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N. LIKHANOV
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Institute for Problems of Information Transmission, Russian Academy of Science, Moscow, Russia
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INTRODUCTION
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High-quality traf c measurements indicate that actual traf c behavior over highspeed networks shows self-similar features. These include an analysis of hundreds of millions of observed packets on several Ethernet LANs [7, 8], and an analysis of a few million observed frame data by variable bit rate (VBR) video services [1]. In these studies, packet traf c appears to be statistically self-similar [2, 11]. Self-similar traf c is characterized by ``burstiness'' across an extremely wide range of time scales [7]. This behavior of aggregate Ethernet traf c is very different from conventional traf c models (e.g., Poisson, batch Poisson, Markov modulated Poisson process [4]). A lot of studies have been made for the design, control, and performance of highspeed and cell-relay networks, using traditional traf c models. It is likely that many of those results need major revision when self-similar traf c models are considered [18]. Self-similarity manifests itself in a variety of different ways: a spectral density that diverges at the origin, a nonsummable autocorrelation function (indicating longrange dependence), an index of dispersion of counts (IDCs) that increases monotonically with the sample time T , and so on [7]. A key parameter characterizing selfsimilar processes is the so-called Hurst parameter, H, which is designed to capture the degree of self-similarity.
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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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Self-similar process models can be derived in different ways. One way is to construct the self-similar process as a sum of independent sources with a special form of the autocorrelation function. If we put the peak rate of each source going to zero as the number of sources goes to in nity, models like those of Mandelbrot [11] and Taqqu and Levy [15] will be obtained. Queueing analysis for these kinds of processes is given in s 4 and 5 in this volume. Another approach is to consider on=off sources with constant peak rate, while the number of sources goes to in nity. In this way, we obtain self-similar processes with sessions arrived as Poisson r.v.'s [9]. Originally this process was proposed by Cox [2] and queueing analysis was done recently by many authors [3, 6, 9, 10, 13, 17]. The main results of these papers are presented in this volume. In 9 we can nd a complete overview of this topic. s 7 and 10, present some particular results for the above model as well as results for the model with nite number of on=off sources. Models close to on=off processes arrived as Poisson r.v.'s are considered in 11. In 6 a queueing system with instantaneous arrivals is given. In this chapter we will nd the class of all self-similar processes with independent sessions arrived as Poission r.v.'s. For the particular case of the Cox model, we will nd asymptotic bounds for buffer over ow probability. Compare with 9, Section 10.4 of 10, and Section 7.4 of 7, where asymptotic bounds are presented for a wide class of processes beyond the self-similar one we will focus on the self-similar case (Pareto distribution of the active period). For this case we present some new bounds, which are more accurate compared to the best-known current bounds [3, 10]. This chapter is organized in the following way. First, we give the de nition of second-order self-similar traf c and some well-known, but useful, relations between variance, autocorrelation, and spectral density functions. This is followed by a construction of a class of second-order self-similar processes. Finally, asymptotic queueing behavior for a particular form of the processes from our class is analyzed.
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