Queueing Systems

, Volume 23, Issue 1–4, pp 177–195 | Cite as

Diffusion approximations for re-entrant lines with a first-buffer-first-served priority discipline

  • Hong Chen
  • Hanqin Zhang
Article

Abstract

The diffusion approximation is proved for a class of queueing networks, known as re-entrant lines, under a first-buffer-first-served (FBFS) service discipline. The diffusion limit for the workload process is a semi-martingale reflecting Brownian motion on a nonnegative orthant. This approximation has recently been used by Dai, Yeh and Zhou [21] in estimating the performance measures of the re-entrant lines with a FBFS discipline.

Keywords

Re-entrant lines diffusion approximation multiclass queueing network heavy traffic semi-martingale reflecting Brownian motion 

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References

  1. [1]
    A. Berman and R.J. Plemmons,Nonnegative Matrices in the Mathematical Sciences (Academic Press, New York, 1979).Google Scholar
  2. [2]
    P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968).Google Scholar
  3. [3]
    H. Chen, Fluid approximations and stability of multiclass queueing networks: work-conserving disciplines,Annals of Applied Probability 5 (1995) 637–655.Google Scholar
  4. [4]
    H. Chen and A. Mandelbaum, Stochastic discrete flow networks: diffusion approximations and bottlenecks,Annals of Probability 19 (1991) 1463–1519.Google Scholar
  5. [5]
    H. Chen and A. Mandelbaum, Hierarchical modelling of stochastic networks, Part II: Strong approximations, in:Stochastic Modeling and Analysis of Manufacturing Systems, ed. D.D. Yao (Springer-Verlag, 1994) pp. 107–131.Google Scholar
  6. [6]
    H. Chen and H. Zhang, Diffusion approximations for multiclass FIFO queueing networks,Annals of Applied Probability (submitted).Google Scholar
  7. [7]
    J. Dai and W.Y. Dai, Diffusion approximations for queueing networks with finite capacity, preprint (1995).Google Scholar
  8. [8]
    J. Dai and T.G. Kurtz, A multiclass station with Markovian feedback in heavy traffic,Mathematics of Operations Research 20 (1995) 721–742.Google Scholar
  9. [9]
    J. Dai and V. Nguyen, On the convergence of multiclass queueing networks in heavy traffic,Annals of Applied Probability 4 (1994) 26–42.Google Scholar
  10. [10]
    J. Dai and Y. Wang, Nonexistence of Brownian models of certain multiclass queueing networks,Queueing Systems 13 (1993) 41–46.Google Scholar
  11. [11]
    J. Dai and G. Weiss, Stability and instability of fluid models for certain re-entrant lines,Mathematics of Operations Research 21 (1996) 115–134.Google Scholar
  12. [12]
    J. Dai, D.H. Yeh and C. Zhou, The QNET method for re-entrant queueing networks with priority disciplines,Operations Research (to appear).Google Scholar
  13. [13]
    P.W. Glynn, Diffusion approximations, in:Handbooks in Operations Research and Management Science, II: Stochastic Models, eds. D.P. Heyman and M.J. Sobel (North-Holland, Amsterdam, 1990).Google Scholar
  14. [14]
    J.M. Harrison,Brownian Motion and Stochastic Flow Systems (Wiley, New York, 1985).Google Scholar
  15. [15]
    J.M. Harrison and V. Nguyen, Brownian models of multiclass queueing networks: current status and open problems,Queueing Systems 13 (1993) 5–40.Google Scholar
  16. [16]
    J.M. Harrison and M.I. Reiman, Reflected Brownian motion on an orthant,Annals of Probability 9 (1981) 302–308.Google Scholar
  17. [17]
    J.M. Harrison and R.J. Williams, A multiclass closed queueing network with unconventional heavy traffic behavior,Annals of Applied Probability 6 (1996) 1–47.Google Scholar
  18. [18]
    D.P. Johnson, Diffusion approximations for optimal filtering of jump processes and for queueing networks, PhD Thesis, University of Wisconsin (1983).Google Scholar
  19. [19]
    P.R. Kumar, Re-entrant lines,Queueing Systems 13 (1993) 87–110.Google Scholar
  20. [20]
    A. Mandelbaum and W.A. Massey, Strong approximations for time-dependent queues,Mathematics of Operations Research 20 (1995) 33–64.Google Scholar
  21. [21]
    G. Pats, State-dependent queueing networks: Approximations and applications, PhD Thesis, Technion, Israel (1995).Google Scholar
  22. [22]
    W.P. Peterson, A heavy traffic limit theorem for networks of queues with multiple customer types,Mathematics of Operations Research 16 (1991) 90–118.Google Scholar
  23. [23]
    A. Puhalskii, On the invariance principle for the first passage time,Mathematics of Operations Research 19 (1994) 946–954.Google Scholar
  24. [24]
    M.I. Reiman, Open queueing networks in heavy traffic,Mathematics of Operations Research 9 (1984) 441–458.Google Scholar
  25. [25]
    M.I. Reiman, A multiclass feedback queue in heavy traffic,Advances in Applied Probability 20 (1988) 179–207.Google Scholar
  26. [26]
    H.L. Royden,Real Analysis (Macmillan, New York, 1988).Google Scholar
  27. [27]
    A.V. Skorohod, Limit theorems for stochastic processes,Theory of Probability and Its Applications 1 (1956) 261–290.Google Scholar
  28. [28]
    L.M. Taylor and R.J. Williams, Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant,Probability Theory and Related Fields 96 (1993) 283–317.Google Scholar
  29. [29]
    W. Whitt, Some useful functions for functional limit theorems,Mathematics of Operations Research 5 (1980) 67–85.Google Scholar
  30. [30]
    K. Yamada, Diffusion approximations for open state-dependent queueing networks under heavy traffic situation, Technical Report, Institute of Information Science and Electronics, University of Tsukuba, Japan (1993)Google Scholar
  31. [31]
    P. Yang, Least controls for a class of constrained linear stochastic systems,Mathematics of Operations Research 18 (1993) 275–291.Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1996

Authors and Affiliations

  • Hong Chen
    • 1
  • Hanqin Zhang
    • 2
  1. 1.Faculty of Commerce and Business AdministrationUniversity of British ColumbiaVancouverCanada
  2. 2.Institute of Applied MathematicsAcademia SinicaBeijingPR China

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