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Scheduling on identical machines: How good is LPT in an on-line setting? (English) Zbl 0892.90098
Summary: We consider a parallel machine scheduling problem where jobs arrive over time. A set of independent jobs has to be scheduled on $m$ identical machines, where preemption is not allowed and the number of jobs is unknown in advance. Each job becomes available at its release date, which is not known in advance, and its processing time becomes known at its arrival. We deal with the problem of minimizing the makespan, which is the time by which all jobs have been finished. We propose and analyze the following on-line LPT algorithm: At any time a machine becomes available for processing, schedule an available job with the largest processing time. We prove that this algorithm has a performance guarantee of ${3\over 2}$, and that this bound is tight. Furthermore, we show that any on-line algorithm will have a performance bound of at least 1.3473. This bound is improved to $(5- \sqrt 5)/2\approx 1.3820$ for $m=2$.

##### MSC:
 90B35 Scheduling theory, deterministic
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##### References:
 [1] Graham, R. L.: Bounds on multiprocessing timing anomalies. SIAM J. Appl. math. 17, 416-429 (1969) · Zbl 0188.23101 [2] Hall, L. A.; Shmoys, D. B.: Approximation schemes for constrained scheduling problems. Proc. 30th ann. IEEE symp. On foundations of computer science, 134-139 (1989) [3] Hong, K. S.; Leung, J. Y. -T.: On-line scheduling of real-time tasks. IEEE trans. Comput. 41, 1326-1331 (1992) [4] J. Sgall, On-line scheduling -- a survey, in: A. Fiat and G.J. Woeginger (eds.), On-line Algorithms, Lecture Notes in Computer Science, to appear. [5] Shmoys, D. B.; Wein, J.; Williamson, D. P.: Scheduling parallel machines on-line. SIAM J. Comput. 24, 1313-1331 (1995) · Zbl 0845.68042 [6] A.P.A. Vestjens, Scheduling uniform machines on-line requires nondecreasing speed ratios, Math. Programming (B), to appear. · Zbl 0920.90077