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Within the models for the 12 hour time period a clear periodicity is recognizable, a periodicity that has also been observed within other datasets. During the rst half of a day the Pareto and lognormal models give a better t than the exponential one, while the situation is reversed for the second half of the day. An explanation for this is that during night hours many fewer actions are initiated and, as such, the number of long interarrival times is larger, thus explaining why the heavy-tailed Pareto distribution yields a better t than the exponential distribution. The Weibull distribution yields an even better t than either the Pareto or the exponential one. We also consider the interarrival time distribution of all HTTP connections over all 1204 possible one hour time periods that contain suf cient observations. The Weibull models are statistically better than the Pareto models in 92.9% of the instances, better than the exponential models in 92.3% of the instances, and better than the lognormal in 94.5% of the instances. In only 0.7% of the instances is the Pareto model better than the Weibull one and in only 0.2% is the exponential model better than the Weibull model. (Even though the Weibull model contains the

An associated FJTaskRunnerGroup class provides control and entry points into this framework A FJTaskRunnerGroup is constructed with a given number of worker threads that should ordinarily be equal to the number of CPUs on a system The class supports method invoke that starts up a main task, which will in turn normally create many others

^ Fig. 15.13 (a) Con dence intervals for l2 for the tted Weibull, lognormal, exponential, and Pareto model of TELNET interarrival times within each 12 hour time frame for the 29 30 June 1995 dataset. (b) Histogram of Weibull shape parameter c of the tted Weibull distributions for the HTTP application for one hour time periods for the external AT&T dataset from 16 to 30 March. (c) Histogram of Weibull shape parameter c of the tted Weibull distributions for the applications TELNET, X, FTP, HTTP, RFS, SMTP, and FINGER and two hour time periods for all CMU datasets.

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exponential model, the exponential model can sometimes result in a better t than the Weibull model. This is possible since the parameter estimation for the Weibull model is more complex and can be less precise than for the exponential model.) Similar observations apply to the CMU datasets and other applications. Given that the Weibull model is superior for many applications and that its shape parameter c may vary quite substantially from one time period to the next, it is of interest to look at the distribution of the shape parameter c of the tted Weibull distributions. Figure 15.13(b) gives the histogram of Weibull shape parameter c for the tted Weibull distribution of the external AT&T dataset from 16 to 31 March 1996 and Fig. 15.13(c) does the same for the tted Weibull distributions for the applications TELNET, X, FTP, HTTP, RFS, SMTP, and FINGER over two hour time periods for all CMU datasets. That the number of Weibull models that have a shape parameter around one is small is another indication that the exponential model is inappropriate to model these interarrival times. That more than 50% of the shape parameters are less than 0.65 indicates the inherent burstiness of the request sequences.

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To underscore the observation that the Weibull distribution gives good ts especially for HTTP TCP connection interarrival times, we again consider traces from AT&T's WorldNet. The rst trace was collected on the 23 July 1997 starting at 7 p.m. for a duration of seven hours on an FDDI ring carrying traf c from roughly 420 modems used by WorldNet dialup customers (for more details see Feldmann et al. [13, 15]). For this dataset we consider the interarrival time distribution of HTTP connections over all 36 possible 12 minute time periods. In all instances the Weibull models are statistically better than the Pareto and exponential models, and better than the lognormal in 91.6% of the case. Never is any other model statistically better than the Weibull one. Similar observations apply to other subsets. One of the problems in this dataset is that it includes 371,972 HTTP connections. With this many connections arriving in a bursty fashion the timer resolution can become a potential problem. This can disturb small interarrival times and therefore undersample them. Figure 15.14(a) shows the histogram of the interarrival times for a two hour sample of the data. The spikes in the range of 0.0001 0.001 second have two origins: timer inaccuracy and browsers almost simultaneously opening multiple TCP connections. Nevertheless, the data, except for the spikes, is still well modeled by a Weibull distribution. Figure 15.14(b) shows the histogram of Weibull shape parameter c for the tted Weibull distributions. The backbone AT&T WorldNet dataset contains so many HTTP requests (1,187,865 over one hour) that the timer resolution on the monitoring machine is not suf cient to resolve the small time scale effects. Therefore we do not attempt to model the interarrival times for this dataset.

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