Queueing Systems

, Volume 60, Issue 3–4, pp 227–246 | Cite as

Steady state approximations of limited processor sharing queues in heavy traffic

  • Jiheng Zhang
  • Bert Zwart


We investigate steady state properties of limited processor sharing queues in heavy traffic. Our analysis builds on previously obtained process limit theorems, and requires the interchange of steady state and heavy traffic limits, which are established by a coupling argument. The limit theorems yield explicit approximations of the steady state queue length and response time distribution in heavy traffic, of which the quality is supported by simulation experiments.


Limited processor sharing Measure-valued process Steady state Heavy traffic Queue size Delay probability Response time 

Mathematics Subject Classification (2000)

60K25 68M20 90B22 68M07 


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  1. 1.
    Asmussen, S.: Applied Probability and Queues, 2nd edn. Applications of Mathematics (New York), vol. 51. Springer, New York (2003) Google Scholar
  2. 2.
    Avi-Itzhak, B., Halfin, S.: Expected response times in a non-symmetric time sharing queue with a limited number of service positions. In: Proceedings of the 12th International Teletraffic Congress. Torino (1988) Google Scholar
  3. 3.
    Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York (1999) Google Scholar
  4. 4.
    Blake, R.: Optimal control of thrashing. In: Proceedings of the 1982 ACM SIGMETRICS Conference on Measurements and Modeling of Computer Systems. Seattle, WA (1982) Google Scholar
  5. 5.
    Budhiraja, A., Lee, C.: Stationary distribution convergence for generalized Jackson networks in heavy traffic. Technical Report, University of North Carolina at Chapel Hill (2008).
  6. 6.
    Denning, P.J., Kahn, K.C., Leroudier, J., Potier, D., Suri, R.: Optimal multiprogramming. Acta Inform. 7, 197–216 (1976) CrossRefGoogle Scholar
  7. 7.
    Elnikety, S., Nahum, E., Tracy, J., Zwaenepoel, W.: A method for transparent admission control and request scheduling in e-commerce web sites. In: World-Wide-Web Conference (2004) Google Scholar
  8. 8.
    Gamarnik, D., Zeevi, A.: Validity of heavy traffic steady-state approximation in generalized Jackson networks. Ann. Appl. Probab. 16(1), 56–90 (2006) CrossRefGoogle Scholar
  9. 9.
    Grishechkin, S.: GI/G/1 processor sharing queue in heavy traffic. Adv. Appl. Probab. 26(2), 539–555 (1994) CrossRefGoogle Scholar
  10. 10.
    Gromoll, H.C.: Diffusion approximation for a processor sharing queue in heavy traffic. Ann. Appl. Probab. 14(2), 555–611 (2004) CrossRefGoogle Scholar
  11. 11.
    Gromoll, H.C., Kruk, Ł.: Heavy traffic limit for a processor sharing queue with soft deadlines. Ann. Appl. Probab. 17(3), 1049–1101 (2007) CrossRefGoogle Scholar
  12. 12.
    Gromoll, H.C., Puha, A.L., Williams, R.J.: The fluid limit of a heavily loaded processor sharing queue. Ann. Appl. Probab. 12(3), 797–859 (2002) CrossRefGoogle Scholar
  13. 13.
    Gromoll, H.C., Robert, P., Zwart, B.: Fluid limits for processor sharing queues with impatience. Math. Oper. Res. 33(2), 375–402 (2008) CrossRefGoogle Scholar
  14. 14.
    Gupta, V., Dai, J.G., Harchol-Balter, M., Zwart, B.: On the inapproximability of M/G/K: Why two moments of job size distribution are not enough. Technical report, Carnegie Mellon University (2007) Google Scholar
  15. 15.
    Heiss, H.-U., Wagner, R.: Adaptive load control in transaction processing systems. In: Proceedings of the 17th International Conference on Large Data Bases (1991) Google Scholar
  16. 16.
    Kamra, A., Misra, V., Nahum, E.M.: Yaksha: A self-tuning controller for managing the performance of 3-tiered web sites. In: Twelfth IEEE International Workshop on Quality of Service (2004) Google Scholar
  17. 17.
    Kleinrock, L.: Queueing Systems. Computer Applications, vol. II. Wiley, New York (1976) Google Scholar
  18. 18.
    Maulik, K., Zwart, B.: An extension of the square root law of TCP. Ann. Oper. Res. (2008, to appear) Google Scholar
  19. 19.
    Nuyens, M., van der Weij, W.: The limited processor sharing queue. Technical report, CWI, Amsterdam (2007) Google Scholar
  20. 20.
    Ritchie, D.M., Thompson, K.: The Unix time-sharing system. J. Assoc. Comput. Mach. 17(7), 365–375 (1974) Google Scholar
  21. 21.
    Schroeder, B., Harchol-Balter, M., Iyengar, A., Nahum, E., Wierman, A.: How to determine a good multi-programming level for external scheduling. In: Proceedings of the 22nd International Conference on Data Engineering. Atlanta, GA (2006) Google Scholar
  22. 22.
    Sigman, K., Wolff, R.W.: A review of regenerative processes. SIAM Rev. 35(2), 269–288 (1993) CrossRefGoogle Scholar
  23. 23.
    Zhang, F., Lipsky, L.: Modelling restricted processor sharing. In: Proc. of the 2006 Int’l Conf. on Parallel and Distributed Processing Techniques and Applications (PDPTA06) (2006) Google Scholar
  24. 24.
    Zhang, F., Lipsky, L.: An analytical model for computer systems with non-exponential service times and memory thrashing overhead. In: Proc. of the 2007 Int’l Conf. on Parallel and Distributed Processing Techniques and Applications (PDPTA07) (2007) Google Scholar
  25. 25.
    Zhang, J., Dai, J.G., Zwart, B.: Diffusion limits of limited processor sharing queues. Technical report, Georgia Institute of Technology (2007).
  26. 26.
    Zhang, J., Dai, J.G., Zwart, B.: Law of large number limits of limited processor sharing queues. Technical report, Georgia Institute of Technology (2007).

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.H. Milton Stewart School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA

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