Abstract
This paper considers two parallel machine scheduling problems, where the objectives of both problems are to minimize the makespan, and the jobs arrive over time, on two uniform machines with speeds 1 and s (s ≥ 1), and on m identical machines, respectively. For the first problem, the authors show that the on-line LPT algorithm has a competitive ratio of (1 + \(\sqrt 5 \))/2 ≈ 1.6180 and the bound is tight. Furthermore, the authors prove that the on-line LPT algorithm has the best possible competitive ratio if s ≥ 1.8020. For the second problem, the authors present a lower bound of (15 − \(\sqrt {17} \))/8 ≈ 1.3596 on the competitive ratio of any deterministic on-line algorithm. This improves a previous result of 1.3473.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Graham R L, Bounds for certain multiprocessing anomalies, Bell System Technical Journal, 1966, 45(9): 1563–1581.
Epstein L, Noga J, Seiden S, et al., Randomized on-line scheduling on two uniform machines, Journal of Scheduling, 2001, 4(2): 71–92.
Li R and Huang H C, On-line scheduling for jobs with arbitrary release times, Computing, 2004, 73(1): 79–97.
Hall L A and Shmoys D B, Approximation schemes for constrained scheduling problems//Foundations of Computer Science, 1989, 30th Annual Symposium on IEEE, 1989, 134–139.
Shmoys D B, Wein J, and Williamson D P, Scheduling parallel machines on-line, SIAM Journal on Computing, 1995, 24(6): 1313–1331.
Graham R L, Bounds on multiprocessing timing anomalies, SIAM Journal on Applied Mathematics, 1969, 17(2): 416–429.
Chen B and Vestjens A, Scheduling on identical machines: How good is LPT in an on-line setting?, Operations Research Letters, 1997, 21(4): 165–169.
Noga J and Seiden S S, An optimal online algorithm for scheduling two machines with release times, Theoretical Computer Science, 2001, 268(1): 133–143.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the Special Funds of the National Natural Science Foundation of China under Grant No. 61340045, the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20123705110003, and Innovation Project of Shangdong Graduate Education under Grant No. SDYC13036.
This paper was recommended for publication by Editor FANG Yong.
Rights and permissions
About this article
Cite this article
Li, S., Zhang, Y. On-Line Scheduling on Parallel Machines to Minimize the Makespan. J Syst Sci Complex 29, 472–477 (2016). https://doi.org/10.1007/s11424-015-3252-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-015-3252-8