Queueing Systems

, Volume 30, Issue 1–2, pp 27–88 | Cite as

Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse

  • R.J. Williams
Article

Abstract

Certain diffusion processes known as semimartingale reflecting Brownian motions (SRBMs) have been shown to approximate many single class and some multiclass open queueing networks under conditions of heavy traffic. While it is known that not all multiclass networks with feedback can be approximated in heavy traffic by SRBMs, one of the outstanding challenges in contemporary research on queueing networks is to identify broad categories of networks that can be so approximated and to prove a heavy traffic limit theorem justifying the approximation. In this paper, general sufficient conditions are given under which a heavy traffic limit theorem holds for open multiclass queueing networks with head-of-the-line (HL) service disciplines, which, in particular, require that service within each class is on a first-in-first-out (FIFO) basis. The two main conditions that need to be verified are that (a) the reflection matrix for the SRBM is well defined and completely- S, and (b) a form of state space collapse holds. A result of Dai and Harrison shows that condition (a) holds for FIFO networks of Kelly type and their proof is extended here to cover networks with the HLPPS (head-of-the-line proportional processor sharing) service discipline. In a companion work, Bramson shows that a multiplicative form of state space collapse holds for these two families of networks. These results, when combined with the main theorem of this paper, yield new heavy traffic limit theorems for FIFO networks of Kelly type and networks with the HLPPS service discipline.

