Limits for cumulative input processes to queues

Probability in the Engineering and Informational Sciences

Volumn 14, Issue 2, 2000, Pages 123-150

  Whitt, Ward ^a  

^a AT AND T LABS RESEARCH   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

QUEUEING THEORY; STABILITY CRITERIA; STOCHASTIC SYSTEMS; TOPOLOGY;

BUFFER CONTENTS; CONTINUOUS FUNCTIONS; FUNCTION SPACES; FUNCTIONAL CENTRAL LIMIT THEOREM; HEAVY TRAFFICS; PERIOD DISTRIBUTIONS; REFLECTION MAPS; STABLE PROCESS;

PROBABILITY DISTRIBUTIONS;

EID: 0034552533     PISSN: 02699648     EISSN: None     Source Type: Journal    
DOI: 10.1017/s0269964800142019     Document Type: Article

References (44)

1. Barford, P. & Crovella, M. (1998). Generating representative web workloads for network and server performance evaluation. Proceedings of the 1998 ACM Sigmetrics, pp. 151-160.
2. Berger, A.W. & Whitt, W. (1992). The Brownian approximation for rate control throttles and the G/G/1/C queue. Journal of Discrete Event Dynamic Systems 2: 7-60.
3. Billingsley, P. (1968). Convergence of probability measures. New York: Wiley.
4. Boxma, O.J. & Cohen, J.W. (1999). Heavy-traffic analysis for the GI/G/1 queue with heavy-tailed distributions. Queueing Systems (to appear).
5. Boxma, O.J. & Cohen, J.W. (1999). The M/G/1 queue: heavy tails and heavy traffic. In K. Park & W. Willinger (eds.), Self-similar network traffic and performance evaluation. New York: Wiley.
6. Cohen, J.W. (1982). The single-server queue. Amsterdam: North-Holland.
7. Cohen, J.W. (1998). A heavy-traffic theorem for the GI/G/1 queue with a Pareto-type service time distribution. Journal of Applied Mathematics and Stochastic Analysis 11: 247-254; special issue dedicated to R. Syski.
8. Crovella, M.E. & Bestavros, A. (1996). Self-similarity in World Wide Web traffic: Evidence and possible causes. Proceedings of ACM Sigmetrics '96, pp. 160-169.
9. Dudley, R.M. (1968). Distances of probability measures and random variables. Annals of Mathematical Statistics 39: 1563-1572.
10. Dupius, A. & Guillemin, F. (1997). Etude des phenomenes transitores de congestion Lies an multi-plexage statistique en boucle ouverte sur un lien, ATM. France Telecom CNET Report.
11. Ethier, S.N. & Kurtz, T.G. (1986). Markov processes. Characterization and convergence. New York: Wiley.
12. Feller, W. (1971). An introduction to probability theory and its applications, Vol. II, 2nd ed. New York: Wiley.
13. Furrer, H., Michna, Z., & Weron, A. (1997). Stable Lévy motion approximation in collective risk theory. Insurance: Mathematics and Economics 20: 97-114.
14. Heyman, D.P. & O'Leary, D.P. (1995). What is fundamental for Markov chains: First passage times, fundamental matrices and group generalized inverses. In W.J. Stewart (ed.), Computations with Markov chains. Boston: Kluwer, pp. 151-161.
15. Jacod, J. & Shiryaev, A.N. (1987). Limit theorems for stochastic processes. New York: Springer-Verlag.
16. Kella, O. (1993). Parallel and tandem fluid networks with dependent Lévy inputs. Annals of Applied Probability 3: 682-695.
17. Kella, O. & Whitt, W. (1992). A tandem fluid network with Lévy input. In I. Basawa & U. Bhat (eds.), Queues and related models. Oxford: Oxford University Press, pp. 112-128.
18. Kella, O. & Whitt, W. (1996). Stability and structural properties of stochastic storage networks. Journal of Applied Probability 33: 1169-1180.
19. Kemeny, J.G. & Snell, J.L. (1960). Finite Markov chains. Princeton, NJ: Van Nostrand.
