The M/G/1 queue with heavy-tailed service time distribution

IEEE Journal on Selected Areas in Communications

Volumn 16, Issue 5, 1998, Pages 749-763

  Boxma, Onno J ^a,b,c,d   Cohen, J W ^a,b,d,e,f,g  

^a CWI   (Netherlands)

^b UTRECHT UNIVERSITY   (Netherlands)

^c TILBURG UNIVERSITY   (Netherlands)

^d Dept Probability Networks A ^*

^e DELFT UNIVERSITY OF TECHNOLOGY   (Netherlands)

^f PHILIPS RESEARCH LABORATORIES   (Germany)

^g TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

Asymptotic series for tail probabilities; Heavy tailed service time distribution; Heavy traffic approximation; M G 1 model; Regular variation; Waiting time distribution

Indexed keywords

APPROXIMATION THEORY; MATHEMATICAL MODELS; PROBABILITY; QUEUEING THEORY;

HEAVY TAILED SERVICE TIME DISTRIBUTION;

TELECOMMUNICATION TRAFFIC;

EID: 0032099217     PISSN: 07338716     EISSN: None     Source Type: Journal    
DOI: 10.1109/49.700910     Document Type: Article

References (29)

1. J. Abate, G. L. Choudhury, and W. Whitt, "Waiting time tail probabilities in queues with long-tail service-time distributions," Queueing Syst., vol. 16, pp. 311-338, 1994.
2. J. Abate and W. Whitt, "The Fourier-series method for inverting transforms of probability distributions," Queueing Syst., vol. 10, pp. 5-88, 1992.
3. _, "Numerical inversion of Laplace transforms of probability distributions," ORSA J. Computing, vol. 7, pp. 36-43, 1995.
4. J. Beran, R. Sherman, M. S. Taqqu, and W. Willinger, "Long-range dependence in variable-bit-rate video," IEEE Trans. Commun., vol. 43, pp. 1566-1579, 1995.
5. N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation. Cambridge, UK: Cambridge University Press, 1987.
6. O. J. Boxma and J. W. Cohen, "Heavy-traffic analysis for the GI/G/1 queue with heavy-tailed distributions," Tech. Rep. PNA-R9710, CWI, Amsterdam, 1997.
7. _, "The single server queue: Heavy tails and heavy traffic," to appear in Self-Similar Network Traffic and Performance Evaluation, K. Park and W. Willinger, Eds. New York: Wiley, 1998.
8. O. J. Boxma, J. W. Cohen, and Q. Deng, "Heavy-traffic analysis of the M/G/1 queue with priority classes," Report Tilburg Univ., Faculty Economics, 1998.
9. O. J. Boxma and V. Dumas, "Fluid queues with long-tailed activity period distributions," Tech. Rep. PNA-R9705, CWI, Amsterdam, The Netherlands, 1997; invited survey, to appear in Computer Communications.
10. J. W. Cohen, "Some results on regular variation for distributions in queueing and fluctuation theory," J. Appl. Probab., vol. 10, pp. 343-353, 1973.
11. _, "Superimposed renewal processes and storage with gradual input," Stochastic Processes Appl., vol. 2, pp. 31-58, 1974.
12. _, "On the M/G/1 queue with heavy-tailed service time distributions," Tech. Rep. PNA-R9702, CWI, Amsterdam, The Netherlands, 1997.
13. _, "A heavy-traffic theorem for the GI/G/1 queue with a Pareto-type service time distribution," Tech. Rep. PNA-R9709, CWI, Amsterdam, 1997, to appear in J. Appl. Math. Stoch. Anal., special issue in honor of R. Syski.
14. _, "The M/G/1 fluid model with heavy-tailed message length distributions," Tech. Rep. PNA-R9714, CWI, Amsterdam, 1997.
15. _, "Heavy-traffic limit theorems for the heavy-tailed GI/G/1 queue," Tech. Rep. PNA-R9719, CWI, Amsterdam, The Netherlands, 1997.
16. G. Doetsch, Handbuch der Laplace-Transformation Vol. I, II, III. Basel: Birkhäuser Verlag, 1955.
17. A. Erdélyi et al., Higher Transcendental Functions, Vol. 1, 2, 3. New York: McGraw-Hill, 1953.
18. A. Erdélyi et al., Tables of Integral Transforms. New York: McGrawHill, 1954.
19. H. Furrer, "Risk processes perturbed by α-stable Lévy motion," Rep. Dept. Math., ETHZ, 1996.
20. H. Furrer, Z. Michna, and A. Weron, "Stable Levy motion approximation in collective risk theory," Rep. Dept. Math. Stat., Lund Univ., 1996, to appear in Insurance Math. Economics.
21. J. F. C. Kingman, "The heavy traffic approximation in the theory of queues," in Proc. Symp. Congestion Theory, W. L. Smith and W. E. Wilkinson, Eds., 1965, pp. 137-159.
22. W. E. Leland, M. S. Taqqu, W. Willinger, and D. V. Wilson, "On the self-similar nature of Ethernet traffic" (extended version), IEEE/ACM Trans. Networking, vol. 2, pp. 1-15, 1994.
23. N. W. McLachlan, Complex Variable Theory and Transform Calculus,rev. ed. Cambridge, UK: Cambridge University Press, 1955.
24. A. G. Pakes, "On the tails of waiting-time distributions," J. Appl. Probab., vol. 12, pp. 555-564, 1975.
25. V. Paxson and S. Floyd, "Wide area traffic: The failure of Poisson modeling," IEEE/ACM Trans. Networking, vol. 3, pp. 226-244, 1995.
26. J. W. Roberts, U. Mocci, and J. Virtamo, Eds., Broadband Network Teletraffic (Final Report of Action COST 242). Berlin, Germany: SpringerVerlag, 1996.
27. W. Whitt, "Heavy traffic limit theorems for queues: A survey," Mathematical Methods in Queueing Theory, A. B. Clarke, Ed. Berlin, Germany: Springer, 1974, pp. 307-350.
28. E. Willekens and J. Teugels, "Asymptotic expansions for waiting time probabilities in an M/G/1 queue with long-tailed service time distributions," Queueing Syst., vol. 10, pp. 295-312, 1992.
29. W. Willinger, M. S. Taqqu, W. E. Leland, and D. V. Wilson, "Self-similarity in high-speed packet traffic: Analysis and modeling of ethernet traffic measurements," Stat. Sci., vol. 10, pp. 67-85, 1995.