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pgbench: Change terminology from "threshold" to "parameter".

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Per a recommendation from Tomas Vondra, it's more helpful to refer to
the value that determines how skewed a Gaussian or exponential
distribution is as a parameter rather than a threshold.

Since it's not quite too late to get this right in 9.5, where it was
introduced, back-patch this.  Most of the patch changes only comments
and documentation, but a few pgbench messages are altered to match.

Fabien Coelho, reviewed by Michael Paquier and by me.
This commit is contained in:
Robert Haas
2015-12-18 13:24:51 -05:00
parent 6e7b335930
commit 3c7042a7d7
2 changed files with 78 additions and 60 deletions
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67
doc/src/sgml/ref/pgbench.sgml
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@ -788,7 +788,7 @@ pgbench <optional> <replaceable>options</> </optional> <replaceable>dbname</>
<varlistentry>
<term>
<literal>\setrandom <replaceable>varname</> <replaceable>min</> <replaceable>max</> [ uniform | { gaussian | exponential } <replaceable>threshold</> ]</literal>
<literal>\setrandom <replaceable>varname</> <replaceable>min</> <replaceable>max</> [ uniform | { gaussian | exponential } <replaceable>parameter</> ]</literal>
</term>
<listitem>
@ -804,54 +804,63 @@ pgbench <optional> <replaceable>options</> </optional> <replaceable>dbname</>
By default, or when <literal>uniform</> is specified, all values in the
range are drawn with equal probability. Specifying <literal>gaussian</>
or <literal>exponential</> options modifies this behavior; each
requires a mandatory threshold which determines the precise shape of the
requires a mandatory parameter which determines the precise shape of the
distribution.
</para>
<para>
For a Gaussian distribution, the interval is mapped onto a standard
normal distribution (the classical bell-shaped Gaussian curve) truncated
at <literal>-threshold</> on the left and <literal>+threshold</>
at <literal>-parameter</> on the left and <literal>+parameter</>
on the right.
Values in the middle of the interval are more likely to be drawn.
To be precise, if <literal>PHI(x)</> is the cumulative distribution
function of the standard normal distribution, with mean <literal>mu</>
defined as <literal>(max + min) / 2.0</>, then value <replaceable>i</>
between <replaceable>min</> and <replaceable>max</> inclusive is drawn
with probability:
<literal>
(PHI(2.0 * threshold * (i - min - mu + 0.5) / (max - min + 1)) -
PHI(2.0 * threshold * (i - min - mu - 0.5) / (max - min + 1))) /
(2.0 * PHI(threshold) - 1.0)</>.
Intuitively, the larger the <replaceable>threshold</>, the more
defined as <literal>(max + min) / 2.0</>, with
<literallayout>
f(x) = PHI(2.0 * parameter * (x - mu) / (max - min + 1)) /
(2.0 * PHI(parameter) - 1.0)
</literallayout>
then value <replaceable>i</> between <replaceable>min</> and
<replaceable>max</> inclusive is drawn with probability:
<literal>f(i + 0.5) - f(i - 0.5)</>.
Intuitively, the larger <replaceable>parameter</>, the more
frequently values close to the middle of the interval are drawn, and the
less frequently values close to the <replaceable>min</> and
<replaceable>max</> bounds.
About 67% of values are drawn from the middle <literal>1.0 / threshold</>
and 95% in the middle <literal>2.0 / threshold</>; for instance, if
<replaceable>threshold</> is 4.0, 67% of values are drawn from the middle
quarter and 95% from the middle half of the interval.
The minimum <replaceable>threshold</> is 2.0 for performance of
the Box-Muller transform.
<replaceable>max</> bounds. About 67% of values are drawn from the
middle <literal>1.0 / parameter</>, that is a relative
<literal>0.5 / parameter</> around the mean, and 95% in the middle
<literal>2.0 / parameter</>, that is a relative
<literal>1.0 / parameter</> around the mean; for instance, if
<replaceable>parameter</> is 4.0, 67% of values are drawn from the
middle quarter (1.0 / 4.0) of the interval (i.e. from
<literal>3.0 / 8.0</> to <literal>5.0 / 8.0</>) and 95% from
the middle half (<literal>2.0 / 4.0</>) of the interval (second and
third quartiles). The minimum <replaceable>parameter</> is 2.0 for
performance of the Box-Muller transform.
</para>
<para>
For an exponential distribution, the <replaceable>threshold</>
parameter controls the distribution by truncating a quickly-decreasing
exponential distribution at <replaceable>threshold</>, and then
For an exponential distribution, <replaceable>parameter</>
controls the distribution by truncating a quickly-decreasing
exponential distribution at <replaceable>parameter</>, and then
projecting onto integers between the bounds.
To be precise, value <replaceable>i</> between <replaceable>min</> and
To be precise, with
<literallayout>
f(x) = exp(-parameter * (x - min) / (max - min + 1)) / (1.0 - exp(-parameter))
</literallayout>
Then value <replaceable>i</> between <replaceable>min</> and
<replaceable>max</> inclusive is drawn with probability:
<literal>(exp(-threshold*(i-min)/(max+1-min)) -
exp(-threshold*(i+1-min)/(max+1-min))) / (1.0 - exp(-threshold))</>.
Intuitively, the larger the <replaceable>threshold</>, the more
<literal>f(x) - f(x + 1)</>.
Intuitively, the larger <replaceable>parameter</>, the more
frequently values close to <replaceable>min</> are accessed, and the
less frequently values close to <replaceable>max</> are accessed.
The closer to 0 the threshold, the flatter (more uniform) the access
distribution.
The closer to 0 <replaceable>parameter</>, the flatter (more uniform)
the access distribution.
A crude approximation of the distribution is that the most frequent 1%
values in the range, close to <replaceable>min</>, are drawn
<replaceable>threshold</>% of the time.
The <replaceable>threshold</> value must be strictly positive.
<replaceable>parameter</>% of the time.
<replaceable>parameter</> value must be strictly positive.
</para>
<para>