Skip to main content
SpringerLink
Log in

Stability and Heavy Traffic Limits for Queueing Networks

  • Chapter
Perplexing Problems in Probability

Part of the book series: Progress in Probability ((PRPR,volume 44))

Abstract

We discuss here two topics of recent interest in queueing theory. The first is the question of when strictly subcritical queueing networks are stable. Namely, given a network whose stations all serve customers more quickly than the long-term rate at which customers visit the system, when is the underlying Markov process positive recurrent? The other topic is the existence of heavy traffic limits for queueing networks. That is, when does a sequence of networks, under diffusive scaling, converge to a reflecting Brownian motion?

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baskett, F., Chandy, K.M., Muntz, R.R., and Palacios, F.G. (1975). Open, closed and mixed networks of queues with different classes of customers. J. ACM 22 248–260.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bertsimas, D., Gamarnik, D., and Tsitsiklis, J.N. (1996). Stability conditions for multiclass fluid queueing networks. IEEE Trans. Autom. Control 41 1618–1631. Correction (1997): 42 128.

    MathSciNet  Google Scholar 

  3. Bramson, M. (1994a). Instability of FIFO queueing networks. Ann. Appl. Probab. 4 414–431.

    Article  MathSciNet  MATH  Google Scholar 

  4. Bramson, M. (1994b). Instability of FIFO queueing networks with quick service times. Ann. Appl. Probab. 4 693–718.

    Article  MathSciNet  MATH  Google Scholar 

  5. Bramson, M. (1996). Convergence to equilibria for fluid models of FIFO queueing networks. Queueing Systems 22 5–45.

    Article  MathSciNet  MATH  Google Scholar 

  6. Bramson, M. (1997). Convergence to equilibria for fluid models of head-of-the-line proportional processor sharing queueing networks. Queueing Systems 23 1–26.

    Article  MathSciNet  Google Scholar 

  7. Bramson, M. (1998a). Stability of two families of queueing networks and a discussion of fluid limits. Queueing Systems 28 7–31.

    Article  MathSciNet  MATH  Google Scholar 

  8. Bramson, M. (1998b). State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Systems 30 89–148.

    Article  MathSciNet  MATH  Google Scholar 

  9. Bramson, M. (1999). A stable queueing network with unstable fluid model. Ann. Appl. Probab., to appear.

    Google Scholar 

  10. Bramson, M. and Dai, J. (1999). Heavy traffic limits for some queueing networks, in preparation.

    Google Scholar 

  11. Chen, H. and Zhang, H. (1996). Diffusion approximations for re-entrant lines with a first-buffer-first-served priority discipline. Queueing Systems 23 177–195.

    Article  MATH  Google Scholar 

  12. Dai, J. (1995). On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid models. Ann. AppL Probab. 5 49–77.

    Article  MathSciNet  MATH  Google Scholar 

  13. Dai, J. (1996). A fluid-limit criterion for instability of multiclass queueing networks. Ann. Appl. Probab. 6 751–757.

    Article  MathSciNet  MATH  Google Scholar 

  14. Dai, J. and Harrison, J.M. (1993). The QNET method for two-moment analysis of closed manufacturing systems. Ann. Appl. Probab. 3 968–1012.

    Article  MathSciNet  MATH  Google Scholar 

  15. Dai, J., Hasenbein, J., and Vande Vate, J. (1999). Stability of a three-station fluid network. Queueing Systems, to appear.

    Google Scholar 

  16. Dai, J. and Vande Vate, J. (1999). The stability of two-station multi-type fluid networks. Operations Research, to appear.

    Google Scholar 

  17. Dai, J. and Wang, Y. (1993). Nonexistence of Brownian models for certain multiclass queueing networks. Queueing Systems 13 41–46.

    Article  MathSciNet  MATH  Google Scholar 

  18. Dumas, V. (1997). A multiclass network with non-linear, non-convex, non-monotone stability conditions. Queueing Systems 25 1–43.

    Article  MathSciNet  MATH  Google Scholar 

  19. Harrison, J.M. (1978). The diffusion approximation for tandem queues in heavy traffic. Adv. Appl. Probab. 10 886–905.

    Article  MATH  Google Scholar 

  20. Harrison, J.M. and Nguyen, V. (1990). The QNET method for two-moment analysis of open queueing networks. Queueing Systems 6 1–32.

    Article  MathSciNet  MATH  Google Scholar 

  21. Harrison, J.M. and Nguyen, V. (1993). Brownian models of multiclass queueing networks: current status and open problems. Queueing Systems 13 5–40.

    Article  MathSciNet  MATH  Google Scholar 

  22. Harrison, J.M. and Reiman, M.I. (1981). Reflected Brownian motion on an orthant. Ann. Probab. 9 302–308.

    Article  MathSciNet  MATH  Google Scholar 

  23. Harrison, J.M. and Williams, R.J. (1992). Brownian models of feedforward queueing networks: quasireversibility and product form solutions. Ann. App. Probab. 2 263–293.

