Abstract
We consider the problem to minimize the total weighted completion time of a set of jobs with individual release dates which have to be scheduled on identical parallel machines. The durations of jobs are realized on-line according to given probability distributions, and the aim is to find a scheduling policy that minimizes the objective in expectation. We present a polyhedral relaxation of the corresponding performance space, and then derive the first constant-factor performance guarantees for priority policies which are guided by optimum LP solutions, thus generalizing previous results from deterministic scheduling. In the absence of release dates, our LP-based analysis also yields an additive performance guarantee for the WSEPT rule which implies both a worst-case performance ratio and a result on its asymptotic optimality.
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References
Bertsimas, D., Niño-Mora, J.: Conservation laws, extended polymatroids and multi-armed bandit problems: A polyhedral approach to indexable systems. Mathematics of Operations Research 21, 257–306 (1996)
Bruno, J.L., Coffman Jr., E.G., Sethi, R.: Scheduling independent tasks to reduce mean finishing time. Communications of the Association for Computing Machinery 17, 382–387 (1974)
Dacre, M., Glazebrook, K.D., Niño-Mora, J.: The achievable region approach to the optimal control of stochastic systems. Journal of the Royal Statistical Society (to appear)
Eastman, W.L., Even, S., Isaacs, I.M.: Bounds for the optimal scheduling of n jobs on m processors. Management Science 11, 268–279 (1964)
K. D. Glazebrook, Personal communication (January 1999)
Glazebrook, K.D., Niño-Mora, J.: Scheduling multiclass queueing networks on parallel servers: Approximate and heavy-traffic optimality of Klimov’s rule. In: Burkard, R.E., Woeginger, G.J. (eds.) ESA 1997. LNCS, vol. 1284, pp. 232–245. Springer, Heidelberg (1997)
Graham, R.L., Lawler, E.L., Lenstra, J.K., Rinnoy Kan, A.H.G.: Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics 5, 287–326 (1979)
L. A. Hall, A. S. Schulz, D. B. Shmoys, and J. Wein, Scheduling to minimize average completion time: O_-line and on-line approximation algorithms, Mathematics of Operations Research, 22 (1997), pp. 513{544.
Hall, W.J., Wellner, J.A.: Mean residual life. In: Csörgöo, M., Dawson, D.A., Rao, J.N.K., Saleh, A.K.Md.E. (eds.) Statistics and Related Topics, Proceedings of the International Symposium on Statistics and Related Topics, pp. 169–184. North-Holland, Amsterdam (1981)
Kämpke, T.: On the optimality of static priority policies in stochastic scheduling on parallel machines. Journal of Applied Probability 24, 430–448 (1987)
Kawaguchi, T., Kyan, S.: Worst case bound on an LRF schedule for the mean weighted flow-time problem. SIAM Journal on Computing 15, 1119–1129 (1986)
Lawler, E.L., Lenstra, J.K., Rinnoy Kan, A.H.G., Shmoys, D.B.: Sequencing and scheduling: Algorithms and complexity. In: Logistics of Production and Inventory. Handbooks in Operations Research and Management Science, vol. 4, pp. 445–522. North-Holland, Amsterdam (1993)
Möhring, R.H., Radermacher, F.J., Weiss, G.: Stochastic scheduling problems I: General strategies. ZOR - Zeitschrift für Operations Research 28, 193–260 (1984)
Möhring, R.H., Radermacher, F.J., Weiss, G.: Stochastic scheduling problems II: Set strategies. ZOR - Zeitschrift für Operations Research 29, 65–104 (1985)
Möhring, R.H., Schulz, A.S., Uetz, M.: Approximation in stochastic scheduling: The power of LP-based priority policies, Tech. Rep. 595/1998, Department of Mathematics, Technical University of Berlin (1998)
Phillips, C.A., Stein, C., Wein, J.: Minimizing average completion time in the presence of release dates. WADS 1995 82, 199–223 (1998); A preliminary version of this paper (Scheduling jobs that arrive over time). In: Mathamaticial Programming. LNCS, vol. 955, pp. 86–97. Springer, Heidelberg (1995)
Rothkopf, M.H.: Scheduling with random service times. Management Science 12, 703–713 (1966)
Schulz, A.S.: Scheduling to minimize total weighted completion time: Performance guarantees of LP-based heuristics and lower bounds, in Integer Programming and Combinatorial Optimization. In: Cunningham, W.H., Queyranne, M., McCormick, S.T. (eds.) IPCO 1996. LNCS, vol. 1084, pp. 301–315. Springer, Heidelberg (1996)
Sgall, J.: On-line scheduling, in Online Algorithms: The State of the Art. In: Fiat, A. (ed.) Dagstuhl Seminar 1996. LNCS, vol. 1442, pp. 196–231. Springer, Heidelberg (1998)
Smith, W.E.: Various optimizers for single-stage production. Naval Research and Logistics Quarterly 3, 59–66 (1956)
Spaccamela, A.M., Rhee, W.S., Stougie, L., van de Geer, S.: Probabilistic analysis of the minimum weighted flowtime scheduling problem. Operations Research Letters 11, 67–71 (1992)
Weber, R.R., Varaiya, P., Walrand, J.: Scheduling jobs with stochastically ordered processing times on parallel machines to minimize expected flowtime. Journal of Applied Probability 23, 841–847 (1986)
Weiss, G.: Approximation results in parallel machines stochastic scheduling. Annals of Operations Research 26, 195–242 (1990)
Weiss, G.: Turnpike optimality of Smith’s rule in parallel machines stochastic scheduling. Mathematics of Operations Research 17, 255–270 (1992)
Weiss, G.: Personal communication (January 1999)
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Möhring, R.H., Schulz, A.S., Uetz, M. (1999). Stochastic Machine Scheduling: Performance Guarantees for LP-Based Priority Policies. In: Hochbaum, D.S., Jansen, K., Rolim, J.D.P., Sinclair, A. (eds) Randomization, Approximation, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 1999 1999. Lecture Notes in Computer Science, vol 1671. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-48413-4_16
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DOI: https://doi.org/10.1007/978-3-540-48413-4_16
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