Single-Station Queues

  • Hong Chen
  • David D. Yao
Part of the Stochastic Modelling and Applied Probability book series (SMAP, volume 46)


The main subject of this chapter is limit theorems for the queue-length and the workload processes in the G/G/1 queue. The limit theorems include the functional strong law of large numbers (FSLLN), the functional law of the iterated logarithm (FLIL), the functional central limit theorem (FCLT) and the strong approximation. In addition, we also establish an exponential rate of convergence result for the fluid approximation. The limit of the FSLLN and the FLIL is a single station fluid model. Because of this, the FSLLN is often known as the fluid approximation. Similarly, the limit of the FCLT and the strong approximation takes the form of a one-dimensional reflected Brownian motion. Since this limit is a diffusion process, the FCLT has been conventionally known as diffusion approximation. Throughout this and the following chapters we shall follow the convention to use the terms “fluid approximation” and “diffusion approximation” interchangeably with their underlying limit theorems. (In contrast, in Chapter 10, where the proposed approximations are not necessarily supported by limit theorems, we shall use the term “Brownian approximation,” in keeping with the Brownian network models in the research literature as approximations for queueing networks.)


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  1. [1]
    Chen, H. (1996). Rate of convergence of the fluid approximation for generalized Jackson networks. Journal of Applied Probability, 33, 3, 804–814.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Chen, H. and A. Mandelbaum. (1991). Stochastic discrete flow networks: diffusion approximations and bottlenecks. Annals of Probability, 19, 4, 1463–1519.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Chen, H. and A. Mandelbaum. (1994). Hierarchical modelling of stochastic networks:, Part II: strong approximations. In Stochastic Modelling and Analysis of Manufacturing Systems, ed. Chen, H. and A. Mandelbaum, 107–131, Springer-Verlag.Google Scholar
  4. [4]
    Chen, H. and H. Zhang. (1996). Diffusion approximations for multi-class re-entrant lines under a first-buffer-first-served discipline. Queueing Systems, Theory and Applications, 23, 177–195.MATHGoogle Scholar
  5. [5]
    Csörgö, M., P. Deheuvels, and L. Horvath. (1987). An approximation of stopped sums with applications in queueing theory. Advances in Applied Probability. 19, 674–690.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Glynn, P.W. and W. Whitt. (1991a). A new view of the heavy-traffic limit theorem for infinite-server queues. Advances in Applied Probability, 23, 188–209.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Glynn, P.W. and W. Whitt. (1991b). Departures from many queues in series. Annals of Applied Probability, 1, 546–572.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Johnson, D.P. (1983). Diffusion approximations for optimal filtering of jump processes and for queueing networks, Ph.D. dissertation, University of Wisconsin.Google Scholar
  9. [9]
    Harrison, J.M. (1985). Brownian Motion and Stochastic Flow Systems,Wiley.Google Scholar
  10. [10]
    Iglehart, D.L. (1971). Multiple channel queues in heavy traffic, IV: Laws of the iterated logarithm. Zeitschrift Wahrscheinlichkeitstheorie, 17, 168–180.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Iglehart, D.L. and W. Whitt. (1970a). Multiple channel queues in heavy traffic, I. Advances in Applied Probability, 2, 150–177.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    Iglehart, D.L. and W. Whitt. (1970b). Multiple channel queues in heavy traffic, II. Advances in Applied Probability, 2, 355–364.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Kingman, J.F.C. (1965). The heavy traffic approximation in the theory of queues. In Proceedings of Symposium on Congestion Theory, W. Smith and W. Wilkinson (eds.), University of North Carolina Press, Chapel Hill, 137–159.Google Scholar
  14. [14]
    Whitt, W. (1971). Weak convergence theorems for priority queues: preemptive-resume discipline. Journal of Applied Probability, 8, 74–94.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Whitt, W. (1974). Heavy traffic theorems for queues: a survey. In Mathematical Methods in Queueing Theory, ed. A.B. Clarke, Springer-Verlag.Google Scholar
  16. [16]
    Whitt, W. (1980). Some useful functions for functional limit theorems. Mathematics of Operations Research, 5, 67–85.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Zhang, H., G. Hsu, and R. Wang. (1990). Strong approximations for multiple channel queues in heavy traffic. Journal of Applied Probability, 28, 658–670.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Hong Chen
    • 1
  • David D. Yao
    • 2
  1. 1.Faculty of Commerce and Business AdministrationUniversity of British ColumbiaVancouverCanada
  2. 2.Department of Operations Research and Industrial EngineeringColumbia UniversityNew YorkUSA

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