A Near Optimal Mechanism for Energy Aware Scheduling

  • Antonios Antoniadis
  • Andrés Cristi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11059)


With the increased popularity of cloud computing it is of paramount importance to understand energy-efficiency from a game-theoretic perspective. An important question is how the operator of a server should deal with combining energy-efficiency and the particular interests of the users. Consider a cloud server, where clients/agents can submit jobs for processing. The quality of service that each agent perceives is given by a non-decreasing function of the completion time of her job which is private information. The server has to process the jobs and charge each agent while trying to optimize the social cost, defined as the energy expenditure plus the sum of the values of the cost functions of the agents. The operator would like to design a mechanism in order to optimize this objective, which ideally is computationally tractable, charges the users “fairly” and induces a game with an equilibrium.

We describe and analyze one such mechanism called modAVR, which relies on an adaption of the well-known Average Rate (AVR) algorithm for scheduling the jobs. We prove that modAVR combines the aforementioned properties with a constant Price of Anarchy, i.e., despite the fact that it is based on an algorithm designed for optimizing the energy alone, every equilibrium it results in is near-optimal for the total social cost as well. The existence of a Nash equilibrium is proven for both mixed strategies and (in a slightly more restricted setting) pure strategies.

A further interesting feature of modAVR is that it is indirect: each user needs only to declare an upper bound on the completion time of her job, and not the cost function.

Additionally, we prove that for the corresponding mechanism that uses the classical YDS algorithm for scheduling the jobs no pure Nash equilibrium can exist for a very broad and natural class of cost functions. Finally, we are able to extend several of our results for modAVR to a mechanism based on a slight variation of the YDS algorithm. This variation is known also to not admit Nash equilibria in pure strategies.



We would like to thank José Correa, Dimitris Fotakis, Martin Hoefer, Ruben Hoeksma, Minming Li, and Sebastian Ott for interesting discussions related to this work.


  1. 1.
    Albers, S.: Energy-efficient algorithms. Commun. ACM 53(5), 86–96 (2010)CrossRefGoogle Scholar
  2. 2.
    Albers, S., Antoniadis, A.: Race to idle: new algorithms for speed scaling with a sleep state. ACM Trans. Algorithms 10(2), 9:1–9:31 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Albers, S., Antoniadis, A., Greiner, G.: On multi-processor speed scaling with migration. J. Comput. Syst. Sci. 81(7), 1194–1209 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Albers, S., Fujiwara, H.: Energy-efficient algorithms for flow time minimization. ACM Trans. Algorithms 3(4), 49 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Angel, E., Bampis, E., Chau, V., Thang, N.K.: Throughput maximization in multiprocessor speed-scaling. In: Ahn, H.-K., Shin, C.-S. (eds.) ISAAC 2014. LNCS, vol. 8889, pp. 247–258. Springer, Cham (2014). Scholar
  6. 6.
    Antoniadis, A., Huang, C., Ott, S.: A fully polynomial-time approximation scheme for speed scaling with sleep state. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, pp. 1102–1113. SIAM (2015)Google Scholar
  7. 7.
    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. LNCS, vol. 5125, pp. 409–420. Springer, Heidelberg (2008). Scholar
  8. 8.
    Bansal, N., Kimbrel, T., Pruhs, K.: Speed scaling to manage energy and temperature. J. ACM 54(1), 3:1–3:39 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bansal, N., Pruhs, K., Stein, C.: Speed scaling for weighted flow time. SIAM J. Comput. 39(4), 1294–1308 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chan, H., Chan, W., Lam, T.W., Lee, L., Mak, K., Wong, P.W.H.: Energy efficient online deadline scheduling. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, pp. 795–804. SIAM (2007)Google Scholar
  11. 11.
    Chan, H., Edmonds, J., Lam, T.W., Lee, L., Marchetti-Spaccamela, A., Pruhs, K.: Nonclairvoyant speed scaling for flow and energy. Algorithmica 61(3), 507–517 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dürr, C., Jez, L., Vásquez, O.C.: Scheduling under dynamic speed-scaling for minimizing weighted completion time and energy consumption. Discrete Appl. Math. 196, 20–27 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dürr, C., Jez, L., Vásquez, O.C.: Mechanism design for aggregating energy consumption and quality of service in speed scaling scheduling. Theor. Comput. Sci. 695, 28–41 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Glicksberg, I.L.: A further generalization of the kakutani fixed theorem, with application to nash equilibrium points. Proc. Am. Math. Soc. 3(1), 170 (1952)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Irani, S., Pruhs, K.: Algorithmic problems in power management. SIGACT News 36(2), 63–76 (2005)CrossRefGoogle Scholar
  16. 16.
    Lam, T.W., Lee, L., To, I.K., Wong, P.W.H.: Online speed scaling based on active job count to minimize flow plus energy. Algorithmica 65(3), 605–633 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Megow, N., Verschae, J.: Dual techniques for scheduling on a machine with varying speed. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013. LNCS, vol. 7965, pp. 745–756. Springer, Heidelberg (2013). Scholar
  18. 18.
    Mehta, A., Roughgarden, T., Sundararajan, M.: Beyond moulin mechanisms. In: Proceedings of the 8th ACM Conference on Electronic Commerce, EC 2007, pp. 1–10 (2007)Google Scholar
  19. 19.
    Moulin, H.: Incremental cost sharing: characterization by coalition strategy-proofness. Soc. Choice Welfare 16(2), 279–320 (1999)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Roughgarden, T.: Intrinsic robustness of the price of anarchy. In: Proceedings of the Forty-first Annual ACM Symposium on Theory of Computing, STOC 2009, pp. 513–522. ACM, New York (2009)Google Scholar
  21. 21.
    Roughgarden, T.: The price of anarchy in games of incomplete information. ACM Trans. Econ. Comput. 3(1), 6:1–6:20 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Yao, F.F., Demers, A.J., Shenker, S.: A scheduling model for reduced CPU energy. In: 36th Annual Symposium on Foundations of Computer Science, Milwaukee, Wisconsin, 23–25 October 1995, pp. 374–382. IEEE Computer Society (1995)Google Scholar

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Authors and Affiliations

  1. 1.Universität des Saarlandes and Max-Planck-Institut für InformatikSaarbrückenGermany
  2. 2.Universidad de ChileSantiagoChile

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