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

Queueing Systems

Volumn 30, Issue 1-2, 1998, Pages 27-88

  Williams, R J ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Completely S; Diffusions; FIFO Kelly type; Head of the line proportional processor sharing; Heavy traffic; Multiclass queueing networks; Semimartingale reflecting Brownian motions

Indexed keywords

BROWNIAN MOVEMENT; DIFFUSION; QUEUEING THEORY;

COMPLETELY-S; FIFO KELLY TYPE; HEAVY TRAFFICS; MULTI-CLASS QUEUEING NETWORKS; PROCESSOR SHARING; REFLECTING BROWNIAN MOTION;

QUEUEING NETWORKS;

EID: 0032262848     PISSN: 02570130     EISSN: 15729443     Source Type: Journal    
DOI: 10.1023/a:1019108819713     Document Type: Article

References (55)

1. A. Berman and R.J. Plemmons, Non-negative Matrices in the Mathematical Sciences (SIAM, Philadelphia, PA, 1994).
2. n, Stochastics and Stochastics Rep. 34 (1991) 149-167.
3. D. Bertsekas and R. Gallagher, Data Networks (Prentice-Hall, Englewood Cliffs, NJ, 1992).
4. P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968).
5. M. Bramson, Instability of FIFO queueing networks, Ann. Appl. Probab. 4 (1994) 414-431.
6. M. Bramson, Instability of FIFO queueing networks with quick service times, Ann. Appl. Probab. 4 (1994) 693-718.
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.
8. M. Bramson, Convergence to equilibria for fluid models of FIFO queueing networks, Queueing Systems 22 (1996) 5-45.
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.
10. M. Bramson, Stability of two families of queueing networks and a discussion of fluid limits, Queueing Systems 28 (1998) 7-31.
11. M. Bramson, State space collapse with application to heavy traffic limits for multiclass queueing networks, Queueing Systems 30 (1998) 89-148.
12. H. Carlsson and O. Nerman, An alternative proof of Lorden's renewal inequality, Adv. in Appl. Probab. 18 (1986) 1015-1016.
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.
14. H. Chen and W. Whitt, Diffusion approximations for open queueing networks with service interruptions, Queueing Systems 13 (1993) 335-359.
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.
16. H. Chen and H. Zhang, Diffusion approximations for some multiclass queueing networks with FIFO service disciplines, Preprint (1997).
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.
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.
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).
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.
21. J.G. Dai and V. Nguyen, On the convergence of multiclass queueing networks in heavy traffic, Ann. Appl. Probab. 4 (1994) 26-42.
22. J.G. Dai and Y. Wang, Nonexistence of Brownian models of certain multiclass queueing networks, Queueing Systems 13 (1993) 41-46.
23. J.G. Dai, G. Wang and Y. Wang, Private communication (1992).
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.
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.
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.
27. S.N. Ethier and T.G. Kurtz, Markov Processes: Characterization and Convergence (Wiley, New York, 1986).
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.
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.
30. J.M. Harrison and V. Nguyen, Brownian models of multiclass queueing networks: Current status and open problems, Queueing Systems 13 (1993) 5-40.
31. J.M. Harrison and M.I. Reiman, Reflected Brownian motion on an orthant, Ann. Probab. 9 (1981) 302-308.
32. J.M. Harrison and R.J. Williams, Brownian models of feedforward queueing networks: Quasire-versibility and product form solutions, Ann. Appl. Probab. 2 (1992) 263-293.
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.
34. D.L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic I, Adv. in Appl. Probab. 2 (1970) 150-177.
35. D.L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic II, Adv. in Appl. Probab. 2 (1970) 355-364.
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.
37. F.P. Kelly and R.J. Williams, eds., Stochastic Networks, IMA Volumes in Mathematics and Its Applications 71 (Springer, New York, 1995).
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.
39. T. Lindvall, Lectures on the Coupling Method (Wiley, New York, 1992).
40. A. Mandelbaum, The dynamic complementarity problem, Preprint (1992).
41. A. Mandelbaum and L. Van der Heyden, Complementarity and reflection (1987, unpublished work).
42. W.P. Peterson, Diffusion approximations for networks of queues with multiple customer types, Math. Oper. Res. 9 (1991) 90-118.
43. Y.V. Prokhorov, Convergence of random processes and limit theorems in probability theory, Theory Probab. Appl. 1 (1956) 157-214.
44. M.I. Reiman, Open queueing networks in heavy traffic, Math. Oper. Res. 9 (1984) 441-458.
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.
46. M.I. Reiman, A multiclass feedback queue in heavy traffic, Adv. in Appl. Probab. 20 (1988) pp. 179-207.
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.
48. 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.
49. A.V. Skorokhod, Limit theorems for stochastic processes, Theory Probab. Appl. 1 (1956) 261-290.
50. 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.
51. W. Whitt, Weak convergence theorems for priority queues: Preemptive resume discipline, J. Appl. Probab. 8 (1971) 74-94.
52. W. Whitt, Large fluctuations in a deterministic multiclass network of queues, Managm. Sci. 39 (1993) 1020-1028.
53. 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.
54. R.J. Williams, An invariance principle for semimartingale reflecting Brownian motions in an orthant, Queueing Systems 30 (1998) 5-25.
55. D.D. Yao, ed., Stochastic Modeling and Analysis of Manufacturing Systems (Springer, New York, 1994).