20. Konstantopoulos, T. & Lin, S.-J. (1998). Macroscopic models for long-range dependent network traffic. Queueing Systems 28: 215-243.
21. Kurtz, T.G. (1996). Limit theorems for workload input models. In F.P. Kelly, S. Zachary, & I. Ziedins (eds.), Stochastic networks. Theory and applications. Oxford: Clarendon Press, pp. 119-139.
22. Leland, W.E., Taqqu, M., Willinger, W., & Wilson, D.V. (1994). On the self-similar nature of ethernet traffic. IEEE/ACM Transactions on Networking 2: 1-15.
23. Neuts, M.F. (1989). Structured stochastic matrices of M/G/1 type and their applications. New York: Marcel Dekker.
24. Puhalskii, A.A. & Whitt, W. (1997). Functional large deviation principles for first-passage-time processes. Annals of Applied Probability 7: 362-381.
25. Reiman, M.I. (1984). Open queueing networks in heavy traffic. Mathematics of Operations Research 9: 441-458.
26. Resnick, S. & Rootzén, H. (1998). Self-similar communication models and very heavy tails. Technical report, Cornell University.
27. Resnick, S. & Samorodnitsky, G. (1998). A heavy traffic limit theorem for workload processes with heavy tailed service requirements. Technical report, Cornell University.
28. Resnick, S. & van den Berg, E. (1999). Weak convergence of high-speed network traffic models. Technical report, Cornell University.
29. Samorodnitsky, G. & Taqqu, M. (1994). Stable non-Gaussian random processes. New York: Chapman & Hall.
30. Skorohod, A.V. (1956). Limit theorems for stochastic processes. Theory of Probability and Its Applications 1: 261-290.
31. Takács, L. (1967). Combinatorial methods in the theory of stochastic processes. New York: Wiley.
32. Takács, L. (1971). Letter to the Editor. Journal of Applied Probability 8: 848-849.
33. Takagi, H. (1993). Queueing analysis. Vol. 2: Finite systems. Amsterdam: North-Holland.
34. Taqqu, M.S., Willinger, W., & Sherman, R. (1997). Proof of a fundamental result in self-similar traffic modeling. Computer Communication Review 27: 5-22.
35. Tsoukatos, K.P. & Makowski, A.M. (1997). Heavy-traffic analysis of a multiplexer driven by M/GI/∞ input processes. In V. Ramaswami & P.E. Wirth (eds.), Teletraffic contributions for the information age. Proceedings of ITC 15. Amsterdam: Elsevier, pp. 497-506.
36. Tsoukatos, K.P. & Makowski, A.M. (1999). Interpolation approximations for M/GI/∞ arrival processes. Technical report, University of Maryland.
37. Tsoukatos, K.P. & Makowski, A.M. (2000). Heavy traffic limits associated with M/GI/∞ input processes. Queueing Systems (to appear).
38. Whitt, W. (1971). Weak convergence theorem for priority queues: Preemptive-resume discipline. Journal of Applied Probability 8: 74-94.
39. Whitt, W. (1980). Some useful functions for functional limit theorems. Mathematics of Operations Research 5: 67-85.
40. Whitt, W. (1998). Overview of non-Brownian FCLTs for the single-server queue. Technical report, AT&T Labs.
41. Whitt, W. (1999). On the Skorohod M topologies. Technical report, AT&T Labs.
42. Whitt, W. (1999). The reflection map with discontinuities. Technical report, AT&T Labs. To appear in Mathematics of Operations Research.
43. Willinger, W., Taqqu, M.S., Sherman, R., & Wilson, D.V. (1997). Self-similarity through high variability: Statistical analysis of ethernet LAN traffic at the source level. IEEE/ACM Transactions on Networking 5: 71-86.
44. Zolotarev, V.M. (1986). One-dimensional stable distributions, Vol. 65. Providence, RI: American Mathematical Society.