Reinhard Vicinus reported that the reported average response times cannot
be larger than 16s due to the double multiply being performed by
swrate_add() which causes an overflow very quickly. Indeed, with N=512,
the highest average value is 16448.
One solution proposed by Reinhard is to turn to long long, but this
involves 64x64 multiplies and 64->32 divides, which are extremely
expensive on 32-bit platforms.
There is in fact another way to avoid the overflow without using larger
integers, it consists in avoiding the multiply using the fact that
x*(n-1)/N = x-(x/N).
Now it becomes possible to store average values as large as 8.4 millions,
which is around 2h18mn.
Interestingly, this improvement also makes the code cheaper to execute
both on 32 and on 64 bit platforms :
Before :
00000000 <swrate_add>:
0: 8b 54 24 04 mov 0x4(%esp),%edx
4: 8b 0a mov (%edx),%ecx
6: 89 c8 mov %ecx,%eax
8: c1 e0 09 shl $0x9,%eax
b: 29 c8 sub %ecx,%eax
d: 8b 4c 24 0c mov 0xc(%esp),%ecx
11: c1 e8 09 shr $0x9,%eax
14: 01 c8 add %ecx,%eax
16: 89 02 mov %eax,(%edx)
After :
00000020 <swrate_add>:
20: 8b 4c 24 04 mov 0x4(%esp),%ecx
24: 8b 44 24 0c mov 0xc(%esp),%eax
28: 8b 11 mov (%ecx),%edx
2a: 01 d0 add %edx,%eax
2c: 81 c2 ff 01 00 00 add $0x1ff,%edx
32: c1 ea 09 shr $0x9,%edx
35: 29 d0 sub %edx,%eax
37: 89 01 mov %eax,(%ecx)
This fix may be backported to 1.6.
While the current functions report average event counts per period, we are
also interested in average values per event. For this we use a different
method. The principle is to rely on a long tail which sums the new value
with a fraction of the previous value, resulting in a sliding window of
infinite length depending on the precision we're interested in.
The idea is that we always keep (N-1)/N of the sum and add the new sampled
value. The sum over N values can be computed with a simple program for a
constant value 1 at each iteration :
N
,---
\ N - 1 e - 1
> ( --------- )^x ~= N * -----
/ N e
'---
x = 1
Note: I'm not sure how to demonstrate this but at least this is easily
verified with a simple program, the sum equals N * 0.632120 for any N
moderately large (tens to hundreds).
Inserting a constant sample value V here simply results in :
sum = V * N * (e - 1) / e
But we don't want to integrate over a small period, but infinitely. Let's
cut the infinity in P periods of N values. Each period M is exactly the same
as period M-1 with a factor of ((N-1)/N)^N applied. A test shows that given a
large N :
N - 1 1
( ------- )^N ~= ---
N e
Our sum is now a sum of each factor times :
N*P P
,--- ,---
\ N - 1 e - 1 \ 1
> v ( --------- )^x ~= VN * ----- * > ---
/ N e / e^x
'--- '---
x = 1 x = 0
For P "large enough", in tests we get this :
P
,---
\ 1 e
> --- ~= -----
/ e^x e - 1
'---
x = 0
This simplifies the sum above :
N*P
,---
\ N - 1
> v ( --------- )^x = VN
/ N
'---
x = 1
So basically by summing values and applying the last result an (N-1)/N factor
we just get N times the values over the long term, so we can recover the
constant value V by dividing by N.
A value added at the entry of the sliding window of N values will thus be
reduced to 1/e or 36.7% after N terms have been added. After a second batch,
it will only be 1/e^2, or 13.5%, and so on. So practically speaking, each
old period of N values represents only a quickly fading ratio of the global
sum :
period ratio
1 36.7%
2 13.5%
3 4.98%
4 1.83%
5 0.67%
6 0.25%
7 0.09%
8 0.033%
9 0.012%
10 0.0045%
So after 10N samples, the initial value has already faded out by a factor of
22026, which is quite fast. If the sliding window is 1024 samples wide, it
means that a sample will only count for 1/22k of its initial value after 10k
samples went after it, which results in half of the value it would represent
using an arithmetic mean. The benefit of this method is that it's very cheap
in terms of computations when N is a power of two. This is very well suited
to record response times as large values will fade out faster than with an
arithmetic mean and will depend on sample count and not time.
Demonstrating all the above assumptions with maths instead of a program is
left as an exercise for the reader.
Some freq counters will have to work on periods different from 1 second.
The original freq counters rely on the period to be exactly one second.
The new ones (freq_ctr_period) let the user define the period in ticks,
and all computations are operated over that period. When reading a value,
it indicates the amount of events over that period too.
The rate-limit was applied to the smoothed value which does a special
case for frequencies below 2 events per period. This caused irregular
limitations when set to 1 session per second.
The proper way to handle this is to compute the number of remaining
events that can occur without reaching the limit. This is what has
been added. It also has the benefit that the frequency calculation
is now done once when entering event_accept(), before the accept()
loop, and not once per accept() loop anymore, thus saving a few CPU
cycles during very high loads.
With this fix, rate limits of 1/s are perfectly respected.
With this change, all frontends, backends, and servers maintain a session
counter and a timer to compute a session rate over the last second. This
value will be very useful because it varies instantly and can be used to
check thresholds. This value is also reported in the stats in a new "rate"
column.
