Slow Down and Sleep for Profit in Online Deadline Scheduling

  • Peter Kling
  • Andreas Cord-Landwehr
  • Frederik Mallmann-Trenn
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7659)


We present and study a new model for energy-aware and profit-oriented scheduling on a single processor. The processor features dynamic speed scaling as well as suspension to a sleep mode. Jobs arrive over time, are preemptable, and have different sizes, values, and deadlines. On the arrival of a new job, the scheduler may either accept or reject the job. Accepted jobs need a certain energy investment to be finished in time, while rejected jobs cause costs equal to their values. Here, power consumption at speed s is given by P(s) = s α  + β and the energy investment is power integrated over time. Additionally, the scheduler may decide to suspend the processor to a sleep mode in which no energy is consumed, though awaking entails fixed transition costs γ. The objective is to minimize the total value of rejected jobs plus the total energy.

Our model combines aspects from advanced energy conservation techniques (namely speed scaling and sleep states) and profit-oriented scheduling models. We show that rejection-oblivious schedulers (whose rejection decisions are not based on former decisions) have - in contrast to the model without sleep states - an unbounded competitive ratio w.r.t. the processor parameters α and β. It turns out that the worst-case performance of such schedulers depends linearly on the jobs’ value densities (the ratio between a job’s value and its work). We give an algorithm whose competitiveness nearly matches this lower bound. If the maximum value density is not too large, the competitiveness becomes α α  + 2. Also, we show that it suffices to restrict the value density of low-value jobs only. Using a technique from [13] we transfer our results to processors with a fixed maximum speed.


Competitive Ratio Online Algorithm Sleep Mode Sleep State Energy Investment 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albers, S.: Algorithms for Dynamic Speed Scaling. In: Proc. of the 28th International Symp. On Theoretical Aspects of Computer Science (STACS), Schloss Dagstuhl, pp. 1–11 (2011)Google Scholar
  2. 2.
    Albers, S.: Energy-Effcient Algorithms. Comm. of the ACM 53(5), 86–96 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Albers, S., Antoniadis, A.: Race to Idle: New Algorithms for Speed Scaling with a Sleep State. In: Proceedings of the 23rd Symposium on Discrete Algorithms, SODA (2012)Google Scholar
  4. 4.
    Albers, S., Antoniadis, A., Greiner, G.: On Multi-Processor Speed Scaling with Migration. In: Proceedings of the 23rd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pp. 279–288. ACM (2011)Google Scholar
  5. 5.
    Bansal, N., Chan, H.-L., Pruhs, K., Katz, D.: Improved Bounds for Speed Scaling in Devices Obeying the Cube-Root Rule. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 144–155. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Bansal, N., Chan, H.-L., Lam, T.-W., Lee, L.-K.: Scheduling for Speed Bounded Processors. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 409–420. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Bansal, N., Kimbrel, T., Pruhs, K.: Speed Scaling to Manage Energy and Temperature. Journal of the ACM 54(1), 1–39 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Baptiste, P.: Scheduling Unit Tasks to Minimize the Number of Idle Periods: A Polynomial Time Algorithm for Online Dynamic Power Management. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithm (SODA), pp. 364–367. ACM (2006)Google Scholar
  9. 9.
    Baptiste, P., Chrobak, M., Dürr, C.: Polynomial Time Algorithms for Minimum Energy Scheduling. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 136–150. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  10. 10.
    Barroso, L.A., Hölzle, U.: The Case for Energy-Proportional Computing. Computer 40(12), 33–37 (2007)CrossRefGoogle Scholar
  11. 11.
    Baruah, S., Koren, G., Mishra, B., Raghunathan, A., Rosier, L., Shasha, D.: Online Scheduling in the Presence of Overload. In: Proc. of the 32nd Symp. on Foundations of Computer Science (FOCS), pp. 100–110 (1991)Google Scholar
  12. 12.
    Chan, H.-L., Chan, W.-T., Lam, T.-W., Lee, L.-K., Mak, K.-S., Wong, P.W.H.: Energy Efficient Online Deadline Scheduling. In: Proceedings ofthe 18th Symposium on Discrete Algorithms (SODA), pp. 795–804. SIAM (2007)Google Scholar
  13. 13.
    Chan, H.-L., Lam, T.-W., Li, R.: Tradeoff between Energy and Throughput for Online Deadline Scheduling. In: Jansen, K., Solis-Oba, R. (eds.) WAOA 2010. LNCS, vol. 6534, pp. 59–70. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  14. 14.
    Cord-Landwehr, A., Kling, P., Mallmann-Trenn, F.: Slow Down & Sleep for Profit in Online Deadline Scheduling. arXiv:1209.2848 [cs.DS] (2012)Google Scholar
  15. 15.
    Han, X., Lam, T.-W., Lee, L.-K., To, I.K.K., Wong, P.W.H.: Deadline Scheduling and Power Management for Speed Bounded Processors. Theoretical Computer Science 411(42), 3587–3600 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Irani, S., Shukla, S., Gupta, R.: Algorithms for Power Savings. ACM Transactions on Algorithm 3(4) (2007)Google Scholar
  17. 17.
    Pruhs, K., Stein, C.: How to Schedule When You Have to Buy Your Energy. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX 2010, LNCS, vol. 6302, pp. 352–365. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  18. 18.
    Yao, F.F., Demers, A.J., Shenker, S.: A Scheduling Model for Reduced CPU Energy. In: Proceedings of the 36th Annual Symposium on Foundations of Computer Science (FOCS), pp. 374–382 (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Peter Kling
    • 1
  • Andreas Cord-Landwehr
    • 1
  • Frederik Mallmann-Trenn
    • 1
  1. 1.Heinz Nixdorf Institute and Computer Science DepartmentUniversity of PaderbornGermany

Personalised recommendations