Skip to main content
SpringerLink
Log in

State-space collapse in stationarity and its application to a multiclass single-server queue in heavy traffic

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

Recently Gamarnik and Zeevi (Ann. Appl. Probab. 16:56–90, 2006) and Budhiraja and Lee (Math. Oper. Res. 34:45–56, 2009) established that, under suitable conditions, a sequence of the stationary scaled queue lengths in a generalized Jackson queueing network converges to the stationary distribution of multidimensional reflected Brownian motion in the heavy-traffic regime. In this work we study the corresponding problem in multiclass queueing networks (MQNs).

In the first part of this work we consider the MQNs for which the fluid stability is valid, state-space collapse is exhibited under suitable initial conditions and a heavy traffic limit theorem holds. For such MQNs we establish that, under the assumption of the tightness of a sequence of stationary scaled workloads, the sequence converges to the stationary distribution of semimartingale reflecting Brownian motion in the heavy-traffic regime. The key to the proof is to show that state-space collapse occurs in the heavy-traffic regime in stationarity under the assumption of tightness.

In the second part, using the result obtained, it is shown that such a convergence of stationary workload holds for a multiclass single-server queue with feedback routing, where the tightness is proved by the Lyapunov function method developed in (Gamarnik and Zeevi in Ann. Appl. Probab. 16:56–90, 2006).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Asmussen, S.: Applied Probability and Queues. Wiley, New York (1987) (Second edition: Springer, 2003)

    Google Scholar 

  2. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  5. Bramson, M.: State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Syst. 28, 7–31 (1998)

    Article  Google Scholar 

  6. Bramson, M., Dai, J.G.: Heavy traffic limits for some queueing networks. Ann. Appl. Probab. 11, 49–90 (2001)

    Article  Google Scholar 

  7. Bremaud, P.: Point Processes and Queues: Martingale Dynamics. Springer, Berlin (1984)

    Google Scholar 

  8. Budhiraja, A., Lee, C.: Stationary distribution convergence for generalized Jackson networks in heavy traffic. Math. Oper. Res. 34, 45–56 (2009)

    Article  Google Scholar 

  9. Chen, H.: A sufficient condition for the positive recurrence of a semimartingale reflecting Brownian motion in an orthant. Ann. Appl. Probab. 6, 758–765 (1996)

    Article  Google Scholar 

  10. Chung, K.L.: A Course in Probability Theory. Academic Press, New York (1974)

    Google Scholar 

  11. Dai, J.G.: On positive Harris recurrence for multiclass queueing networks: a unified approach via fluid limit models. Ann. Appl. Probab. 5, 49–77 (1995)

    Article  Google Scholar 

  12. Dai, J.G., Meyn, S.P.: Stability and convergence of moments for multiclass queueing networks via fluid limit models. IEEE Trans. Autom. Control 40, 1889-1904 (1995)

    Article  Google Scholar 

  13. Dai, J.G., Wang, Y.: Nonexistence of Brownian models of certain multiclass queueing networks. Queueing Syst. 13, 41–46 (1993)

    Article  Google Scholar 

  14. Davis, M.H.A.: Piecewise deterministic Markov processes: a general class of nondiffusion stochastic models. J. R. Stat. Soc. Ser. B 46, 353–388 (1984)

    Google Scholar 

  15. Dupuis, P., Williams, R.J.: Lyapunov functions for semimartingale reflecting Brownian motions. Ann. Probab. 22, 680–702 (1994)

    Article  Google Scholar 

  16. Gamarnik, D., Zeevi, A.: Validity of heavy traffic steady-state approximations in generalized Jackson networks. Ann. Appl. Probab. 16, 56–90 (2006)

    Article  Google Scholar 

  17. Harrison, J.M.: Balanced fluid models of multiclass queueing networks: a heavy traffic conjecture. In: Stochastic Networks. IMA Volumes in Mathematics and its Applications, vol. 71, pp. 1–20. Springer, Berlin (1995)

    Google Scholar 

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

    Article  Google Scholar 

  19. Kaspi, H., Mandelbaum, A.: Regenerative closed queueing networks. Stoch. Stoch. Rep. 39, 239–258 (1992)

    Google Scholar 

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

    Article  Google Scholar 

  21. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 2nd edn. Springer, Berlin (1994)

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  24. Williams, R.J.: Some recent developments for queueing networks. In: Accardi, L., Heyde, C.C. (eds.) Probability Towards 2000. Lecture Notes in Statistics, vol. 128, pp. 340–356. Springer, Berlin (1998)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. QC com., 1-20-21 Kitano, Yasu-city, Shiga, 520-2361, Japan

    Toshiyuki Katsuda

Authors
  1. Toshiyuki Katsuda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Toshiyuki Katsuda.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Katsuda, T. State-space collapse in stationarity and its application to a multiclass single-server queue in heavy traffic. Queueing Syst 65, 237–273 (2010). https://doi.org/10.1007/s11134-010-9178-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11134-010-9178-x

Keywords

Search

Navigation