Re: [tcpm] alpha_cubic (was: Concluding WGLC for draft-ietf-tcpm-rfc8312bis-03)

[Yoshifumi Nishida <[e-mail]>]

Hi Bob,

Thanks for the very detailed explanation.
I agree with some of your points. But, I have two comments on this.

The first point is about RTT variation.
I think you have a point there, but I personally think the paper presumes
very shallow queue so that the model can be very simple.
In this case, we don't have to think about min or max of RTT as they will
be mostly the same.
I think we can probably argue the accuracy of the model, but I think it is
too broad as a scope of this document. It looks a very deep research topic
to me.
I think the model is basically derived from Equation-based Congestion
Control paper in sigcomm 2000. If we want to update or renew it, we might
need another RFC5348.
So, I think it would be better for this doc to use the model as it is and
leave its enhancements for future discussions.

The second point is α in the draft might be too big.
During my WGLC review, I read [FHP00] once and I had similar thoughts.
But, after I re-read the paper several times, I start having a different
interpretation.

First off, it is very clear that β=0.7 is more aggressive than β =0.5 when
there's no other traffic.
As cwnd growth is linear, if there's enough long time for the data
transfer, the average cwnd for β =0.5 will be (1.0 + 0.5)/2 = 0.75 Wmax and
it will be (1.0 + 0.7)/2 = 0.85 Wmax for β=0.7.
So, it is obvious that this model does not aim for the cases where there's
no other traffic.

I believe the choice of (α,  β) in [FHP00] is designed to be more or less
fair only when it competes with (α=1.0,  β =0.5)
Hence, I think we should not apply the loss rate for no-other-traffic
situation to the formula, which can lead to α3 = 0.20
Instead, I am thinking we can think in this way..

(α=1.0,  β =0.5) model oscillates between 0.5 W1 and 1.0 W1 in a congestion
epoch.
(α=X,  β =0.7) model oscillates between 0.7 W2 and 1.0 W2 in the same
congestion epoch.
  where W1 is max window for (α=1.0,  β =0.5), W2 is max window for (α=X,
β =0.7)
When these two models have the same loss ratio, it should satisfy (1.0 +
0.5) W1 = (1.0 + 0.7) W2

Also, in one congestion epoch, (α=1.0,  β =0.5) increases 0.5 W1 while
(α=X,  β =0.7) increases 0.3 W2
Now, if we define the congestion epoch as e RTTs, e = 0.5 W1 / 1.0  and
also e = 0.3 W2 / X
This means, X should satisfy
  0.5 W1 /1.0  = 0.3 * 1.5/1.7 W1/ X
then we get X = 0.529.

This will mean if we launch (α=1.0,  β =0.5) and (α=0.529,  β =0.7) at the
same time and reduce them when the sum of their windows becomes Wmax, the
throughput of them will mostly be the same.
--
Yoshi

On Tue, Sep 7, 2021 at 5:17 AM Bob Briscoe <[e-mail]> wrote:

