, Operations Research Letters 36 (2008) 250-254].Assume that the speeds of 2 uniform machines are 1 and s ≥ 1, respectively. For P L where we can reassign the last k jobs of the sequence, we show a lower bound of competitive ratio 1 + 1 1+s . For P A where we can reassign arbitrary k jobs, we show a lower bound of competitive ratio (s+1) 2 s 2 +s+1 . We propose a s+1 s -competitive algorithm HSF-1 for both P L and P A . For P A , we propose a (s+1) 2 s+2 -competitive algorithm EX-RA, which is superior to HSF-1 when 1 ≤ s ≤ √ 2.

Makespan minimization in online scheduling with machine eligibility

“…They provide lower and upper bounds with an algorithm for the competitive ratio as a function of the ratio of the two speeds. Lee et al (2009) point out an error in Liu et al (2009) and improve both lower and upper bounds. Finally, Tan and Zhang (2010a) provide an optimal algorithm.…”

“…Thus, their algorithm is optimal for three identical machines. Liu et al (2009) consider problem Q2 | M i (GoS) | C max with the two machines having different speeds. They provide lower and upper bounds with an algorithm for the competitive ratio as a function of the ratio of the two speeds.…”

Online scheduling with one rearrangement at the end: Revisited

“…We focus on the third problem P A and define K = 1 in this paper. Our contribution: In this paper, we propose an improved algorithm for 1 s 2, which is better than previous result [7]. The results are summarized in Table 1.…”

Optimal semi-online algorithms for scheduling problems with reassignment on two identical machines

“…Such a case is defined as chines, three optimal algorithms with competitive ratio 3 2 , √ 2, 4 3 are presented. Liu et al [4] generalized P A to two uniform machines (denoted it by Q A ), obtained a lower bound (s+1) 2 s 2 +s+1 and upper bound min{ (s+1) 2 s+2 , s+1 s }. Cao and Liu [5] made a further investigation on the generalized version of P E (denoted it by Q E ) and presented an optimal algorithm with a competitive ratio min{…”