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Figure 11.7 displays the distribution of queue size. The results are obtained by sampling 20,000 windows with a time unit size of 10. If we select a logarithmic scale for the horizontal axis, the curve log pn versus log n is asymptotically linear and illustrates of a polynomial tail distribution.
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Fig. 11.7 tion.
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Simulated queue size distribution (bottom, logarithmic scale), for on=off aggrega-
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Remark. We can extend this result in a similar manner as for the parallel queue analysis, that is, making the singularity set a little more complicated. In principle, it is possible to obtain tail distributions with oscillating terms. We can also treat more general cases. The artifact condition l > 1 can also be eliminated. But the most promising work item left in the present study is the estimation of a suitable upper bound for the queue length tail distribution, since the present work is limited to lower bounds.
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A simple model is presented of arrivals based on the aggregation of independent memoryless on=off sources. It is shown that appropriate tuning of parameters produces long-term dependence and polynomial tail distributions of buffer occupancy. Such a model may give rise to a new source of interesting insight about longterm dependence in Web traf c. Future work can progress in two different directions. The rst direction is to nd less restrictive conditions on parameter tuning. The second direction is to nd an exact estimate of the asymptotic tail distribution instead of a lower bound.
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1. W. Willinger, M. Taqqu, W. Leland, and D. Wilson. Self-similarity in high-speed packet traf c: analysis and modeling of Ethernet traf c measurement. Statist. Sci., 10(1):67 85, 1995. 2. W. Feller. An Introduciton to Probability Theory and Its Applications. Wiley, New York, 1971. 3. F. Baccelli. Effect of cross traf c on the performance of TCP. Workshop on the Modeling of TCP, Ecole Normale Superieure, 1998. 4. N. Duf eld and N. O'Connell. Large deviations and over ow probabilities for the general single server queue, with applications. DIAS Technical Report No DIAS-STP-93-30, 1993. 5. J. Beran. Statistics for Long Memory Processes, Vol. 61 of Monographs on Statistics and Applied Probability. Chapman and Hall, New York, 1994. 6. P. Jacquet. Analytic information theory in service of queueing with aggregated exponential on=off arrivals. In Proceedings of the 35th Annual Allerton Conference on Communication, Control and Computing, pp. 242 251, 1997. 7. O. Boxma and V. Dumas. Fluid queues with long-tailed activity period distribution. In Computer Communications, invited survey in the special issue on stochastic analysis and optimization of communication systems, 1998. 8. B. Tsybakov and N. Georganaas. Over ow probability in ATM queue with self-similar input traf c. In Proceedings IEEE International Communications Conference (ICC '97), 1997.
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9. P. Flajolet, X. Gourdon, and P. Dumas. Mellin transform and asymptotics: harmonic sums. Theor. Comput. Sci., 144(1 2):3 58, 1995. 10. P. Flajolet, M. Regnier, and D. Sotteau. Algebraic methods for trie statistics. Ann. Discrete Math., 25:145 188, 1985. 11. P. Jacquet and W. Szpankowski. Entropy computations via analytic depoisonization, IEEE Trans. Inf. Theory, 1072 1081, 1999. 12. W. Szpankowski. Techniques of the average case analysis of algorithms. In M. Atallah, ed., Handbook on Algorithms and Theory of Computation. CRC Press, Boca Raton, FL, 1997. 13. M. Neuts. Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. John Hopkins University Press, Baltimore, 1981. 14. P. Jacquet and N. Vvedenskaya. OnOff sources in an interconnection network: performance analysis when packets are routed to the shortest queue of two randomly selected nodes. INRIA Research Report RR-3570, 1998. 15. P. Flajolet and A. Odlyzko. Singularity analysis of generating functions. SIAM J. Discuss. Math., 3(2):216 240, 1990.
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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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