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3.2 HEAVY-TAILED DISTRIBUTIONS
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in which the positive constant k represents the smallest possible value of the random variable. In practice, random variables that follow heavy-tailed distributions are characterized as exhibiting many small observations mixed in with a few large observations. In such data sets, most of the observations are small, but most of the contribution to the sample mean or variance comes from the few large observations. This effect can be seen in Fig. 3.1, which shows 10,000 synthetically generated observations drawn from a Pareto distribution with a 1:2 and mean m 6. In Fig. 3.1(a) the scale allows all observations to be shown; in Fig. 3.1(b) the y axis is expanded to show the region from 0 to 200. These gures show the characteristic, visually striking behavior of heavy-tailed random variables. From plot (a) it is clear that a few large observations are present, some on the order of hundreds to one thousand; while from plot (b) it is clear that most observations are quite small, typically on the order of tens or less.
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Fig. 3.1
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Sample data from heavy-tailed distribution with a 1:2.
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Fig. 3.2
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Running mean of data from Fig. 3.1.
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An example of the effect of this variability on sample statistics is shown in Fig. 3.2. This gure shows the running sample mean of the data points from Fig. 3.1, as well as a level line showing the mean of the underlying distribution (6). Note that the sample mean starts out well below the distributional mean, and that even after 10,000 observations it is not close in relative terms to the distributional mean. 3.2.2 Heavy Tails in Computing Systems
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A number of recent studies have shown evidence indicating that aspects of computing and telecommunication systems can show heavy-tailed distributions. Measurements of computer network traf c have shown that autocorrelations are often related to heavy tails; this is the phenomenon of self-similarity [5, 10]. Measurements of le sizes in the Web [1, 3] and in I=O patterns [13] have shown evidence that le sizes can show heavy-tailed distributions. In addition, the CPU time demands of UNIX processes have also been shown to follow heavy-tailed distributions [7, 9]. The presence of heavy-tailed distributions in measured data can be assessed in a number of ways. The simplest is to plot the ccdf on log log axes and visually inspect the resulting curve for linearity over a wide range (several orders of magnitude). This is based on Eq. (3.1), which can be recast as  d log F x a; x3I d log x lim so that for large x, the ccdf of a heavy-tailed distribution should appear to be a straight line on log log axes with slope a. An example empirical data set is shown in Fig. 3.3, which is taken from Crovella and Bestavros [3]. This gure is the ccdf of le sizes transferred through the network due to the Web, plotted on log log axes. The gure shows that the le size distribution appears to show power-law behavior over approximately three orders of magnitude. The slope of the line t to the upper tail is approximately 1:2, ^ yielding a % 1:2.
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Fig. 3.3
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Log log complementary distribution of sizes of les transferred through the Web.
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As heavy-tailed distributions are increasingly used to represent workload characteristics of computing systems, researchers interested in simulating such systems are beginning to use heavy-tailed inputs to simulations. For example, Paxson [12] describes methods for generating self-similar time series for use in simulating network traf c and Park et al. [11] use heavy-tailed le sizes as inputs to a network simulation. However, an important question arises: How stable are such simulations This can be broken down into two questions: 1. How long until such simulations reach steady state 2. How variable is system performance at steady state In this section we will show that if simulation outputs are dependent on all the moments of the distribution F, then the answers to the above questions can be surprising. Essentially, we show that such simulations can take a very long time to reach steady state; and that such simulations can be much more variable at steady state than is typical for traditional systems. Note that some simulation statistics may not be affected directly by all the moments of the distribution F, and our conclusions do not necesssarily apply to those cases. For example, the mean number of customers in an M =G=I queueing system may not show unusual behavior even if the service time distribution F is heavy tailed because that statistic only depends on the mean of F. Since not all simulation statistics will be affected by heavy-tailed workloads, we choose a simple statistic to show the generality of our observations: the sample mean of the heavy-tailed inputs. Since our results apply to the sample mean of the input, we expect that any system property that behaves like the sample mean should show similar behavior. For example, assume we want to achieve steady state in a particular simulation. This implies that the measured system utilization l (where l 1 is the x
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