A best online algorithm for scheduling on two parallel batch machines

Tian, Jing; Fu, Ruyan; Yuan, Jinjiang

doi:10.1016/j.tcs.2009.02.011

Cited by 12 publications

(6 citation statements)

References 6 publications

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“…Based on the above observation, we develop an online algorithm named A ∞ α below following the idea of Modified-Sleepy algorithm proposed in Tian et al (2009). Let U (t) be the set of all released and unscheduled jobs available at time t. Let J (t) with release time r (t) and processing time p (t) be the longest job in U (t), and ties are broken by choosing one with the latest release time.…”

Section: An Optimal Algorithmmentioning

confidence: 99%

“…In the unbounded model, they developed an optimal online algorithm with a competitive ratio 1 + β m where β m is the positive solution of equation considered online scheduling on two parallel batch processing machines, and provided an algorithm with a competitive ratio √ 2. Tian et al (2009) presented a matching lower bound of √ 2 for the problem as well as an optimal algorithm with a tighter structure. In this paper, we focus on the unbounded model, and study a semi-online scenario such that each job is of processing time at least α (≥ 1) times of that of its preceding job, i.e., jobs arrive in a non-decreasing order of processing times.…”

Section: Introductionmentioning

confidence: 99%

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Optimal Semi-Online Algorithm for Scheduling on Two Parallel Batch Processing Machines

Liu

Zheng

Zhu

et al. 2014

Asia Pac. J. Oper. Res.

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1450038-1 Asia Pac. J. Oper. Res. 2014.31. Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only. M. Liu et al.to minimize the makespan. Given any job J j together with its following job J j+1 , it is assumed that their processing times satisfy p j+1 ≥ αp j where α ≥ 1 is a constant. That is, jobs arrive in a non-decreasing order of processing times. We mainly propose an optimal φ-competitive online algorithm where φ ≥ 1 is a solution of equation Proof. ByStep 2 of the algorithm, we have the desired result.Proof. By Corollary 1, r n > S n−1 and then C max (π) ≥ r n + p n > S n−1 + p n . Since J n is regular, we have C max (σ) = S n + p n = max{φ(S n−1 + p n ), r n + p n }, implying C max (σ)/C max (π) ≤ φ. The lemma follows.By the above lemma, we focus on proving C max (σ)/C max (π) ≤ φ in the other case where J n is a delayed job in the remainder. By symmetry of two machines, we assume without loss of generality that J n is scheduled on M 1 . Lemma 4. n ≥ 3. 1450038-7 Asia Pac. J. Oper. Res. 2014.31. Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only. M. Liu et al.Proof. We already have n ≥ 2. Assume otherwise that n = 2. On the arrival of the second job J 2 , there is only one of the two machines processing the first job J 1 , implying that the other machine is being idle from the beginning and thus J 2 cannot be postponed by job J 1 . A contradiction to the assumption. The lemma follows. Theorem 2. AlgorithmProof. We prove the theorem by contradiction. Assume otherwise C max (σ)/ C max (π) > φ. Consider the last three jobs J n−2 , J n−1 and J n in σ with C n−2 < C n−1 < C n due to Corollary 2. There are two cases below.Case 1. J n−1 is scheduled on M 1 . We further discuss two subcases according to the scheduling of job J n−2 .Case 1.1. J n−2 is scheduled on M 1 . By Corollary 1, r n > S n−1 , which is not smaller than C n−2 due to case condition that both J n−2 and J n−1 are scheduled on M 1 with S n−2 < S n−1 . Together with Corollary 2, we claim that r n > C n−2 > C i for 1 ≤ i < n − 2, and thus machine M 2 is idle and available from time r n , implying J n is a regular job. By Lemma 3, C max (σ)/C max (π) ≤ φ. A contradiction.Case 1.2. J n−2 is scheduled on M 2 . By Corollary 2, we have C n−2 < C n−1 and thus J n must be scheduled on M 2 by Step 2.1 of the algorithm. A contradiction to the fact that J n is scheduled on machine M 1 .Case 2. J n−1 is scheduled on M 2 . We similarly divide this case into two subcases.Case 2.1. J n−2 is scheduled on M 2 . Let J l (1 ≤ l < n − 2) be the preceding job of J n on machine M 1 . We claim such job J l does exist since otherwise it contradicts the fact that J n is a delayed job. With similar reasoning to Case 1.1, we have r n > S n−1 ≥ C n−2 > C l , implying that J n is a regular job. A contradiction again.Case 2.2. J n−2 is scheduled on M 1 . Since J n−2 and J n are both scheduled on M 1 , combining C n−2 < C n−1 and the fact that J n is a delayed job, we have S n = C...

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Section: An Optimal Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Optimal Semi-Online Algorithm for Scheduling on Two Parallel Batch Processing Machines

Liu

Zheng

Zhu

et al. 2014

Asia Pac. J. Oper. Res.

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show abstract

“…Zhang et al [17] gave a lower bound of competitive ratio Tian et al [13], respectively, provided different online algorithms with the same competitive ratio √ 2, and the latter authors also provided a lower bound of competitive ratio √ 2. So, both algorithms provided in [9] and [13] are best possible.…”

Section: Introductionmentioning

confidence: 99%

“…So, both algorithms provided in [9] and [13] are best possible. The result for m = 2 was generalized to general m by Liu et al [6] and Tian et al [14].…”

Section: Introductionmentioning

confidence: 99%

Online scheduling on unbounded parallel-batch machines to minimize maximum flow-time

Yuan

2011

Information Processing Letters

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“…Specifically, they gave a lower bound √ 2 on the competitive ratio, and presented a dense-algorithm with a competitive ratio √ 5+1 2 . When all the jobs have equal processing times, Zhang et al [15] established two best possible online algorithms with competitive ratios 1 + ξ m and Nong et al [9] and Tian et al [5] provided different online algorithms with the same competitive ratio √ 2; and the latter authors also provided a lower bound √ 2 on the competitive ratio. So both algorithms provided in [9] and [5] are best possible.…”

Section: Introductionmentioning

confidence: 99%