mirror of
https://git.kernel.org/pub/scm/network/iproute2/iproute2.git
synced 2024-11-15 14:05:22 +08:00
96 lines
4.3 KiB
Plaintext
96 lines
4.3 KiB
Plaintext
I. About the distribution tables
|
|
|
|
The table used for "synthesizing" the distribution is essentially a scaled,
|
|
translated, inverse to the cumulative distribution function.
|
|
|
|
Here's how to think about it: Let F() be the cumulative distribution
|
|
function for a probability distribution X. We'll assume we've scaled
|
|
things so that X has mean 0 and standard deviation 1, though that's not
|
|
so important here. Then:
|
|
|
|
F(x) = P(X <= x) = \int_{-inf}^x f
|
|
|
|
where f is the probability density function.
|
|
|
|
F is monotonically increasing, so has an inverse function G, with range
|
|
0 to 1. Here, G(t) = the x such that P(X <= x) = t. (In general, G may
|
|
have singularities if X has point masses, i.e., points x such that
|
|
P(X = x) > 0.)
|
|
|
|
Now we create a tabular representation of G as follows: Choose some table
|
|
size N, and for the ith entry, put in G(i/N). Let's call this table T.
|
|
|
|
The claim now is, I can create a (discrete) random variable Y whose
|
|
distribution has the same approximate "shape" as X, simply by letting
|
|
Y = T(U), where U is a discrete uniform random variable with range 1 to N.
|
|
To see this, it's enough to show that Y's cumulative distribution function,
|
|
(let's call it H), is a discrete approximation to F. But
|
|
|
|
H(x) = P(Y <= x)
|
|
= (# of entries in T <= x) / N -- as Y chosen uniformly from T
|
|
= i/N, where i is the largest integer such that G(i/N) <= x
|
|
= i/N, where i is the largest integer such that i/N <= F(x)
|
|
-- since G and F are inverse functions (and F is
|
|
increasing)
|
|
= floor(N*F(x))/N
|
|
|
|
as desired.
|
|
|
|
II. How to create distribution tables (in theory)
|
|
|
|
How can we create this table in practice? In some cases, F may have a
|
|
simple expression which allows evaluating its inverse directly. The
|
|
Pareto distribution is one example of this. In other cases, and
|
|
especially for matching an experimentally observed distribution, it's
|
|
easiest simply to create a table for F and "invert" it. Here, we give
|
|
a concrete example, namely how the new "experimental" distribution was
|
|
created.
|
|
|
|
1. Collect enough data points to characterize the distribution. Here, I
|
|
collected 25,000 "ping" roundtrip times to a "distant" point (time.nist.gov).
|
|
That's far more data than is really necessary, but it was fairly painless to
|
|
collect it, so...
|
|
|
|
2. Normalize the data so that it has mean 0 and standard deviation 1.
|
|
|
|
3. Determine the cumulative distribution. The code I wrote creates a table
|
|
covering the range -10 to +10, with granularity .00005. Obviously, this
|
|
is absurdly over-precise, but since it's a one-time only computation, I
|
|
figured it hardly mattered.
|
|
|
|
4. Invert the table: for each table entry F(x) = y, make the y*TABLESIZE
|
|
(here, 4096) entry be x*TABLEFACTOR (here, 8192). This creates a table
|
|
for the ("normalized") inverse of size TABLESIZE, covering its domain 0
|
|
to 1 with granularity 1/TABLESIZE. Note that even with the granularity
|
|
used in creating the table for F, it's possible not all the entries in
|
|
the table for G will be filled in. So, make a pass through the
|
|
inverse's table, filling in any missing entries by linear interpolation.
|
|
|
|
III. How to create distribution tables (in practice)
|
|
|
|
If you want to do all this yourself, I've provided several tools to help:
|
|
|
|
1. maketable does the steps 2-4 above, and then generates the appropriate
|
|
header file. So if you have your own time distribution, you can generate
|
|
the header simply by:
|
|
|
|
maketable < time.values > header.h
|
|
|
|
2. As explained in the other README file, the somewhat sleazy way I have
|
|
of generating correlated values needs correction. You can generate your
|
|
own correction tables by compiling makesigtable and makemutable with
|
|
your header file. Check the Makefile to see how this is done.
|
|
|
|
3. Warning: maketable, makesigtable and especially makemutable do
|
|
enormous amounts of floating point arithmetic. Don't try running
|
|
these on an old 486. (NIST Net itself will run fine on such a
|
|
system, since in operation, it just needs to do a few simple integral
|
|
calculations. But getting there takes some work.)
|
|
|
|
4. The tables produced are all normalized for mean 0 and standard
|
|
deviation 1. How do you know what values to use for real? Here, I've
|
|
provided a simple "stats" utility. Give it a series of floating point
|
|
values, and it will return their mean (mu), standard deviation (sigma),
|
|
and correlation coefficient (rho). You can then plug these values
|
|
directly into NIST Net.
|