> Yoshi,
>
> You recently asked Markku for an explanation of his point #4.
>
> Coincidentally, a few weeks ago, I was checking the formula in RFC8312 for
> Cubic in TCP-Friendly mode (called "C-Reno" in this email for brevity), and
> I also checked back on that preliminary paper that Markku refers to. I
> think Markku has a different way of explaining the flaw in the equation,
> but below I'll try to explain what I think the problems are. See [BB]
> inline...
>
> If you don't read HTML email, I'm afraid the maths is probably going to
> look like gobbledygook.
>
> On 30/08/2021 17:33, Markku Kojo wrote:
>
> Hi Yoshi, all,
>
> On Wed, 18 Aug 2021, Yoshifumi Nishida wrote:
>
> 4. Fairness to AIMD congestion control
>
>    The equation on page 12 to derive increase factor α_cubic that
>    intends to achieve the same average window as AIMD TCP seems to
>    have its origins in a preliminary paper that states that the
>    authors do not have an explanation to the discrepancy between
>    their AIMD model and experimental results, which clearly deviate.
>    It seems to have gone unnoticed that the equation assumes equal
>    drop probability for the different values of the increase factor
>    and multiplicative decrease factor but the drop probability
>    changes when these factors change.
>
>
> [BB] The source of eqn (4) in RFC8312 is eqn (4) in [FHP00]. This paper
> has been cited 250 times but it is not peer reviewed.
>
> Equations (2) & (3) of [FHP00] that lead up to (4) use an RTT (R) as if it
> doesn't vary. But it does, because the sawteeth vary the queue. But the
> real problem is that the model (silently) assumes that R is an average RTT
> sitting part-way up the sawteeth, whatever the link rate or base RTT. This
> overlooks the fact that, if the tips of the sawteeth are always at the same
> queue depth (as in a tail-drop buffer) the average R will be higher if the
> sawteeth have smaller amplitude. This sounds picky, but the assumption on
> whether sawteeth vary about their bottom, top or middle greatly alters the
> outcome:
>
> I've taken three possible alternative assumptions, with their resulting
> additive increase factors for theoretical equality with Reno (I've given
> all the derivations at the end of this email):
> #1) All sawteeth have the same Rmin: α ~= (1/β2 - 1) / 3;        Example:
> β = 0.7; α ~= 0.35
> #2) All sawteeth have the same Rmax: α ~= 4(1-β2) / 3;
> Example: β = 0.7; α ~= 0.68
> #3) All sawteeth have the same Ravg: α ~= 3(1-β) / (1+β);        Example:
> β = 0.7; α ~= 0.53    <== used in [FHP00] and therefore in RFC8312
>
>
> Which assumption is most applicable?
> #2 models a tail-drop queue.
> #3 (or somewhere between #3 and #2) models a probabilistic AQM, such as
> PIE or RED.
> #1 is not applicable to the current Internet, but it would be if a magic
> AQM was invented that minimized the standing queue.
> I don't think any of the assumptions are good models of C-Reno's
> interaction with CoDel.
>
> So which one should draft-ietf-tcpm-rfc8312bis recommend?
> You might think #2, given one suspects the vast majority of queues are
> still tail drop...
> However, it may be 'none of the above', 'cos the paper also says it
> assumes deterministic marking; i.e. that that each flow experiences just
> one drop or mark at the top of each sawtooth. If that applies at all, it
> only applies to AQMs, not to tail-drop buffers that tend to drop more than
> one packet at a time. Nonetheless, for the purpose of rate comparison, all
> the flows will share the same bottleneck. So if flow rates are equal under
> a certain AQM, they might be equal under tail-drop too (see {Note 1} at the
> very end).
>
> I would still like to say 'none of the above' because Linux C-Reno has
> been widely used with α=1, β=0.7 for many years now without the sky
> falling. So it's questionable whether friendliness should be defined
> relative to Reno. Otherwise, from day 1, RFC8312bis will be stating the
> forlorn hope that Linux C-Reno should regress to become less aggressive
> than its former self, just to be friendly to Reno, which apparently barely
> exists any more (according to the Great TCP Congestion Control Census of
> 2019 [MSJ+19]).
>
> Nonetheless, we need to bear in mind that BSD variants of Cubic (e.g.
> Apple's) are also widely used and I believe they comply with the α for
> C-Reno defined in RFC8312.
>
> (FAQ: Why does C-Reno add <1 segment per RTT to be friendly to Reno, which
> adds 1 segment per RTT?
> C-Reno in the RFC multiplicatively decreases cwnd to 0.7*cwnd on a loss
> (multiplicative decrease factor β=0.7), whereas Reno uses β=0.5. So,
> because C-Reno reduces less, it's also meant to increase less per round so
> that the sawteeth don't reach the max window again more quickly than Reno
> would. That is, it's meant to use a smaller AI factor (α) than 1 segment.
> Using α<1 doesn't necessarily slow down Cubic's acceleration into unused
> capacity, because it can switch to a convex (hockey stick) Cubic curve once
> its slow linear increase has made the queue large enough to come out of its
> 'TCP-Friendly' mode.)
>
> Whatever, rfc8312bis certainly shouldn't refer solely to [FHP00] without
> caveats.
>
> Then we have the problem that none of the above formulae are verified
> against reality.
> Even in the [FHP00] paper, which uses assumption #3, the simulations are
> out by about 2x. The paper used β=7/8. So, from the theory, for each of the
> above three assumptions, the additive increase factor would have had to be:
> α1 = 0.10
> α2 = 0.31
> α3 = 0.20
> They used a RED AQM (which should be somewhere between assumption #2 & #3).
> But they had to configure C-Reno with α=0.4 to get close to equal flow