multiclass queueing networks heavy traffic FIFO Kelly type head-of-the-line-proportional processor sharing semimartingale reflecting Brownian motions diffusions completely-S 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Berman and R.J. Plemmons, Non-negative Matrices in the Mathematical Sciences (SIAM, Philadelphia, PA, 1994).Google Scholar
  2. [2]
    A. Bernard and A. El Kharroubi, Régulation de processus dans le premier orthant de Rn, Stochastics and Stochastics Rep. 34 (1991) 149-167.Google Scholar
  3. [3]
    D. Bertsekas and R. Gallagher, Data Networks (Prentice-Hall, Englewood Cliffs, NJ, 1992).Google Scholar
  4. [4]
    P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968).Google Scholar
  5. [5]
    M. Bramson, Instability of FIFO queueing networks, Ann. Appl. Probab. 4 (1994) 414-431.Google Scholar
  6. [6]
    M. Bramson, Instability of FIFO queueing networks with quick service times, Ann. Appl. Probab. 4 (1994) 693-718.Google Scholar
  7. [7]
    M. Bramson, Two badly behaved queueing networks, in: Stochastic Networks, IMA Volumes in Mathematics and Its Applications 71, eds. F.P. Kelly and R.J. Williams (Springer, New York, 1995) pp. 105-116.Google Scholar
  8. [8]
    M. Bramson, Convergence to equilibria for fluid models of FIFO queueing networks, Queueing Systems 22 (1996) 5-45.CrossRefGoogle Scholar
  9. [9]
    M. Bramson, Convergence to equilibria for fluid models of head-of-the-line proportional processor sharing queueing networks, Queueing Systems 23 (1996) 1-26.CrossRefGoogle Scholar
  10. [10]
    M. Bramson, Stability of two families of queueing networks and a discussion of fluid limits, Queueing Systems 28 (1998) 7-31.CrossRefGoogle Scholar
  11. [11]
    M. Bramson, State space collapse with application to heavy traffic limits for multiclass queueing networks, Queueing Systems 30 (1998) 89-148.CrossRefGoogle Scholar
  12. [12]
    H. Carlsson and O. Nerman, An alternative proof of Lorden's renewal inequality, Adv. in Appl. Probab. 18 (1986) 1015-1016.CrossRefGoogle Scholar
  13. [13]
    H. Chen and A. Mandelbaum, Leontief systems, RBV's and RBM's, in: Applied Stochastic Analysis, eds. M.H.A. Davis and R.J. Elliott (Gordon and Breach, New York, 1991) pp. 1-43.Google Scholar
  14. [14]
    H. Chen and W. Whitt, Diffusion approximations for open queueing networks with service interruptions, Queueing Systems 13 (1993) 335-359.CrossRefGoogle Scholar
  15. [15]
    H. Chen and H. Zhang, Diffusion approximations for re-entrant lines with a first-buffer-first-served priority discipline, Queueing Systems 23 (1996) 177-195.CrossRefGoogle Scholar
  16. [16]
    H. Chen and H. Zhang, Diffusion approximations for some multiclass queueing networks with FIFO service disciplines, Preprint (1997).Google Scholar
  17. [17]
    J.G. Dai, On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid limit models, Ann. Appl. Probab. 5 (1995) 49-77.Google Scholar
  18. [18]
    J.G. Dai, Stability of open multiclass queueing networks via fluid models, in: Stochastic Networks, IMA Volumes in Mathematics and Its Applications 71, eds. F.P. Kelly and R.J. Williams (Springer, New York, 1995) pp. 71-90.Google Scholar
  19. [19]
    J.G. Dai and W. Dai, A heavy traffic limit theorem for a class of open queueing networks with finite buffers, submitted to Queueing Systems (1997).Google Scholar
  20. [20]
    J.G. Dai and J.M. Harrison, The QNET method for two-moment analysis of closed manufacturing systems, Ann. Appl. Probab. 3 (1993) 968-1012.Google Scholar
  21. [21]
    J.G. Dai and V. Nguyen, On the convergence of multiclass queueing networks in heavy traffic, Ann. Appl. Probab. 4 (1994) 26-42.Google Scholar
  22. [22]
    J.G. Dai and Y. Wang, Nonexistence of Brownian models of certain multiclass queueing networks, Queueing Systems 13 (1993) 41-46.CrossRefGoogle Scholar
  23. [23]
    J.G. Dai, G. Wang and Y. Wang, Private communication (1992).Google Scholar
  24. [24]
    J.G. Dai and R.J. Williams, Existence and uniqueness of semimartingale reflecting Brownian motions in convex polyhedrons, Theory Probab. Appl. 40 (1995) 1-40.CrossRefGoogle Scholar
  25. [25]
    J.G. Dai, D.H. Yeh and C. Zhou, The QNET method for re-entrant queueing networks with priority disciplines, Oper. Res. 45 (1997) 610-623.Google Scholar
  26. [26]
    P. Dupuis and H. Ishii, On the Lipschitz continuity of the solution mapping to the Skorokhod problem, Stochastics and Stochastics Rep. 35 (1991) 31-62.Google Scholar
  27. [27]
    S.N. Ethier and T.G. Kurtz, Markov Processes: Characterization and Convergence (Wiley, New York, 1986).Google Scholar
  28. [28]
