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where h p; z z 1 a* p 1 b* p z : p z p 1 a* p b* p z
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Note that @z h p; z jz 0 , which is the key to the calculation of these moments, is symmetric with respect to a p and b* p . With the help of expansions (5.30) for a* p and b* z p , the above expressions for ma p and mb p can be expanded * * for small positive p. The appplication of Tauberian Theorem 5.7.1 recalled in Section 5.7.3 then provides us nally with the asymptotics for large t of the conditional moments ma t and mb t in the form nt O t 2 q with q min r; s (we omit here some easy but lengthy calculations). It can be veri ed, in particular, that m t $ if r < s and m t $ bs b 1 n 2 s t G 3 s ar an 2 r t G 3 r
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if s < r, and m t is asymptotic to the sum of the two latter equivalents if r s. As a consequence, m t $ m0 t 2 q for large t, where constant m0 is easily shown to equal m0 3 q l=2s2 in view of the above results and de nition (5.34) of constant l. REFERENCES
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1. S. Asumussen. Applied Probability and Queues. Wiley, New York, 1987. 2. B. Bensaou, J. Guibert, J. W. Roberts, and A. Simonian. Performance of an ATM multiplexer queue in the uid approximation using the Benes approach. Ann. Oper. Res. 49:137 160, 1994. 3. P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968. 4. O. Boxma and V. Dumas. Fluid queues with long-tailed activity period distributions. Comput. Commun., 21:1509 1529, 1998. 5. F. Brichet, J. W. Roberts, A. Simonian, and D. Veitch. Heavy traf c analysis of a storage model with long-range dependent on=off sources. Queueing Syst., 23:197 215, 1996. 6. D. R. Cox. Long-range dependence. In H. A. David and H. T. David, eds., A Review, Statistics: An Appraisal, Proceedings 50th Anniversary Conference, Iowa State Statistical Laboratory. The Iowa State University Press, Ames, 1984. 7. G. Doetsch. Handbuch der Laplace-transformation (in German). Birkhauser Verlag, Basel, 1955. 8. V. Dumas and A. Simonian. Asymptotic bounds for the uid queue fed by sub-exponential on=off sources. To appear in Adv. Appl. Prob., 32(1), March 2000.
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REFERENCES
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9. W. Feller. An Introduction to Probability Theory and Its Applications, Volume II. Wiley, New York, 1971. 10. T. Kurtz. Limit theorem for workload input models. In F. Kelly, S. Zachary, and I. Ziedins, eds., Stochastic Networks: Theory and Applications. Oxford University Press, New York, 1996. 11. T. Lindvall. Lectures on the Coupling Method. Wiley, New York, 1992. 12. J. D. Murray. Asymptotic Analysis. Springer-Verlag, New York, 1984. 13. I. Norros. A storage model with self-similar input. Queueing Syst., 16:387 396, 1994. 14. G. Samorodnitsky and M. S. Taqqu. Stable Non Gaussian Random Processes. Chapman and Hall, London, 1994. 15. A. N. Shiryayiev. Probability. Springer-Verlag, New York, 1984. 16. A. Simonian and J. Virtamo. Transient and stationary distributions for uid queues and input processes with a density. SIAM J. Appl. Math., 51(6):1732 1739, December 1991.
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Self-Similar Network Traf c and Performance Evaluation, Edited by Kihong Park and Walter Willinger Copyright # 2000 by John Wiley & Sons, Inc. Print ISBN 0-471-31974-0 Electronic ISBN 0-471-20644-X
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THE SINGLE SERVER QUEUE: HEAVY TAILS AND HEAVY TRAFFIC
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Recently, there has been much interest in the behavior of queues with heavy-tailed service time distributions. This interest has been triggered by a large number of traf c measurements on modern communication network traf c (e.g., see Willinger et al. [38] for Ethernet LAN traf c, Paxson and Floyd [30] for WAN traf c, and Beran et al. [3] for VBR video traf c; see also various chapters in this volume). These measurements and their statistical analysis (e.g., see Leland et al. [26]) suggest that modern communication traf c often possesses the properties of selfsimilarity and long-range dependence. A natural possibility to introduce long-range dependence in an input traf c process is to take a uid queue and to assume that at least one of the input quantities I (on or off periods in the uid queue fed by on=off sources) has the following ``heavy-tail'' behavior: P I > t $ hn t n ;
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Self-Similar Network Traf c and Performance Evaluation, Edited by Kihong Park and Walter Willinger ISBN 0-471-31974-0 Copyright # 2000 by John Wiley & Sons, Inc.
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