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.

Author: robertmhaas

doc/src/sgml/ref/pgbench.sgml

@@ -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>

src/bin/pgbench/pgbench.c

@@ -90,7 +90,7 @@ static int	pthread_join(pthread_t th, void **thread_return);
 #define LOG_STEP_SECONDS	5	/* seconds between log messages */
 #define DEFAULT_NXACTS	10		/* default nxacts */
 
-#define MIN_GAUSSIAN_THRESHOLD		2.0 /* minimum threshold for gauss */
+#define MIN_GAUSSIAN_PARAM		2.0 /* minimum parameter for gauss */
 
 int			nxacts = 0;			/* number of transactions per client */
 int			duration = 0;		/* duration in seconds */
@@ -488,47 +488,47 @@ getrand(TState *thread, int64 min, int64 max)
 
 /*
  * random number generator: exponential distribution from min to max inclusive.
- * the threshold is so that the density of probability for the last cut-off max
- * value is exp(-threshold).
+ * the parameter is so that the density of probability for the last cut-off max
+ * value is exp(-parameter).
  */
 static int64
-getExponentialRand(TState *thread, int64 min, int64 max, double threshold)
+getExponentialRand(TState *thread, int64 min, int64 max, double parameter)
 {
 	double		cut,
 				uniform,
 				rand;
 
-	Assert(threshold > 0.0);
-	cut = exp(-threshold);
+	Assert(parameter > 0.0);
+	cut = exp(-parameter);
 	/* erand in [0, 1), uniform in (0, 1] */
 	uniform = 1.0 - pg_erand48(thread->random_state);
 
 	/*
-	 * inner expresion in (cut, 1] (if threshold > 0), rand in [0, 1)
+	 * inner expresion in (cut, 1] (if parameter > 0), rand in [0, 1)
 	 */
 	Assert((1.0 - cut) != 0.0);
-	rand = -log(cut + (1.0 - cut) * uniform) / threshold;
+	rand = -log(cut + (1.0 - cut) * uniform) / parameter;
 	/* return int64 random number within between min and max */
 	return min + (int64) ((max - min + 1) * rand);
 }
 
 /* random number generator: gaussian distribution from min to max inclusive */
 static int64
-getGaussianRand(TState *thread, int64 min, int64 max, double threshold)
+getGaussianRand(TState *thread, int64 min, int64 max, double parameter)
 {
 	double		stdev;
 	double		rand;
 
 	/*
-	 * Get user specified random number from this loop, with -threshold <
-	 * stdev <= threshold
+	 * Get user specified random number from this loop,
+	 * with -parameter < stdev <= parameter
 	 *
 	 * This loop is executed until the number is in the expected range.
 	 *
-	 * As the minimum threshold is 2.0, the probability of looping is low:
+	 * As the minimum parameter is 2.0, the probability of looping is low:
 	 * sqrt(-2 ln(r)) <= 2 => r >= e^{-2} ~ 0.135, then when taking the
 	 * average sinus multiplier as 2/pi, we have a 8.6% looping probability in
-	 * the worst case. For a 5.0 threshold value, the looping probability is
+	 * the worst case. For a parameter value of 5.0, the looping probability is
 	 * about e^{-5} * 2 / pi ~ 0.43%.
 	 */
 	do
@@ -553,10 +553,10 @@ getGaussianRand(TState *thread, int64 min, int64 max, double threshold)
 		 * over.
 		 */
 	}
-	while (stdev < -threshold || stdev >= threshold);
+	while (stdev < -parameter || stdev >= parameter);
 
-	/* stdev is in [-threshold, threshold), normalization to [0,1) */
-	rand = (stdev + threshold) / (threshold * 2.0);
+	/* stdev is in [-parameter, parameter), normalization to [0,1) */
+	rand = (stdev + parameter) / (parameter * 2.0);
 
 	/* return int64 random number within between min and max */
 	return min + (int64) ((max - min + 1) * rand);
@@ -1483,7 +1483,7 @@ doCustom(TState *thread, CState *st, instr_time *conn_time, FILE *logfile, AggVa
 			char	   *var;
 			int64		min,
 						max;
-			double		threshold = 0;
+			double		parameter = 0;
 			char		res[64];
 
 			if (*argv[2] == ':')
@@ -1554,41 +1554,49 @@ doCustom(TState *thread, CState *st, instr_time *conn_time, FILE *logfile, AggVa
 				{
 					if ((var = getVariable(st, argv[5] + 1)) == NULL)
 					{
-						fprintf(stderr, "%s: invalid threshold number: \"%s\"\n",
+						fprintf(stderr, "%s: invalid parameter: \"%s\"\n",
 								argv[0], argv[5]);
 						st->ecnt++;
 						return true;
 					}
-					threshold = strtod(var, NULL);
+					parameter = strtod(var, NULL);
 				}
 				else
-					threshold = strtod(argv[5], NULL);
+					parameter = strtod(argv[5], NULL);
 
 				if (pg_strcasecmp(argv[4], "gaussian") == 0)
 				{
-					if (threshold < MIN_GAUSSIAN_THRESHOLD)
+					if (parameter < MIN_GAUSSIAN_PARAM)
 					{
-						fprintf(stderr, "gaussian threshold must be at least %f (not \"%s\")\n", MIN_GAUSSIAN_THRESHOLD, argv[5]);
+						fprintf(stderr, "gaussian parameter must be at least %f (not \"%s\")\n", MIN_GAUSSIAN_PARAM, argv[5]);
 						st->ecnt++;
 						return true;
 					}
 #ifdef DEBUG
-					printf("min: " INT64_FORMAT " max: " INT64_FORMAT " random: " INT64_FORMAT "\n", min, max, getGaussianRand(thread, min, max, threshold));
+					printf("min: " INT64_FORMAT " max: " INT64_FORMAT " random: " INT64_FORMAT "\n",
+						   min, max,
+						   getGaussianRand(thread, min, max, parameter));
 #endif
-					snprintf(res, sizeof(res), INT64_FORMAT, getGaussianRand(thread, min, max, threshold));
+					snprintf(res, sizeof(res), INT64_FORMAT,
+							 getGaussianRand(thread, min, max, parameter));
 				}
 				else if (pg_strcasecmp(argv[4], "exponential") == 0)
 				{
-					if (threshold <= 0.0)
+					if (parameter <= 0.0)
 					{
-						fprintf(stderr, "exponential threshold must be greater than zero (not \"%s\")\n", argv[5]);
+						fprintf(stderr,
+							"exponential parameter must be greater than zero (not \"%s\")\n",
+							argv[5]);
 						st->ecnt++;
 						return true;
 					}
 #ifdef DEBUG
-					printf("min: " INT64_FORMAT " max: " INT64_FORMAT " random: " INT64_FORMAT "\n", min, max, getExponentialRand(thread, min, max, threshold));
+					printf("min: " INT64_FORMAT " max: " INT64_FORMAT " random: " INT64_FORMAT "\n",
+						   min, max,
+						   getExponentialRand(thread, min, max, parameter));
 #endif
-					snprintf(res, sizeof(res), INT64_FORMAT, getExponentialRand(thread, min, max, threshold));
+					snprintf(res, sizeof(res), INT64_FORMAT,
+							 getExponentialRand(thread, min, max, parameter));
 				}
 			}
 			else	/* this means an error somewhere in the parsing phase... */
@@ -2282,8 +2290,9 @@ process_commands(char *buf, const char *source, const int lineno)
 		if (pg_strcasecmp(my_commands->argv[0], "setrandom") == 0)
 		{
 			/*
-			 * parsing: \setrandom variable min max [uniform] \setrandom
-			 * variable min max (gaussian|exponential) threshold
+			 * parsing:
+			 *   \setrandom variable min max [uniform]
+			 *   \setrandom variable min max (gaussian|exponential) parameter
 			 */
 
 			if (my_commands->argc < 4)
@@ -2308,7 +2317,7 @@ process_commands(char *buf, const char *source, const int lineno)
 				if (my_commands->argc < 6)
 				{
 					syntax_error(source, lineno, my_commands->line, my_commands->argv[0],
-					 "missing threshold argument", my_commands->argv[4], -1);
+					 "missing parameter", my_commands->argv[4], -1);
 				}
 				else if (my_commands->argc > 6)
 				{