Queueing Systems

, Volume 65, Issue 3, pp 237–273 | Cite as

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

  • Toshiyuki Katsuda


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).


Brownian approximation Heavy traffic Multiclass queueing network Stationary distribution Semimartingale reflecting Brownian motion State space collapse Multiclass single-server queue Lyapunov function 
60K25 60F17 90B22 60J25 93E15 


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  1. 1.
    Asmussen, S.: Applied Probability and Queues. Wiley, New York (1987) (Second edition: Springer, 2003) Google Scholar
  2. 2.
    Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968) Google Scholar
  3. 3.
    Bramson, M.: Convergence to equilibria for fluid models of FIFO queueing networks. Queueing Syst. 22, 5–45 (1996) CrossRefGoogle Scholar
  4. 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) CrossRefGoogle Scholar
  5. 5.
    Bramson, M.: State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Syst. 28, 7–31 (1998) CrossRefGoogle Scholar
  6. 6.
    Bramson, M., Dai, J.G.: Heavy traffic limits for some queueing networks. Ann. Appl. Probab. 11, 49–90 (2001) CrossRefGoogle Scholar
  7. 7.
    Bremaud, P.: Point Processes and Queues: Martingale Dynamics. Springer, Berlin (1984) Google Scholar
  8. 8.
    Budhiraja, A., Lee, C.: Stationary distribution convergence for generalized Jackson networks in heavy traffic. Math. Oper. Res. 34, 45–56 (2009) CrossRefGoogle Scholar
  9. 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) CrossRefGoogle Scholar
  10. 10.
    Chung, K.L.: A Course in Probability Theory. Academic Press, New York (1974) Google Scholar
  11. 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) CrossRefGoogle Scholar
  12. 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) CrossRefGoogle Scholar
  13. 13.
    Dai, J.G., Wang, Y.: Nonexistence of Brownian models of certain multiclass queueing networks. Queueing Syst. 13, 41–46 (1993) CrossRefGoogle Scholar
  14. 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. 15.
    Dupuis, P., Williams, R.J.: Lyapunov functions for semimartingale reflecting Brownian motions. Ann. Probab. 22, 680–702 (1994) CrossRefGoogle Scholar
  16. 16.
    Gamarnik, D., Zeevi, A.: Validity of heavy traffic steady-state approximations in generalized Jackson networks. Ann. Appl. Probab. 16, 56–90 (2006) CrossRefGoogle Scholar
  17. 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. 18.
    Harrison, J.M., Nguyen, V.: Brownian models of multiclass queueing networks: current status and open problems. Queueing Syst. 13, 5–40 (1993) CrossRefGoogle Scholar
  19. 19.
    Kaspi, H., Mandelbaum, A.: Regenerative closed queueing networks. Stoch. Stoch. Rep. 39, 239–258 (1992) Google Scholar
  20. 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) CrossRefGoogle Scholar
  21. 21.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 2nd edn. Springer, Berlin (1994) Google Scholar
  22. 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) CrossRefGoogle Scholar
  23. 23.
    Williams, R.J.: Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse. Queueing Syst. 30, 27–88 (1998) CrossRefGoogle Scholar
  24. 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

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.QC com.ShigaJapan

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