Analysis of an M/M/1 Retrial Queue with Speed Scaling

  • Tuan Phung-Duc
  • Wouter RogiestEmail author
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 383)


Recently, queues with speed scaling have received considerable attention due to their applicability to data centers, enabling a better balance between performance and energy consumption. This paper proposes a new model where blocked customers must leave the service area and retry after a random time, with retrial rate either varying proportionally to the number of retrying customers (linear retrial rate) or non-varying (constant retrial rate). For both, we study the case without and with setup time. In all four cases, we obtain an exact solution for the stationary queue length distribution. This document presents the resulting expressions as well as their derivation.


Data center Energy efficiency Speed scaling Setup time Retrial queue 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Artalejo, J.R., Economou, A., Lopez-Herrero, M.J.: Analysis of a multiserver queue with setup times. Queueing Systems 51(1–2), 53–76 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Artalejo, J.R., Phung-Duc, T.: Markovian Retrial Queues with Two Way Communication. Journal of Industrial and Management Optimization 8(4), 781–806 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Barroso, L.A., Holzle, U.: The case for energy-proportional computing. Computer 40(12), 33–37 (2007)CrossRefGoogle Scholar
  4. 4.
    Van Heddeghem, W., Lambert, S., Lannoo, B., Colle, D., Pickavet, M., Demeester, P.: Trends in worldwide ICT electricity consumption from 2007 to 2012. Computer Communications 50(1), 64–76 (2014)CrossRefGoogle Scholar
  5. 5.
    Conway, R.W., Maxwell, W.L.: A Queueing Model with State Dependent Service Rate. Journal of Industrial Engineering 12, 132–136 (1961)Google Scholar
  6. 6.
    Yao, F., Demers, A., Shenker, S.: A scheduling model for reduced CPU energy. In: Proceedings 36th Annual Symposium on Foundations of Computer Science, pp. 374–382 (1995)Google Scholar
  7. 7.
    Wierman, A., Andrew, L., Tang, A.: Power-aware speed scaling in processor sharing systems. In: Proceedings of IEEE INFOCOM 2009, pp. 2007–2015 (2009)Google Scholar
  8. 8.
    Phung-Duc, T., Rogiest, W., Takahashi, Y., Bruneel, H.: Retrial queues with balanced call blending: analysis of single-server and multiserver model. Annals of Operations Research (2014). doi:  10.1007/s10479-014-1598-2 CrossRefzbMATHGoogle Scholar
  9. 9.
    Phung-Duc, T.: An explicit solution for a tandem queue with retrials and losses. Operational Research 12(2), 189–207 (2012)CrossRefGoogle Scholar
  10. 10.
    Phung-Duc, T.: Impatient customers in power-saving data centers. In: Sericola, B., Telek, M., Horváth, G. (eds.) ASMTA 2014. LNCS, vol. 8499, pp. 185–199. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  11. 11.
    Phung-Duc, T.: Server farms with batch arrival and staggered setup. In: Proceedings of the Fifth Symposium on Information and Communication Technology, pp. 240–247. ACM (2014)Google Scholar
  12. 12.
    Phung-Duc, T.: Exact solution for M/M/\(c\)/Setup queue (2014). Preprint:
  13. 13.
    Phung-Duc, T.: Multiserver queues with finite capacity and setup time. In: Remke, A., Manini, D., Gribaudo, M. (eds.) ASMTA 2015. LNCS, vol. 9081, pp. 173–187. Springer, Heidelberg (2015) CrossRefGoogle Scholar
  14. 14.
    Parthasarathy, P.R., Sudhesh, R.: Time-dependent analysis of a single-server retrial queue with state-dependent rates. Operations Research Letters 35(5), 601–611 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lu, X., Aalto, S., Lassila, P.: Performance-energy trade-off in data centers: impact of switching delay. In: Proceedings of 22nd IEEE ITC Specialist Seminar on Energy Efficient and Green Networking (SSEEGN), pp. 50–55 (2013)Google Scholar
  16. 16.
    Gandhi, A., Harchol-Balter, M., Adan, I.: Server farms with setup costs. Performance Evaluation 67, 1123–1138 (2010)CrossRefGoogle Scholar
  17. 17.
    Gandhi, A., Doroudi, S., Harchol-Balter, M., Scheller-Wolf, A.: Exact analysis of the M/M/k/setup class of markov chains via recursive renewal reward. In: Proceedings of the ACM SIGMETRICS, pp. 153–166 (2013)CrossRefGoogle Scholar
  18. 18.
    Gandhi, A., Doroudi, S., Harchol-Balter, M., Scheller-Wolf, A.: Exact analysis of the M/M/k/setup class of Markov chains via recursive renewal reward. Queueing Systems 77(2), 177–209 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Mitrani, I.: Managing performance and power consumption in a server farm. Annals of Operations Research 202(1), 121–134 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Mathematical and Computing SciencesTokyo Institute of TechnologyTokyoJapan
  2. 2.Department of Telecommunications and Information ProcessingGhent UniversityGentBelgium

Personalised recommendations