Scheduling Periodic Tasks in a Hard Real-Time Environment

  • Friedrich Eisenbrand
  • Nicolai Hähnle
  • Martin Niemeier
  • Martin Skutella
  • José Verschae
  • Andreas Wiese
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6198)

Abstract

We give a rigorous account on the complexity landscape of an important real-time scheduling problem that occurs in the design of software-based aircraft control. The goal is to distribute tasks τi = (ci,pi) on a minimum number of identical machines and to compute offsets ai for the tasks such that no collision occurs. A task τi releases a job of running time ci at each time \(a_i + k\cdot p_i, \, k \in {\mathbb N}_0\) and a collision occurs if two jobs are simultaneously active on the same machine. Our main results are as follows: (i) We show that the minimization problem cannot be approximated within a factor of n1 − ε for any ε> 0. (ii) If the periods are harmonic (for each i,j one has pi |pj or pj |pi), then there exists a 2-approximation for the minimization problem and this result is tight, even asymptotically. (iii) We provide asymptotic approximation schemes in the harmonic case if the number of different periods is constant.

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References

  1. [BBNS02]
    Bar-Noy, A., Bhatia, R., Naor, J., Schieber, B.: Minimizing service and operation costs of periodic scheduling. Math. Oper. Res. 27(3) (2002)Google Scholar
  2. [Bha98]
    Bhatia, R.: Approximation Algorithms for Scheduling Problems. PhD thesis, University of Maryland (1998)Google Scholar
  3. [BHR93]
    Baruah, S.K., Howell, R.R., Rosier, L.E.: Feasibility problems for recurring tasks on one processor. In: Selected papers of the 15th International Symposium on Mathematical Foundations of Computer Science, pp. 3–20. Elsevier, Amsterdam (1993)Google Scholar
  4. [BRTV90]
    Baruah, S., Rousier, L., Tulchinsky, I., Varvel, D.: The complexity of periodic maintenance. In: Proceedings of the International Computer Symposium (1990)Google Scholar
  5. [But04]
    Buttazzo, G.C.: Hard Real-time Computing Systems: Predictable Scheduling Algorithms And Applications. Springer, Heidelberg (2004)Google Scholar
  6. [CSW08]
    Conforti, M., Di Summa, M., Wolsey, L.A.: The mixing set with divisible capacities. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 435–449. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. [EHN+10]
    Eisenbrand, F., Hähnle, N., Niemeier, M., Skutella, M., Verschae, J., Wiese, A.: Scheduling periodic tasks in a hard real-time environment. Technical report, EPF Lausanne & TU Berlin (February 2010), http://disopt.epfl.ch/webdav/site/disopt/shared/PM_EHNSVW10_report.pdf
  8. [FdlVL81]
    de la Fernandez Vega, W., Lueker, G.S.: Bin packing can be solved within 1 + ε in linear time. Combinatorica 1, 349–355 (1981)MATHCrossRefMathSciNetGoogle Scholar
  9. [GJ79]
    Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-Completeness. Freemann, New York (1979)MATHGoogle Scholar
  10. [KAL96]
    Korst, J., Aarts, E., Lenstra, J.K.: Scheduling periodic tasks. INFORMS Journal on Computing 8, 428–435 (1996)MATHCrossRefGoogle Scholar
  11. [KALW91]
    Korst, J., Aarts, E., Lenstra, J.K., Wessels, J.: Periodic multiprocessor scheduling. In: Aarts, E.H.L., van Leeuwen, J., Rem, M. (eds.) PARLE 1991. LNCS, vol. 505, pp. 166–178. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  12. [KK82]
    Karmakar, N., Karp, R.M.: An efficient approximation scheme for the one-dimensional binpacking problem. In: Foundations of Computer Science (FOCS), vol. 23, pp. 312–320 (1982)Google Scholar
  13. [Leu04]
    Leung, J.Y.-T.: Handbook of Scheduling: Algorithms, Models and Performance Analysis. Chapman & Hall/CRC (2004)Google Scholar
  14. [Mar85]
    Marcotte, O.: The cutting stock problem and integer rounding. Mathematical Programming 33, 82–92 (1985)MATHCrossRefMathSciNetGoogle Scholar
  15. [NZM91]
    Niven, I., Zuckerman, H.S., Montgomery, H.L.: An Introduction to the Theory of Numbers, 5th edn. Wiley, Chichester (1991)Google Scholar
  16. [SL94]
    Simchi-Levi, D.: New worst-case results for the bin-packing problem. Naval Research Logistics 41, 579–585 (1994)MATHCrossRefMathSciNetGoogle Scholar
  17. [VA97]
    Verhaegh, W.F.J., Aarts, E.H.L.: A polynomial-time algorithm for knapsack with divisible item sizes. Information Processing Letters 62, 217–221 (1997)CrossRefMathSciNetGoogle Scholar
  18. [WL83]
    Wei, W.D., Liu, C.L.: On a periodic maintenance problem. Operations Research Letters 2, 90–93 (1983)MATHCrossRefGoogle Scholar
  19. [ZIRdF08]
    Zhao, M., de Farias Jr., I.R.: The mixing-MIR set with divisible capacities. Mathematical Programming 115, 73–103 (2008)MATHCrossRefMathSciNetGoogle Scholar
  20. [Zuc07]
    Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory of Computing 3, 103–128 (2007)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Friedrich Eisenbrand
    • 1
  • Nicolai Hähnle
    • 1
  • Martin Niemeier
    • 1
  • Martin Skutella
    • 2
  • José Verschae
    • 2
  • Andreas Wiese
    • 2
  1. 1.EPFLLausanneSwitzerland
  2. 2.TU BerlinGermany

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