Skip to main content
SpringerLink
Log in

Computational geometric approach to submodular function minimization for multiclass queueing systems

  • Original Paper
  • Area 1
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

This paper presents an efficient algorithm for minimizing a certain class of submodular functions that arise in analysis of multiclass queueing systems. In particular, the algorithm can be used for testing whether a given multiclass M/M/1 achieves an expected performance by an appropriate control policy. With the aid of the topological sweeping method for line arrangement, our algorithm runs in O(n 2) time, where n is the cardinality of the ground set. This is much faster than direct applications of general submodular function minimization algorithms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anagnostou, E., Polimenis, V.G., Guibas, L.J.: Topological sweeping in three dimensions. In: Proceedings of International Symposium on Algorithms, pp. 310–317. Springer, Berlin (1990)

  2. Bertsimas D., Paschalidis I.C., Tsitsiklis J.N.: Optimization of multiclass queueing networks: polyhedral and nonlinear characterizations of achievable performance. Ann. Appl. Probab. 4, 43–75 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Coffman E.G. Jr., Mitrani I.: A characterization of waiting time performance realizable by single-server queues. Oper. Res. 28, 810–821 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dacre M., Glazebrook K., Niño-Mora J.: The achievable region approach to the optimal control of stochastic systems. J. Roy. Stat. Soc. B 61, 747–791 (1999)

    Article  MATH  Google Scholar 

  5. Edelsbrunner H.: Algorithms in Combinatorial Geometry. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  6. Edelsbrunner H., Guibas L.J.: Topologically sweeping an arrangement. J. Comput. Syst. Sci. 38, 165–194 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Edelsbrunner H., Guibas L.J.: Topologically sweeping an arrangement—a correction. J. Comput. Syst. Sci. 42, 249–251 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Federgruen A., Groenevelt H.: Characterization and optimization of achievable performance in general queueing systems. Oper. Res. 36, 733–741 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  9. Federgruen A., Groenevelt H.: M/G/c queueing systems with multiple customer classes: Characterization and control of achievable performance under nonpreemptive priority rules. Manag. Sci. 34, 1121–1138 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fujishige S.: Submodular Function and Optimization. North-Holland, Amsterdam (2005)

    Google Scholar 

  11. Gelenbe E., Mitrani I.: Analysis and Synthesis of Computer Systems. Academic Press, New York (1980)

    MATH  Google Scholar 

  12. Grötschel M., Lovász L., Schrijver A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grötschel M., Lovász L., Schrijver A.: Geometric Algorithms and Combinatorial Optimization. Berlin, Springer (1988)

    Book  MATH  Google Scholar 

  14. Iwata S., Fleischer L., Fujishige S.: A combinatorial strongly polynomial algorithm for minimizing submodular functions. J. ACM 48, 761–777 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kumar S., Kumar P.R.: Performance bounds for queueing networks and scheduling policies. IEEE Trans. Autom. Control 39, 1600–1611 (1964)

    Article  Google Scholar 

  16. McCormick, S.T.: Submodular function minimization. In: Aardal, K., Nemhauser, G., Weismantel, R. (eds.) Handbook on Discrete Optimization. Elsevier, Amsterdam (2005)

  17. Orlin J.B.: A faster strongly polynomial time algorithm for submodular function minimization. Math. Program. 118, 237–251 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rafalin, E., Souvaine, D., Streinu, I.: Topological sweep in degenerate cases. In: Proceedings of the 4th International Workshop on Algorithm Engineering and Experiments (ALENEX), pp. 155–165 (2002) (Their implementation is avialable from http://www.cs.tufts.edu/research/geometry/sweep/)

  19. Schrijver A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Comb. Theory B 80, 346–355 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ziegler G.M.: Lectures on Polytopes. Berlin, Springer (1995)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tokyo Research Laboratory, IBM Japan, Yamato, Kanagawa, 242-8502, Japan

    Toshinari Itoko

  2. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Satoru Iwata

Authors
  1. Toshinari Itoko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Satoru Iwata
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Satoru Iwata.

Additional information

A preliminary version of this paper has appeared in Proceedings of the Twelfth International Conference on Integer Programming and Combinatorial Optimization (2007), LNCS 4513, Springer-Verlag, pp. 267–279.

About this article

Cite this article

Itoko, T., Iwata, S. Computational geometric approach to submodular function minimization for multiclass queueing systems. Japan J. Indust. Appl. Math. 29, 453–468 (2012). https://doi.org/10.1007/s13160-012-0074-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-012-0074-0

Keywords

Mathematics Subject Classification (2000)

Search

Navigation