## Processor Flutter - Part 2

Additional analysis based upon the gnuplot data files described in
part 1 is shown below.

The first set of plots shows histogram plots of the
difference data with a fitted density curve. The "rug" of each plot shows
the same increment values previously shown in Part 1. The I7 and Xeon plots
were divided into two ranges to provide better detail. Note that y scale
for the "tail" plots is exaggerated by approximately 10^4.

The bell-shaped density curves in the first two main plots suggest a comparision
with a normal distribution. Data values are the **logarithm** of TICK differences,
so the Q-Q Normal plots below show the fit of the data
to a theoretical log-normal distribution.

The narrative continues after the exhibits.

### Flutter Density

### Q-Q Normal Plots

### Discussion

The frequency plots show three peaks for each processor. For the I7 and Xeon,
the last 2 peaks (the "tail") contain a miniscule portion of the population.
In the Pentium case, approximately 75% of the distribution is concentrated
in the tail.

To roughly paraphrase the Q-Q plots, the log data in the I7 and Xenon plots
appears to be normal for more ~2 SD (standard deviations) on either side
of the mean with excess population outside of that range. The Pentium plot
is harder to interpret. Repeating the qqnorm plot for the Pentium data with
the lower peak omitted (not shown) barely changes the shape of the plot and
does not challenge the approximate plot shown. With some reservations, the
data shown can be best described by a **log normal** distribution.
By definition, the log-normal distribution is the maximum entropy probability
distribution for a random variable for which the mean and variance of the
logarithm are fixed.

A log-normal distribution is often thought of as the multiplicative
product of positive independent random variables. Observed log-normal
behavior in this case suggests that the haveged observed timer variations
can be modeled as the product of independent random timer variations in
the data/instruction cache, branch prediction cache, and the
translation-look-asside cache.

Analysis of the log-normal hypothesis continues in
part 3.

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