    J.M. Harrison, Brownian models of queueing networks with heterogeneous customer populations, in: Stochastic Differential Systems, Stochastic Control Theory and Applications, IMA Volumes in Mathematics and Its Applications, eds. W. Fleming and P.-L. Lions (Springer, New York, 1988) pp. 147-186.Google Scholar
  29. [29]
    J.M. Harrison, Balanced fluid models of multiclass queueing networks: A heavy traffic conjecture, in: Stochastic Networks, IMA Volumes in Mathematics and Its Applications 71, eds. F.P. Kelly and R.J. Williams (Springer, New York, 1995) pp. 1-20.Google Scholar
  30. [30]
    J.M. Harrison and V. Nguyen, Brownian models of multiclass queueing networks: Current status and open problems, Queueing Systems 13 (1993) 5-40.CrossRefGoogle Scholar
  31. [31]
    J.M. Harrison and M.I. Reiman, Reflected Brownian motion on an orthant, Ann. Probab. 9 (1981) 302-308.Google Scholar
  32. [32]
    J.M. Harrison and R.J. Williams, Brownian models of feedforward queueing networks: Quasireversibility and product form solutions, Ann. Appl. Probab. 2 (1992) 263-293.Google Scholar
  33. [33]
    J.W. Harrison and R.J. Williams, A multiclass closed queueing network with unconventional heavy traffic behavior, Ann. Appl. Probab. 6 (1996) 1-47.CrossRefGoogle Scholar
  34. [34]
    D.L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic I, Adv. in Appl. Probab. 2 (1970) 150-177.CrossRefGoogle Scholar
  35. [35]
    D.L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic II, Adv. in Appl. Probab. 2 (1970) 355-364.CrossRefGoogle Scholar
  36. [36]
    D.L. Iglehart and W. Whitt, The equivalence of functional central limit theorems for counting processes and associated partial sums, Ann. Math. Statist. 42 (1971) 1372-1378.Google Scholar
  37. [37]
    F.P. Kelly and R.J. Williams, eds., Stochastic Networks, IMA Volumes in Mathematics and Its Applications 71 (Springer, New York, 1995).Google Scholar
  38. [38]
    P.R. Kumar, Scheduling queueing networks: stability, performance analysis and design, in: Stochastic Networks, IMA Volumes in Mathematics and Its Applications 71, eds. F.P. Kelly and R.J. Williams (Springer, New York, 1995) pp. 21-70.Google Scholar
  39. [39]
    T. Lindvall, Lectures on the Coupling Method (Wiley, New York, 1992).Google Scholar
  40. [40]
    A. Mandelbaum, The dynamic complementarity problem, Preprint (1992).Google Scholar
  41. [41]
    A. Mandelbaum and L. Van der Heyden, Complementarity and reflection (1987, unpublished work).Google Scholar
  42. [42]
    W.P. Peterson, Diffusion approximations for networks of queues with multiple customer types, Math. Oper. Res. 9 (1991) 90-118.Google Scholar
  43. [43]
    Y.V. Prokhorov, Convergence of random processes and limit theorems in probability theory, Theory Probab. Appl. 1 (1956) 157-214.CrossRefGoogle Scholar
  44. [44]
    M.I. Reiman, Open queueing networks in heavy traffic, Math. Oper. Res. 9 (1984) 441-458.CrossRefGoogle Scholar
  45. [45]
    M.I. Reiman, Some diffusion approximations with state space collapse, in: Proc. of the Internat. Seminar on Modeling and Performance Evaluation Methodology, Lecture Notes in Control and Information Sciences, eds. F. Baccelli and G. Fayolle (Springer, New York, 1984) pp. 209-240.Google Scholar
  46. [46]
    M.I. Reiman, A multiclass feedback queue in heavy traffic, Adv. in Appl. Probab. 20 (1988) pp. 179-207.CrossRefGoogle Scholar
  47. [47]
    M.I. Reiman and R.J. Williams, A boundary property of semimartingale reflecting Brownian motions, Probab. Theory Related Fields 77 (1988) 87-97, and 80 (1989) 633.CrossRefGoogle Scholar
  48. [48]
    A.V. Skorokhod, Limit theorems for stochastic processes, Theory Probab. Appl. 1 (1956) 261-290.CrossRefGoogle Scholar
  49. [49]
    L.M. Taylor and R.J. Williams, Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant, Probab. Theory Related Fields 96 (1993) 283-317.CrossRefGoogle Scholar
  50. [50]
    W. Whitt, Weak convergence theorems for priority queues: Preemptive resume discipline, J. Appl. Probab. 8 (1971) 74-94.CrossRefGoogle Scholar
  51. [51]
    W. Whitt, Large fluctuations in a deterministic multiclass network of queues, Managm. Sci. 39 (1993) 1020-1028.Google Scholar
  52. [52]
    R.J. Williams, On the approximation of queueing networks in heavy traffic, in: Stochastic Networks: Theory and Applications, eds. F.P. Kelly, S. Zachary and I. Ziedins (Oxford Univ. Press, Oxford, 1996) pp. 35-56.Google Scholar
  53. [53]
    R.J. Williams, An invariance principle for semimartingale reflecting Brownian motions in an orthant, Queueing Systems 30 (1998) 5-25.CrossRefGoogle Scholar
  54. [54]
    D.D. Yao, ed., Stochastic Modeling and Analysis of Manufacturing Systems (Springer, New York, 1994).Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • R.J. Williams
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaSan Diego, La JollaUSA

Personalised recommendations