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is a nite positive function. When LH is replaced simply by a normalizing constant, Eq. (4.15) is slightly better than the Weibull approximation (4.9). However, it has turned out to be dif cult to obtain more accurate estimates with the Benes presentation. Finally, an approach based on the Girsanov formula for fractional Brownian motion [3, 10, 17] was presented by Norros [14]. The process M t c1 where 1 c1 H 2H 1 B 3 H; H 1 2 2 and B m; n denotes the beta function B m; n 1
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turns out to be a Gaussian martingale with zero mean and variance EMt2 c2 t 2 2H , 2 where c2 is the constant given in Eq. (4.12). Moreover, exp aMt a2 c2 t 2 2H is the 2 Radon Nikodym derivative between probability measures corresponding to fractional Brownian motions on time interval 0; t with drifts a and 0, respectively. Now, the storage occupancy distribution has the presentation   c2 2 2H 2 P V > x E exp MTx Tx ; 2 where Tx is the hitting time Tx inf ft ! 0: Zt xg. Good approximations can be obtained in the case H > 1 by replacing c2 by 1. This leads to 2 1 2H 2 2H P V > x % E exp xTx 1 Tx 2 2 4     2 2H =H 53 1 H =H x 1 x E exp Z1 * ; Z* 2 Z* 1 1 * where the last step follows from the fact that Tx x=Z1 1=H . This can be expected to be rather accurate, but we still have to use some approximation for the unknown distribution of Z1 It is interesting to note that the rough substitution Z1 % jZ1 j gives *. * in fact the same approximation as the simple Benes approximation mentioned after Eq. (4.15). Both the Benes and Girsanov approaches have the appealing feature that they are based on nontrivial exact formulas. However, attempts to use them to obtain inequalities, or alternative derivations of the exact asymptotics (4.14), have not succeeded so far.
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The fractional Brownian storage is a nice mathematical object that offers many interesting challenges for further work. For example, little seems to be known about the nite buffer case, priority queueing, or dependence between busy periods. On the other hand, new and powerful analytical techniques for working with FBM, based on Malliavian calculus, are already waiting for applications see Decreusefond and Ustunel [3]. The problems connected with the motivation of fractional Brownian storage by limit theorems were brie y discussed in Section 4.1. Describing the applicability of this model or, more generally, any Gaussian storage model in quantitative terms is a task that clearly requires further work and innovations.
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1. R. G. Addie, P. Mannersalo, and I. Norros. Performance formulae for queues with Gaussian input. In P. Key and D. Smith, eds., Teletraf c Engineering in a Competitive World. Proceedings of the International Teletraf c Congress-ITC 16.
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2. C. S. Chang, D. D. Yao, and T. Zajic. Large deviations, moderate deviations, and queues with long-range dependent input. Adv. Appl. Probab. 31(1):254 278, 1999. 3. L. Decreusefond and A. S. Ustunel. Stochastic analysis of the fractional Brownian motion. Potential Anal. 10(2):177 214, 1999. 4. J.-D. Deuschel and D. W. Stroock. Large Deviations. Academic Press, Boston, 1989. 5. N. G. Duf eld and N. O'Connell. Large deviations and over ow probabilities for the general single-server queue, with applications. Math. Proc. Cambridge Philos. Soc., 118(2):363 374, 1995. 6. J. Husler and V. Piterbarg. Extremes of a certain class of Gaussian processes. Stoch. Proc. Appl., 83(2): 257 271, 1999. 7. K. R. Krishnan, A. L. Neidhardt, and A. Erramilli. Scaling analysis of traf c management of self-similar processes. In V. Ramaswami and P. E. Wirth, eds., Teletraf c Contributions for the Information Age. Proceedings of the 15th International Teletraf c Congress ITC 15, Washington, DC, pp. 1087 1096. Elsevier, New York, 1997. 8. W. E. Leland, M. S. Taqqu, W. Willinger, and D. V. Wilson. On the self-similar nature of Ethernet traf c (extended version). IEEE=ACM Trans. Networking, 2(1):1 15, February 1994. 9. L. Massoulie and A. Simonian. Large buffer asymptotics for the queue with FBM input. J. Appl. Probab., 36(3), xxx xxx, 1999. 10. G. M. Molchan and Ju. I. Golosov. Gaussian stationary processes with asymptotic power spectrum. Sov. Math. Dokl., 10(1):134 137, 1969. 11. O. Narayan. Exact asymptotic queue length distribution for fractional Brownian traf c. Adv. Perf. Anal., 1:39 63, 1998. 12. I. Norros. A storage model with self-similar input. Queueing Syst., 16:387 396, 1994. 13. I. Norros. On the use of fractional Brownian motion in the theory of connectionless networks. IEEE J. Selected Areas Commun., 13(6), 953 962, August 1995. 14. I. Norros. Four approaches to the fractional Brownian storage. In J. Levy Vehel, E. Lutton, and C. Tricot, eds., Fractals in Engineering. Springer, New York, 1997. 15. I. Norros. Busy periods of fractional Brownian storage: a large deviations approach. Adv. Perf. Anal., 2(7):1 19, 1999. 16. I. Norros, A. Simonian, D. Veitch, and J. Virtamo. A Benes formula for a buffer with fractional Brownian input. In 9th ITC Specialists Seminar '95: Teletraf c Modelling and Measurement, Leidschendam, The Netherlands, 1995. 17. I. Norros, E. Valkeila, and J. Virtamo. An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5(4):571 587, 1999. 18. N. O'Connell and G. Procissi, On the build-up of large queues in a queueing model with fractional Brownian motion input. Technical Report HPL-BRIMS-98-18, Hewlett-Packard's Basic Research Institute in the Mathematical Sciences, 1998. 19. J. Roberts, U. Mocci, and J. Virtamo, eds. Broadband Network Teletraf c. Number 1155 in Lecture Notes in Computer Science. Springer, New York, 1996. 20. A. Shwartz and A. Weiss. Large Deviations for Performance Analysis. Chapman and Hall, London, 1995.
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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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