Re: [aqm] Questioning each PIE heuristic - moving averages and rate measurement

[Bob Briscoe <[e-mail]>]

Michael,

I just started reading your paper. Two comments so far:

*1) PIE rate averaging is an unfortunate example**
*In related work, you mention PIE's measurement of the link rate as an 
example of use of moving averages. This is an unfortunate example, 
because the PIE code screws up the calculation. It averages a fraction 
by averaging the denominator, which I pointed out is wrong in Section 
5.3.2 of my PIE review <http://bobbriscoe.net/pubs.html#piervw_tr>. 
Quoting...

"The [PIE] pseudocode continually measures the sequence of dequeue 
times, t1, t2, t3,... that it takes to transmit constant amounts of 
bytes (DQ_THRESHOLD). Then the exponentially weighted moving average 
(EWMA) of the rate should be:

     ewma(depart_rate) = DQ_THRESHOLD ∗ ewma(1/t1,1/t2,1/t3,...)
                      != DQ_THRESHOLD / ewma(t1,t2,t3,...)
"
PIE uses the second (incorrect) formula. In the review, I discuss how 
wrong this could be, with an example.

*2) Exponential Moving Average is an Approximation**
*In your moving averages paper, you compare your unbiased exponential 
moving average (UEMA) algorithm with this commonly used exponential 
moving average (EMA) algorithm:
     ewma(X) <- (1-a)*X + a*ewma(X)

A few years ago I wondered about this comparison, and wrote a 2-page 
memo for myself to record the proof. I've temporarily uploaded it here: 
Moving Averages <http://bobbriscoe.net/tmp/ewma.pdf>.

I used a Taylor series expansion to show that the iterative formula 
tends to the exact formula (what you call the unbiased EMA or UEMA) as 
the number of readings (n) becomes large.

When n is not large, you could use the formula at the top of my p2 (just 
before the Taylor series approximation) to calculate the ratio between 
the actual (unbiased) UEMA and my regular iterative EMA (REMA). This 
ratio can be pre-calculated, because its independent of the readings:
     REMA/UEMA = (1-a)*(Sum_{j=0}^{n-1} a^j)

where
   "theta" in my formula is "a" in yours
   "Sum_{j=0}^{n-1}" means "the sum from j=0 to j=n-1"
   Your iterative formula (EMA) starts with an exception, whereas I use 
an iterative formula that just starts the same as it carries on. To 
distinguish between them, I have called mine "regular EMA" or REMA.

For instance, if a = 0.75 as in your Fig 1(c), then the correction 
factors for various small numbers of readings, n, are:

n       UEMA/REMA
_____________________
1            2.29
2            1.73
3            1.46
4            1.31
5            1.22
6            1.15
7            1.11
8            1.08
9            1.06
10            1.04
11            1.03
12            1.02
13            1.02
14            1.01
15            1.01
16            1.01
17            1.01
18            1.00
19            1.00
20            1.00


Cheers


Bob

On 30/03/17 10:57, Michael Menth wrote:
> Hi all,
>
> Am 30.03.2017 um 11:08 schrieb Jonathan Morton:
>>> On 30 Mar, 2017, at 10:46, Bob Briscoe <[e-mail]> wrote:
>>>
>>> For PI2 we removed all but 2 and it worked the same or better than PIE in all our tests. I have been assessing each of the other 7 one by one for reinstatement. So far I've rejected 6. I think I can reject this last one by making the sampling time of the base PI algo dependent on the max link rate. Then when the queue goes idle, the base PI algo will decay drop down to zero no slower than the queue drains, without needing this extra heuristic.
>> That’s fair enough.
>>
>> It sounds like the fairly coarsely discrete time intervals in PIE are the main justification for this particular heuristic, so it might be sufficient to document that WRT PIE itself.  Using finer time intervals is clearly a better choice for the future.
> PIE uses time intervals for measurement purposes and several parameters
> contribute. We've recently done some basic work on measurement
> methodology that facilitates a comparison of different measurement
> approaches and better-informed parametrization by introduction of the
> "memory" concept.
> https://atlas.informatik.uni-tuebingen.de/~menth/papers/Menth17c.pdf
> PIE essentially implements TDRM-DTWMA-UEMA illustrated in Fig. 6d.
>
> The concept of "memory" can also be applied to moving averages which are
> also used in PIE for several purposes. Configuration via a "memory" can
> make some heuristics more intuitive.
>
> Best wishes,
>
> Michael
>
>
>
>>   - Jonathan Morton
>>

-- 
________________________________________________________________
Bob Briscoe                               http://bobbriscoe.net/