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