Advertisement

Stochastic Machine Scheduling: Performance Guarantees for LP-Based Priority Policies

  • Rolf H. Möhring
  • Andreas S. Schulz
  • Marc Uetz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1671)

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.

Keywords

Schedule Problem Completion Time Parallel Machine Performance Guarantee Priority Policy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    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)zbMATHMathSciNetGoogle Scholar
  3. 3.
    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)Google Scholar
  4. 4.
    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)CrossRefMathSciNetGoogle Scholar
  5. 5.
    K. D. Glazebrook, Personal communication (January 1999) Google Scholar
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    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. zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    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)Google Scholar
  10. 10.
    Kämpke, T.: On the optimality of static priority policies in stochastic scheduling on parallel machines. Journal of Applied Probability 24, 430–448 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    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)CrossRefGoogle Scholar
  13. 13.
    Möhring, R.H., Radermacher, F.J., Weiss, G.: Stochastic scheduling problems I: General strategies. ZOR - Zeitschrift für Operations Research 28, 193–260 (1984)zbMATHGoogle Scholar
  14. 14.
    Möhring, R.H., Radermacher, F.J., Weiss, G.: Stochastic scheduling problems II: Set strategies. ZOR - Zeitschrift für Operations Research 29, 65–104 (1985)zbMATHGoogle Scholar
  15. 15.
    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)Google Scholar
  16. 16.
    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)Google Scholar
  17. 17.
    Rothkopf, M.H.: Scheduling with random service times. Management Science 12, 703–713 (1966)MathSciNetGoogle Scholar
  18. 18.
    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)Google Scholar
  19. 19.
    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)Google Scholar
  20. 20.
    Smith, W.E.: Various optimizers for single-stage production. Naval Research and Logistics Quarterly 3, 59–66 (1956)CrossRefMathSciNetGoogle Scholar
  21. 21.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Weiss, G.: Approximation results in parallel machines stochastic scheduling. Annals of Operations Research 26, 195–242 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Weiss, G.: Turnpike optimality of Smith’s rule in parallel machines stochastic scheduling. Mathematics of Operations Research 17, 255–270 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Weiss, G.: Personal communication (January 1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Rolf H. Möhring
    • 1
  • Andreas S. Schulz
    • 2
  • Marc Uetz
    • 1
  1. 1.Fachbereich Mathematik, Sekr. MA 6-1Technische Universität BerlinBerlinGermany
  2. 2.MIT, Sloan School of Management and Operations Research CenterCambridgeUSA

Personalised recommendations