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Convexity Results for Queueing System with Variable Service Rate

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Analytical and Stochastic Modeling Techniques and Applications (ASMTA 2014)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 8499))

Abstract

In this paper we consider a single server queueing system with variable service rate. If the number of customers in system is less than a threshold, the service rate is set in a low value and it also can be switched to a high value once the number reaches to the threshold. We study five performance measures: the probability that the system is empty, the expected number in system and in queue, as well as the expected sojourn time in system and waiting time in queue. And we primarily show that these performance measures have the monotonicity or convexity with respect to the traffic intensity. These results are useful to the optimization problem in queueing system.

This work was supported by the National Natural Science Foundation of China (Nos. 11171019, 71390334), and the Program for New Century Excellent Talents in University (NCET-11-0568).

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References

  1. Rolfe, A.J.: A Note on Marginal Allocation in Multiple-Server Service Systems. Management Science 16, 656–658 (1971)

    Article  Google Scholar 

  2. Dyer, M.E., Proll, L.G.: On the Validity of Marginal Analysis for Allocating Servers in M/M/C Queues. Management Science 23, 1019–1022 (1977)

    Article  MATH  Google Scholar 

  3. Grassmann, W.: The Convexity of the Mean Queue Size of the M/M/C Queue with respect to the Traffic Intensity. Journal of Applied Probability 20, 916–919 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  4. Lee, H.L., Cohen, M.A.: A Note on the Convexity of Performance Measures of M/M/C Queueing Systems. Journal of Applied Probability 20, 920–923 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  5. Harel, A., Zipkin, P.H.: Strong Convexity Results for Queueing System. Operations Research 35, 405–418 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  6. Harel, A., Zipkin, P.H.: Convexity Results for the Erlang Delay and Loss Formulae When the Server Unilization Is Held Constant. Operations Research 59, 1420–1426 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dimitrakopoulos, Y., Burnetas, A.: Customer Equilibrium and Optimal Strategies in an M/M/1 Queue with Dynamic Service Control. arXiv:1112.1372v1

    Google Scholar 

  8. Gross, D., Shortle, J.F., Thompson, J.M., Harris, C.M.: Fundamental of Queueing Theory. A John wiley & Sons, Inc., Publication, Hoboken (2008)

    Book  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Beijing Jiaotong University, 100044, Beijing, China

    Xuelu Zhang & Jinting Wang

  2. College of Science, Qilu University of Technology, 250353, Jinan, China

    Xuelu Zhang

  3. Department of basic courses, Shandong Women’s University, 250002, Jinan, China

    Qing Ma

Authors
  1. Xuelu Zhang
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  2. Jinting Wang
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  3. Qing Ma
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Editor information

Editors and Affiliations

  1. INRIA, Campus de Beaulieu, 35042, Rennes, Cedex, France

    Bruno Sericola

  2. BME-HIT, Magyar tudósok, krt.2, 1117, Budapest, Hungary

    Miklós Telek

  3. BME-HIT, Magyar tudósok krt. 2, Hungary

    Gábor Horváth

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© 2014 Springer International Publishing Switzerland

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Zhang, X., Wang, J., Ma, Q. (2014). Convexity Results for Queueing System with Variable Service Rate. In: Sericola, B., Telek, M., Horváth, G. (eds) Analytical and Stochastic Modeling Techniques and Applications. ASMTA 2014. Lecture Notes in Computer Science, vol 8499. Springer, Cham. https://doi.org/10.1007/978-3-319-08219-6_14

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  • DOI: https://doi.org/10.1007/978-3-319-08219-6_14

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08218-9

  • Online ISBN: 978-3-319-08219-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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