Heavy traffic limits for some queueing networks

Annals of Applied Probability

Volumn 11, Issue 1, 2001, Pages 49-90

  Bramson, Maury ^a   Dai, J G ^b  

^a UNIVERSITY OF MINNESOTA   (United States)

^b GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Brownian model; Diffusion approximation; Heavy traffic; Multiclass queueing network; Reflecting Brownian motion

Indexed keywords

EID: 0035626682     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/aoap/998926987     Document Type: Article

References (56)

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. Assoc. Comput. Mach. 22 248-260.
2. BERTSEKAS, D. and GALLAGER, R. (1992). Data Networks. Prentice-Hall, Englewood Cliffs, NJ.
3. BOROVKOV, A. (1964). Some limit theorems in the theory of mass service, I. Theory Probab. Appl. 9 550-565.
4. BOROVKOV, A. (1965). Some limit theorems in the theory of mass service, II. Theory Probab. Appl. 10 375-400.
5. BRAMSON, M. (1997). Convergence to equilibria for fluid models of head-of-the-line proportional processor sharing queueing networks. Queueing Systems Theory Appl. 23 1-26.
6. BRAMSON, M. (1998). Stability of two families of queueing networks and a discussion of fluid limits. Queueing Systems Theory Appl. 28 7-31.
7. BRAMSON, M. (1998). State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Systems Theory Appl. 30 89-148.
8. CHEN, H. and MANDELBAUM, A. (1994). Hierarchical modeling of stochastic networks II: Strong approximations. In Stochastic Modeling and Analysis of Manufacturing Systems (D. D. Yao, ed.) 107-132. Springer, New York.
9. CHEN, H. and YE, H. (1999). Existence condition for the diffusion approximations of multiclass priority queueing networks. Technical Report, Faculty of Commerce and Business Administration, Univ. British Columbia.
10. CHEN, H. and ZHANG, H. (2000). Diffusion approximations for multiclass FIFO queueing networks. Math. Oper. Res. 25 679-707.
11. CHEN, H. and ZHANG, H. (1997). Diffusion approximations for re-entrant lines with a first-buffer-first-served priority discipline. Queueing Systems Theory Appl. 23 177-195.
12. CHEN, H. and ZHANG, H. (1998). Diffusion approximations for Kumar-Seidman network under a priority service discipline. Oper. Res. Lett. 23 71-81.
13. CHEN, H. and ZHANG, H. (2000). A sufficient condition and a necessary condition for the diffusion approximations of multiclass queueing networks under priority service disciplines. Queueing Systems Theory Appl. To appear.
14. COFFMAN, E. G., JR., PUHALSKII, A. A. and REIMAN, M. I. (1995). Polling systems with zero switchover times: A heavy traffic averaging principle. Ann. Appl. Probab. 5 681-719.
15. DAI, J. G. (1996). A fluid-limit model criterion for instability of multiclass queueing networks. Ann. Appl. Probab. 6 751-757.
16. DAI, J. G. and HARRISON, J. M. (1992). Reflected Brownian motion in an orthant: Numerical methods for steady-state analysis. Ann. Appl. Probab. 2 65-86.
17. DAI, J. G., HASENBEIN, J. and VANDE VATE, J. H. (1999). Stability of a three-station fluid network. Queueing Systems Theory Appl. 33 293-325.
18. DAI, J. G. and KURTZ, T. G. (1995). A multiclass station with Markovian feedback in heavy traffic. Math. Oper. Res. 20 721-742.
19. DAI, J. G. and NGUYEN, V. (1994). On the convergence of multiclass queueing networks in heavy traffic. Ann. Appl. Probab. 4 26-42.
20. DAI, J. G. and VANDEVATE, J. (1996). Global stability of two-station queueing networks. Proceedings of Workshop on Stochastic Networks: Stability and Rare Events. Lecture Notes in Statistics 117 1-26. Springer, New York.
21. DAI, J. G. and VANDEVATE, J. (2000). The stability of two-station multi-type fluid networks. Oper. Res. 48 721-744.
22. DAI, J. G. and WANG, Y. (1993). Nonexistence of Brownian models of certain multiclass queueing networks. Queueing Systems Theory Appl. 13 41-46.
23. DAI, J. G. and WEISS, G. (1996). Stability and instability of fluid models for re-entrant lines. Math. Oper. Res. 21 115-134.
24. DAI, J. G., YEH, D. H. and ZHOU, C. (1997). The QNET method for re-entrant queueing networks with priority disciplines. Oper. Res. 45 610-623.
25. DUMAS, V. (1997). A multiclass network with non-linear, non-convex, non-monotonic stability conditions. Queueing Systems Theory Appl. 25 1-43.
