THE LARGE-TIME SCALING BEHAVIOR OF NETWORK TRAFFIC in Visual Studio .NET

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20.2 THE LARGE-TIME SCALING BEHAVIOR OF NETWORK TRAFFIC
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total number of packets or bytes per time unit (measured at time scale m) generated by all connections, then we can write X m k P Xi m k ; k ! 0; 20:1
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where the sum is over all connections i that are active at time k and where Xi m Xi m k : k ! 0 represents the total number of packets or bytes per time unit (again measured at time scale m) generated by connection i.1 Thus, Eq. (20.1) captures the additive nature of aggregate network traf c by expressing the overall traf c rate process X m as a superposition of the traf c rate processes Xi m of the individual connections. Assuming for simplicity that the individual traf c rate processes Xi m are independent from one another and identically distributed, then under weak regularity conditions on the marginal distribution of the Xi m (including, e.g., the existence of second moments), Eq. (20.1) guarantees that the overall traf c rate process (or its deviations from its mean) exhibits Gaussian marginals, as soon as the traf c is generated by a suf ciently large number of individual connections. 20.2.2 Self-Similarity Through Heavy-Tailed Connections
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Focusing on the temporal dynamics of the individual traf c rate processes Xi m , suppose for simplicity that connection i sends packets or bytes at a constant rate (say, rate 1) for some time (the ``active'' or ``on'' period) and does not send any packets or bytes during the ``idle'' or ``off'' period; we will return to the challenging problem of allowing for more realistic ``within-connection'' packet dynamics in Section 20.3. For example, in a LAN environment, a connection corresponds to an individual hostto-host or source destination pair and the corresponding traf c patterns have been shown in Willinger et al. [38] to conform to an alternating renewal process where the successive pairs of on and off periods de ne the inter-renewal intervals. On the other hand, in the context of wide-area networks or WANs such as the Internet, we associate individual connections with ``sessions,'' where a session starts at some random point in time, generates packets or bytes at a constant rate (say, rate 1) during the lifetime of the connection, and then stops transmitting packets or bytes. Here a session can be an FTP appplication, a TELNET connection, a Web session, sending email, reading Network News, and so on, or any imaginable combination thereof. In fact, over 1 to 1 hour periods, session arrivals on Internet links have been shown to 2 be consistent with a homogeneous Poisson process; for example, see Paxson and Floyd [25] for FTP and TELNET sessions, and see Feldmann et al. [12] for Web sessions. Note that in the present setting, only global connection characteristics (e.g., session arrivals, lifetimes of sessions, durations of the on=off periods) play a role, while the details of how the packets arrive within a connection or within an on
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Note that the processes X m and Xi m are de ned by averaging X and Xi over nonoverlapping blocks of size m.
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NETWORK TRAFFIC DYNAMICS
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period have been conveniently modeled away by assuming that the packets within a connection are generated at a constant rate. To describe the stochastic nature of the overall traf c rate process X m , the only stochastic elements that have not yet been speci ed are the distributions of the lengths of the on=off periods (in the case of the LAN example) or the distribution of the session durations (for the WAN case) associated with the individual traf c rate processes Xi m . Based on measured on=off periods of individual host-to-host pairs in a LAN environment (e.g., see Willinger et al. [38]) and measured session durations from different WAN sites (e.g., see Feldman et al. [12], Paxson and Floyd [25] and Willinger et al. [37]), we choose these distributions to be heavy-tailed with in nite variance. Here, a positive random variable U (or the corresponding distribution function F) is called heavy-tailed with tail index > 0 if it satis es P U > y 1 F y % cy ; as y 3 I; 20:2 where c > 0 is a nite constant that does not depend on y. Such distributions are also called hyperbolic or power-law distributions and include, among others, the wellknown class of Pareto distributions. The case 1 < < 2 is of special interest and concerns heavy-tailed distributions with nite mean but in nite variance. Intuitively, in nite variance distributions allow random variables to take values that vary over a wide range of scales and can be exceptionally large with nonnegligible probabilities. Hence, heavy-tailed distributions with in nite variance allow for compact descriptions of the empirically observed high-variability phenomena that dominate traf crelated measurements at all layers in the networking hierarchy; for example, see Feldman et al. [12]. Mathematically, the heavy-tailed property of, for example, the durations during which individual connections actively generate packets implies that the temporal correlations of the stationary versions of an individual traf c rate processes Xi m and, because of the additivity property (20.1), of the overall traf c rate process X m decay hyperbolically slowly; that is, they exhibit long-range dependence. More precisely, if r m r m k : k ! 0 denotes the autocorrelation function of the stationary version of the overall traf c rate process X m , then property (20.2) can be shown to imply long-range dependence (e.g., see Cox [4] and Willinger et al. [38]; for similar results obtained in the context of a uid queueing system under heavy traf c, see 5 in this volume). That is, for all m ! 1, r m satis es r m k % ck 2H 2 ; as k 3 I; 0:5 < H < 1; 20:3 where the parameter H is called the Hurst parameter and measures the degree of long-range dependence in X m ; in terms of the tail index 1 < < 2 that measures the degree of ``heavy-tailedness'' in Eq. (20.2), H is given by H 3 =2. Intuitively, long-range dependence results in periods of sustained greater-thanaverage or lower-than-average traf c rates, irrespective of the time scale over which the rate is measured. In fact, for a zero-mean covariance-stationary process, Eq. (20.3) implies (and is implied by) asymptotic (second-order) self-similarity; that is, after appropriate rescaling, the overall traf c rate processes X m have identical second-order statistical characteristics and ``look similar'' for all suf ciently large
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