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A Single Server Queue with Workload-Dependent Service Speed and Vacations

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Queueing Theory and Network Applications (QTNA 2019)

Abstract

In modern data centers, the trade-off between processing speed and energy consumption is an important issue. Motivated by this, we consider a queueing system in which the service speed is a function of the workload, and in which the server switches off when the system becomes empty, only to be activated again when the workload reaches a certain threshold. For this system we obtain the steady-state workload distribution. We use this result to choose the activation threshold such that a certain cost function, involving holding costs and activation costs, is minimized.

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References

  1. Bekker, R., Borst, S.C., Boxma, O.J., Kella, O.: Queues with workload-dependent arrival and service rates. Queueing Syst. 46, 537–556 (2004)

    Article  MathSciNet  Google Scholar 

  2. Brill, P.H.: Level Crossing Methods in Stochastic Models. Springer International Publishing (2017)

    Google Scholar 

  3. Brill, P.H., Posner, M.J.M.: Level crossing in point processes applied to queues: single server case. Oper. Res. 25, 662–674 (1977)

    Article  MathSciNet  Google Scholar 

  4. Browne, S., Sigman, K.: Workload-modulated queues with application to storage processes. J. Appl. Probab. 29, 699–712 (1992)

    Article  MathSciNet  Google Scholar 

  5. Cohen, J.W.: On up- and downcrossings. J. Appl. Probab. 14, 405–410 (1977)

    Article  MathSciNet  Google Scholar 

  6. Cohen, J.W.: The Single Server Queue, 2nd edn. North-Holland Publishing Company, Amsterdam (1982)

    MATH  Google Scholar 

  7. Feinberg, E.A., Kella, O.: Optimality of D-policies for an \(M/G/1\) queue with a removable server. Queueing Syst. 42, 355–376 (2002)

    Article  MathSciNet  Google Scholar 

  8. Gandhi, A., Harchol-Balter, M., Adan, I.: Server farms with setup costs. Perform. Eval. 67, 1123–1138 (2010)

    Article  Google Scholar 

  9. 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. ACM SIGMETRICS Perform. Eval. Rev. 41, 153–166 (2013)

    Article  Google Scholar 

  10. Gaver, D.P., Miller, R.G.: Limiting distributions for some storage problems. In: Arrow, K.J., Karlin, S., Scarf, H. (eds.) Studies in Applied Probability and Management Science, pp. 110–126. Stanford University Press, Stanford (1962)

    Google Scholar 

  11. Harrison, J.M., Resnick, S.I.: The stationary distribution and first exit probabilities of a storage process with general release rule. Math. Oper. Res. 1, 347–358 (1976)

    Article  MathSciNet  Google Scholar 

  12. Koops, D., Boxma, O.J., Mandjes, M.R.H.: Networks of \(\cdot /G/\infty \) queues with shot-noise-driven arrival intensities. Queueing Syst. 86, 301–325 (2017)

    Article  MathSciNet  Google Scholar 

  13. Maccio, V.J., Down, D.G.: Structural properties and exact analysis of energy-aware multiserver queueing systems with setup times. Perform. Eval. 121, 48–66 (2018)

    Article  Google Scholar 

  14. Phung-Duc, T.: Exact solutions for M/M/c/setup queues. Telecommun. Syst. 64, 309–324 (2017)

    Article  Google Scholar 

  15. Phung-Duc, T., Rogiest, W., Wittevrongel, S.: Single server retrial queues with speed scaling: analysis and performance evaluation. J. Ind. Manage. Optim. 13, 1927–1943 (2017)

    MathSciNet  MATH  Google Scholar 

  16. Ross, S.M.: Stochastic Processes. Wiley, New York (1983)

    MATH  Google Scholar 

  17. Tricomi, F.G.: Integral Equations. Interscience Publishers, New York (1957)

    MATH  Google Scholar 

  18. Wierman, A., Andrew, L.L.H., Lin, M.: Speed scaling: an algorithmic perspective. In: Handbook of Energy-Aware and Green Computing. Chapman & Hall/CRC Computing and Information Science Series (2011)

    Google Scholar 

  19. Yajima, M., Phung-Duc, T.: Batch arrival single-server queue with variable service speed and setup time. Queueing Syst. 86, 241–260 (2017)

    Article  MathSciNet  Google Scholar 

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Correspondence to Yutaka Sakuma .

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Sakuma, Y., Boxma, O., Phung-Duc, T. (2019). A Single Server Queue with Workload-Dependent Service Speed and Vacations. In: Phung-Duc, T., Kasahara, S., Wittevrongel, S. (eds) Queueing Theory and Network Applications. QTNA 2019. Lecture Notes in Computer Science(), vol 11688. Springer, Cham. https://doi.org/10.1007/978-3-030-27181-7_8

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  • DOI: https://doi.org/10.1007/978-3-030-27181-7_8

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-27180-0

  • Online ISBN: 978-3-030-27181-7

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