Advertisement

Journal of Systems Science and Complexity

, Volume 30, Issue 5, pp 1097–1106 | Cite as

Strong approximation method and the (functional) law of iterated logarithm for GI/G/1 queue

  • Yongjiang GuoEmail author
  • Xiyang Hou
Article
  • 60 Downloads

Abstract

In this paper, a unified method based on the strong approximation (SA) of renewal process (RP) is developed for the law of the iterated logarithm (LIL) and the functional LIL (FLIL), which quantify the magnitude of the asymptotic rate of the increasing variability around the mean value of the RP in numerical and functional forms respectively. For the GI/G/1 queue, the method provides a complete analysis for both the LIL and the FLIL limits for four performance functions: The queue length, workload, busy time and idle time processes, covering three regimes divided by the traffic intensity.

Keywords

GI/G/1 queue renewal process (RP) strong approximation (SA) method the functional LIL (FLIL) the law of the iterated logarithm (LIL) 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Csörgő M and Révész P, Strong Approximations in Probability and Statistics, Academic Press, New York, 1981.zbMATHGoogle Scholar
  2. [2]
    Ethier S N and Kurtz T G, Markov Processes: Characterization and Convergence, Wiley, New York, 1986.CrossRefzbMATHGoogle Scholar
  3. [3]
    Strassen V, An invariance principle for the law of the iterated logarith, Z. Wahrscheinlichkeitstheorie verw. Geb, 1964, 3(3): 211–226.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Horváth L, Strong approximation of renewal processes, Stochastic Process. Appl., 1984, 18(1): 127–138.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Chen H and Yao D D, Fundamentals of Queueing Networks, Springer-Verlag, New York, 2001.CrossRefzbMATHGoogle Scholar
  6. [6]
    Csörgő M, Deheuvels P, and Horváth L, An approximation of stopped sums with applications in queueing theory, Advances in Applied Probability, 1987, 19(3): 674–690.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Chen H and Shen X, Strong approximations for multiclass feedforward queueing networks, Annals of Applied Probability, 2000, 10(3): 828–876.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Horváth L, Strong approximations of open queueing networks, Mathematics of Operations Research, 1992, 17(2): 487–508.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Zhang H and Hsu G H, Strong approximations for priority queues: Head-of-the-line-first discipline, Queueing Systems, 1992, 10(3): 213–234.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Iglehart G L, Multiple channel queues in heavy traffic: IV. Law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie verw. Geb, 1971, 17: 168–180.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Sakalauskas L L and Minkevičius S, On the law of the iterated logarithm in open queueing networks, European Journal of Operational Research, 2000, 120(3): 632–640.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Minkevičius S and Steišūnas S, A law of the iterated logarithm for global values of waiting time in multiphase queues, Statistics and Probability Letters, 2003, 61(4): 359–371.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Guo Y and Liu Y, A law of iterated logarithm for multiclass queues with preemptive priority service discipline, Queueing Systems, 2015, 79(3): 251–291.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Jackson J R, Networks of waiting lines, Operations Research, 1957, 5(4): 518–521.MathSciNetCrossRefGoogle Scholar
  15. [15]
    Dai J G, On the positive Harris recurrence for multiclass queueing networks: A unified approach via fluid limit models, Annals of Applied Probability, 1995, 5(1): 49–77.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.School of ScienceBeijing University of Posts and TelecommunicationsBeijingChina
  2. 2.Automatic SchoolBeijing University of Posts and TelecommunicationsBeijingChina

Personalised recommendations