    Article  MathSciNet  MATH  Google Scholar 

  24. Humes, C. (1994). A regulator stabilization technique: Kumar-Seidman revisited. IEEE Trans. Automat. Control 39 191–196.

    Article  MathSciNet  MATH  Google Scholar 

  25. Iglehart, D.L. and Whitt, W. (1970a). Multiple channel queues in heavy traffic I. Adv. App 1. Probab. 2 150–177.

    Article  MathSciNet  MATH  Google Scholar 

  26. Iglehart, D.L. and Whitt, W. (1970b). Multiple channel queues in heavy traffic II. Adv. App!. Probab. 2 355–364.

    Article  MathSciNet  MATH  Google Scholar 

  27. Kelly, F.P. (1975). Networks of queues with customers of different types. J. Appl. Probab. 12 542–554.

    Article  MATH  Google Scholar 

  28. Kelly, F.P. (1979). Reversibility and Stochastic Networks. Wiley, New York.

    MATH  Google Scholar 

  29. Kumar, P.R. and Seidman, T.I. (1990). Dynamic instabilities and stabilization methods in distributed real-time scheduling of manufacturing systems. IEEE Trans. Automat. Control 35 289–298.

    Google Scholar 

  30. Lu, S.H. and Kumar, P.R. (1991). Distributed scheduling based on due dates and buffer priorities. IEEE Trans. Autom. Control 36 1406–1416.

    Article  Google Scholar 

  31. Meyn, S. (1995). Transience of multiclass queueing networks via fluid limit models. Ann. App!. Probab. 5 946–957.

    Article  MathSciNet  MATH  Google Scholar 

  32. Peterson, W.P. (1991). Diffusion approximations for networks of queues with multiple customer types. Math. Oper. Res. 9 90–118.

    Article  Google Scholar 

  33. Reiman, M.I. (1984a). Open queueing networks in heavy traffic. Math. Oper. Res. 9 441–458.

    Article  MathSciNet  MATH  Google Scholar 

  34. Reiman, M.I. (1984b). Some diffusion approximations with state space collapse. Proceedings International Seminar on Modeling and Performance Evaluation Methodology,Lecture Notes in Control and Informational Sciences, F. Baccelli and G. Fayolle (eds.), Springer-Verlag, New York, 209–240.

    Google Scholar 

  35. Reiman, M.I. and Williams, R.J. (1988). A boundary property of semimartingale reflecting Brownian motions. Probab. Theory Related Fields 77 87–97. (Correction (1989) 80 633.)

    MathSciNet  Google Scholar 

  36. Rybko, S. and Stolyar, A. (1992). Ergodicity of stochastic processes that describe the functioning of open queueing networks. Problems Inform. Trans. 28 3–26 (in Russian).

    MathSciNet  Google Scholar 

  37. Seidman, T.I. (1994). “First come, first served” can be unstable! IEEE Trans. Automat. Control 39 2166–2171.

    Article  MathSciNet  MATH  Google Scholar 

  38. Stolyar, A. (1994). On the stability of multiclass queueing networks. Proc. 2nd Conf. on Telecommunication Systems-Modeling and Analysis, Nashville 1020–1028.

    Google Scholar 

  39. Taylor, L.M. and Williams, R.J. (1993). Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant. Probab. Theory Related Fields 96 283–317.

    Article  MathSciNet  Google Scholar 

  40. Whitt, W. (1971). Weak convergence theorems for priority queues: preemptive-resume discipline. J. Appl. Probab. 8 74–94.

    Article  MathSciNet  MATH  Google Scholar 

  41. Williams, R.J. (1995). Semimartingale reflecting Brownian motions in the orthant. Stochastic Networks, IMA Volumes in Mathematics and its Applications, Vol. 71. Springer-Verlag, New York, 125–138.

    Google Scholar 

  42. Williams, R.J. (1996). On the approximation of queueing networks in heavy traffic. Stochastic Networks, Theory and Applications, Royal Statistical Society Lecture Note Series, F.P. Kelly, S. Zachary, I. Ziedlins (eds.), Clarendon Press, Oxford, 35–56.

    Google Scholar 

  43. Williams, R.J. (1998). Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse. Queueing Systems 30 27–88.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Maury Bramson

Authors
  1. Maury Bramson
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Maury Bramson

  2. Department of Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Rick Durrett

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Birkhäuser Boston

About this chapter

Cite this chapter

Bramson, M. (1999). Stability and Heavy Traffic Limits for Queueing Networks. In: Bramson, M., Durrett, R. (eds) Perplexing Problems in Probability. Progress in Probability, vol 44. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2168-5_14

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-2168-5_14

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4612-7442-1

  • Online ISBN: 978-1-4612-2168-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation