[DCCP] tfrc: Document boundaries and limits of the TFRC lookup table

This adds documentation for the TCP Reno throughput equation which is at
the heart of the TFRC sending rate / loss rate calculations.

It spells out precisely how the values were determined and what they mean.
The equations were derived through reverse engineering and found to be
fully accurate (verified using test programs).

This patch does not change any code.

Signed-off-by: Gerrit Renker <gerrit@erg.abdn.ac.uk>
Signed-off-by: Ian McDonald <ian.mcdonald@jandi.co.nz>
Signed-off-by: Arnaldo Carvalho de Melo <acme@mandriva.com>

net/dccp/ccids/lib/tfrc_equation.c

@@ -18,10 +18,76 @@
 #include "tfrc.h"
 
 #define TFRC_CALC_X_ARRSIZE 500
+#define TFRC_CALC_X_SPLIT   50000	/* 0.05 * 1000000, details below */
 
-#define TFRC_CALC_X_SPLIT 50000
-/* equivalent to 0.05 */
-
+/*
+  TFRC TCP Reno Throughput Equation Lookup Table for f(p)
+
+  The following two-column lookup table implements a part of the TCP throughput
+  equation from [RFC 3448, sec. 3.1]:
+
+	 			     s
+  X_calc  =  --------------------------------------------------------------
+	     R * sqrt(2*b*p/3) + (3 * t_RTO * sqrt(3*b*p/8) * (p + 32*p^3))
+
+  Where:
+	X      is the transmit rate in bytes/second
+	s      is the packet size in bytes
+	R      is the round trip time in seconds
+	p      is the loss event rate, between 0 and 1.0, of the number of loss
+	              events as a fraction of the number of packets transmitted
+	t_RTO  is the TCP retransmission timeout value in seconds
+	b      is the number of packets acknowledged by a single TCP ACK
+
+  We can assume that b = 1 and t_RTO is 4 * R. The equation now becomes:
+
+				     s
+  X_calc  =  -------------------------------------------------------
+	     R * sqrt(p*2/3) + (12 * R * sqrt(p*3/8) * (p + 32*p^3))
+
+  which we can break down into:
+
+                      s
+	X_calc  =  ---------
+	            R * f(p)
+
+  where f(p) is given for 0 < p <= 1 by:
+
+	f(p)  =  sqrt(2*p/3) + 12 * sqrt(3*p/8) *  (p + 32*p^3)
+
+  Since this is kernel code, floating-point arithmetic is avoided in favour of
+  integer arithmetic. This means that nearly all fractional parameters are
+  scaled by 1000000:
+    * the parameters p and R
+    * the return result f(p)
+  The lookup table therefore actually tabulates the following function g(q):
+
+  	g(q)  =  1000000 * f(q/1000000)
+
+  Hence, when p <= 1, q must be less than or equal to 1000000. To achieve finer
+  granularity for the practically more relevant case of small values of p (up to
+  5%), the second column is used; the first one ranges up to 100%.  This split
+  corresponds to the value of q = TFRC_CALC_X_SPLIT. At the same time this also
+  determines the smallest resolution.
+
+  The entire table is generated by:
+    for(i=0; i < TFRC_CALC_X_ARRSIZE; i++) {
+	lookup[i][0]  =  g((i+1) * 1000000/TFRC_CALC_X_ARRSIZE);
+	lookup[i][1]  =  g((i+1) * TFRC_CALC_X_SPLIT/TFRC_CALC_X_ARRSIZE);
+    }
+
+  With the given configuration, we have, with M = TFRC_CALC_X_ARRSIZE-1,
+    lookup[0][0]  =  g(1000000/(M+1)) 		 =  1000000 * f(0.2%)
+    lookup[M][0]  =  g(1000000)			 =  1000000 * f(100%)
+    lookup[0][1]  =  g(TFRC_CALC_X_SPLIT/(M+1))	 =  1000000 * f(0.01%)
+    lookup[M][1]  =  g(TFRC_CALC_X_SPLIT)	 =  1000000 * f(5%)
+
+  In summary, the two columns represent f(p) for the following ranges:
+    * The first column is for   0.002  <= p <= 1.0
+    * The second column is for  0.0001 <= p <= 0.05
+  Where the columns overlap, the second (finer-grained) is given preference,
+  i.e. the first column is used only for p >= 0.05.
+ */
 static const u32 tfrc_calc_x_lookup[TFRC_CALC_X_ARRSIZE][2] = {
 	{     37172,   8172 },
 	{     53499,  11567 },
@@ -525,62 +591,26 @@ static const u32 tfrc_calc_x_lookup[TFRC_CALC_X_ARRSIZE][2] = {
 	{ 243315981, 271305 }
 };
 
-/* Calculate the send rate as per section 3.1 of RFC3448
- 
-Returns send rate in bytes per second
-
-Integer maths and lookups are used as not allowed floating point in kernel
-
-The function for Xcalc as per section 3.1 of RFC3448 is:
-
-X =                            s
-     -------------------------------------------------------------
-     R*sqrt(2*b*p/3) + (t_RTO * (3*sqrt(3*b*p/8) * p * (1+32*p^2)))
-
-where 
-X is the trasmit rate in bytes/second
-s is the packet size in bytes
-R is the round trip time in seconds
-p is the loss event rate, between 0 and 1.0, of the number of loss events 
-  as a fraction of the number of packets transmitted
-t_RTO is the TCP retransmission timeout value in seconds
-b is the number of packets acknowledged by a single TCP acknowledgement
-
-we can assume that b = 1 and t_RTO is 4 * R. With this the equation becomes:
-
-X =                            s
-     -----------------------------------------------------------------------
-     R * sqrt(2 * p / 3) + (12 * R * (sqrt(3 * p / 8) * p * (1 + 32 * p^2)))
-
-
-which we can break down into:
-
-X =     s
-     --------
-     R * f(p)
-
-where f(p) = sqrt(2 * p / 3) + (12 * sqrt(3 * p / 8) * p * (1 + 32 * p * p))
-
-Function parameters:
-s - bytes
-R - RTT in usecs
-p - loss rate (decimal fraction multiplied by 1,000,000)
-
-Returns Xcalc in bytes per second
-
-DON'T alter this code unless you run test cases against it as the code
-has been manipulated to stop underflow/overlow.
-
-*/
+/**
+ * tfrc_calc_x - Calculate the send rate as per section 3.1 of RFC3448
+ *
+ *  @s: packet size          in bytes
+ *  @R: RTT                  scaled by 1000000   (i.e., microseconds)
+ *  @p: loss ratio estimate  scaled by 1000000
+ *  Returns X_calc           in bytes per second (not scaled).
+ *
+ * Note: DO NOT alter this code unless you run test cases against it,
+ *       as the code has been optimized to stop underflow/overflow.
+ */
 u32 tfrc_calc_x(u16 s, u32 R, u32 p)
 {
 	int index;
 	u32 f;
 	u64 tmp1, tmp2;
 
-	if (p < TFRC_CALC_X_SPLIT)
+	if (p < TFRC_CALC_X_SPLIT) 		      /* 0      <= p <  0.05  */
 		index = (p / (TFRC_CALC_X_SPLIT / TFRC_CALC_X_ARRSIZE)) - 1;
-	else
+	else		 			      /* 0.05   <= p <= 1.00  */
 		index = (p / (1000000 / TFRC_CALC_X_ARRSIZE)) - 1;
 
 	if (index < 0)
@@ -599,11 +629,13 @@ u32 tfrc_calc_x(u16 s, u32 R, u32 p)
 	else
 		f = tfrc_calc_x_lookup[index][1];
 
+	/* The following computes X = s/(R*f(p)) in bytes per second. Since f(p)
+	 * and R are both scaled by 1000000, we need to multiply by 1000000^2.
+	 * ==> DO NOT alter this unless you test against overflow on 32 bit   */
 	tmp1 = ((u64)s * 100000000);
 	tmp2 = ((u64)R * (u64)f);
 	do_div(tmp2, 10000);
 	do_div(tmp1, tmp2); 
-	/* Don't alter above math unless you test due to overflow on 32 bit */
 
 	return (u32)tmp1; 
 }
@@ -611,10 +643,10 @@ u32 tfrc_calc_x(u16 s, u32 R, u32 p)
 EXPORT_SYMBOL_GPL(tfrc_calc_x);
 
 /*
- * args: fvalue - function value to match
- * returns: p closest to that value
+ *  tfrc_calc_x_reverse_lookup  -  try to find p given f(p)
  *
- * both fvalue and p are multiplied by 1,000,000 to use ints
+ *  @fvalue: function value to match, scaled by 1000000
+ *  Returns closest match for p, also scaled by 1000000
  */
 u32 tfrc_calc_x_reverse_lookup(u32 fvalue)
 {