26. ETHIER, S. N. and KURTZ, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
27. FOSCHINI, G. J. and SALZ, J. (1978). A basic dynamic routing problem and diffusion. IEEE Trans. Comm. 26 320-327.
28. HARRISON, J. M. and NGUYEN, V. (1993). Brownian models of multiclass queueing networks: Current status and open problems. Queueing Systems Theory Appl. 13 5-40.
29. HARRISON, J. M. and NGUYEN, V. (1995). Some badly behaved closed queueing networks. In Stochastic Networks (F. P. Kelly and R. J. Williams, eds.). 71 117-124. Springer, New York.
30. HARRISON, J. M. and REIMAN, M. I. (1981). Reflected Brownian motion on an orthant. Ann. Probab. 9 302-308.
31. HARRISON, J. M. and WILLIAMS, R. J. (1987). Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Probab. 15 115-137.
32. HARRISON, J. M. and WILLIAMS, R. J. (1996). A multiclass closed queueing network with unconventional heavy traffic behavior. Ann. Appl. Probab. 6 1-47.
33. IGLEHART, D. L. (1965). Limit theorems for queues with traffic intensity one. Ann. Math. Statist. 36 1437-1449.
34. IGLEHART, D. L. and WHITT, W. (1970). Multiple channel queues in heavy traffic I. Adv. in Appl. Probab. 2 150-177.
35. IGLEHART, D. L. and WHITT, W. (1970). Multiple channel queues in heavy traffic II: Sequences, networks, and batches. Adv. in Appl. Probab. 2 355-369.
36. JACKSON, J. R. (1957). Networks of waiting lines. Oper. Res. 5 518-521.
37. JOHNSON, D. P. (1983). Diffusion approximations for optimal filtering of jump processes and for queueing networks. Ph.D. dissertation, Univ. Wisconsin.
38. KELLY, F. P. (1975). Networks of queues with customers of different types. J. Appl. Probab. 12 542-554.
39. KINGMAN, J. F. C. (1961). On queues in heavy traffic. Proc. Cambridge Philos. Soc. 57 902-904.
40. KINGMAN, J. F. C. (1965). The heavy traffic approximation in the theory of queues. In Proceedings of the Symposium on Congestion Theory (W. L. Smith et al., eds.) 137-159. Univ. North Carolina Press, Chapel Hill.
41. LU, S. H. and KUMAR, P. R. (1991). Distributed scheduling based on due dates and buffer priorities. IEEE Trans. Automat. Control 36 1406-1416.
42. PETERSON, W. P. (1991). A heavy traffic limit theorem for networks of queues with multiple customer types. Math. Oper. Res. 16 90-118.
43. PROHOROV, Y. (1963). Transient phenomena in processes of mass service. Litovsk. Mat. Sb. 3 199-205 [in Russian].
44. REIMAN, M. I. (1984). Open queueing networks in heavy traffic. Math. Oper. Res. 9 441-458.
45. REIMAN, M. I. (1984). Some diffusion approximations with state space collapse. In Modeling and Performance Evaluation Methodology (F. Baccelli and G. Fayolle, eds.) 209-240. Springer, Berlin.
46. REIMAN, M. I. (1988). A multiclass feedback queue in heavy traffic. Adv. in Appl. Probab. 20 179-207.
47. REIMAN, M. I. and WILLIAMS, R. J. (1989). A boundary property of semimartingale reflecting Brownian motions. Probab. Theory Related Fields 77 87-97. [Correction (1989) 80 633.]
48. STOLYAR, A. L. (1995). On the stability of multiclass queueing networks: A relaxed sufficient condition via limiting fluid processes. Markov Process. Related Fields 1 491-512.
49. 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.
50. WHITT, W. (1971). Weak convergence theorems for priority queues: Preemptive-resume discipline. J. Appl. Probab. 8 74-94.
51. WHITT, W. (1974). Heavy traffic theorems for queues: A survey. In Mathematical Methods in Queueing Theory (A. B. Clarke, ed.) 307-350. Springer, New York.
52. WHITT, W. (1993). Large fluctuations in a deterministic multiclass network of queues. Management Sciences 39 1020-1028.
53. WILLIAMS, R. J. (1996). On the approximation of queueing networks in heavy traffic. In Stochastic Networks: Theory and Applications (F. P. Kelly, S. Zachary and I. Ziedins, eds.) Oxford Univ. Press.
54. WILLIAMS, R. J. (1998). Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse. Queueing Systems Theory Appl. 30 27-88.
55. WILLIAMS, R. J. (1998). An invariance principle for semimartingale reflecting Brownian motions in an orthant. Queueing Systems Theory Appl. 30 5-25.
56. YAO, D. D. (1994). Stochastic Modeling and Analysis of Manufacturing Systems. Springer, New York.