> rates with Reno
> Using my 'finger in the air' modification to the formulae in {Note 1}, α3
> = sqrt(0.20) = 0.44, which would have been about right (perhaps only
> coincidentally).
>
> The question over what α to put in the RFC could run and run. So here's a
> suggested path through this quagmire:
> These alternatives throw doubt on the wisdom of using assumption#3. So we
> could at least pick a new assumption for a theoretical α. Assumption#2 is
> probably the 'most correct', and also happens to give the highest value of
> α for the 'TCP-Friendly' region (α=0.68 when β=0.7).
> (By my unsubstantiated formula in {Note 1}, α2 = sqrt(0.68) = 0.82.)
>
> If we look for an empirical value of α that gives reasonably equal flow
> rates against Reno, I suspect we will get all sorts of different answers
> depending on the conditions, e.g. AQM or not; high or low multiplexing;
> high or low RTT (shared by all flows); pacing, burstiness, etc,etc. If you
> want to take this path, feel free, but if you're still on it in a year's
> time, don't say I didn't warn you.
>
>
> Derivations of the above three formulae follow at the end.
>
> The equations for the drop
>    probability / the # of packets in one congestion epoch
>    are available in the original paper and one can easily verify
>    this. Therefore, the equations used in CUBIC are not correct
>    and seem to underestimate _W_est_ for AIMD TCP, resulting in
>    moving away from AIMD-Friendly region too early. This gives
>    CUBIC unjustified advantage over AIMD TCP particularly in
>    environments with low level of statistical multiplexing. With
>    high level of multiplexing, drop probability goes higher and
>    differences in the drop probablilities tend to get small. On the
>    other hand, with such high level of competition, the theoretical
>    equations may not be that valid anymore.
>
> /Markku
>
>
> [BB] Derivations of the formulae for α under the three different
> assumptions follow. It's possible/likely that this is all already in a
> paper somewhere, but I haven't been able to find one.
> 0) Preparatory material
> Terminology:
> The subscript for C-Reno is omitted given it would be on every term.
> The recovery time of an AIMD congestion control is the average time for
> additive increase to recover the window reduction after a multiplicative
> decrease.
>
> The overall approach is:
> A) Find a formula in terms of the AI and MD factors (α & β), for the
> recovery time of C-Reno.
> B) Equate this to the recovery time of Reno, which will produce a formula
> for the additive increase factor, α.
>
> The recovery time formula for C-Reno is derived by:
> 1) summing up all the round trips in additive increase, which become
> increasingly large as the queue grows. This sum is stated in terms of the
> number of rounds per cycle, J.
> 2) Then J is found by writing two formulae;
>   2a) one for the average sawtooth decrease
>   2b) and the other for the increase.
> Then by assuming a steady state the two can be equated.
> Here goes...
>
> The following formula gives the recovery time, Tr, from one max window to
> the next, in C-Reno with:
> * additive increase (AI) factor α [segment / RTT]
> * and packet-rate r [packet / s]
> assuming that the minimum window of each sawtooth still fully utilizes the
> link.
>     Tr = ΣJ-1j=0    (Rmin + jα/r)
> where j is the index of each round, and J is the number of round trips of
> additive increase per sawtooth. The rationale for the second term is that
> the queue grows by α fractional packets in each round, and the queue delay
> per packet is 1/r (seconds per packet is reciprocal of packets per second).
> The sum of all the second terms forms an arithmetic progression, the sum of
> which is given by the well-known formula, as follows:
>          = J.Rmin + J(J-1)α/2r
>        ~= J.Rmin + J2α/2r.                                (1)
> The approximation holds as long as J>>1. In some implementations (e.g.
> Linux), cwnd increases continuously, so strictly the limits should be j=1/2
> to J-1/2 (the average half way through the first and last rounds), but
> details like this are insignificant compared to the approximation.
>
> The difference in RTT due to the additional queue delay between the top
> and bottom of the sawtooth can be stated in two ways.
> Either as the sum of all the queue delays of each additive increase:
>     (Rmax - Rmin) = Jα/r,                                (2a)
> Or by the multiplicative relationship between the max and min RTTs, by the
> definition of β: Rmin = β.Rmax.
>     (Rmax - Rmin) = (Rmin / β) - Rmin
>                                = Rmin(1-β)/β                 (2b)
> Equating (2a) and (2b):
>     J = r.Rmin(1-β) / (αβ)                                  (2)
>
> Substituting for J from (2) in (1):
>     Tr = r.Rmin2 (1-β) / (βα) + r.Rmin2 (1-β)2/(2β2α)
>          = r.Rmin2 (1-β) / (βα) * (1 + (1-β)/2β )
>          = r.Rmin2 (1-β)(1+β) / (2β2α)
>          = r.Rmin2 (1-β2) / (2β2α)                                     (3)
> And, because Rmin = β.Rmax.
>     Tr = r.Rmax2 (1-β2) / (2α)                                         (4)
>
> The well-known steady-state Reno formula from [FF99] is:
>     rReno ~= (1/Ravg) sqrt(3 / 2p)
>             (5)
> assuming deterministic marking (which is also assumed for the above model
> of C-Reno).
>
> Assuming a Reno sawtooth is approximately linear, the average RTT of a
> Reno flow,
>     Ravg ~= (Rmax + Rmin)/2
> and
>     Rmin = Rmax/2
> Therefore
>     rReno ~= (1/Rmin) sqrt(2 / 3p)
>             (6)
>     rReno ~= (1/Rmax) sqrt(8 / 3p)
>             (7)
>
> With the preparatory material done, next we take each of the three
> assumptions, one at a time.
> #1) All sawteeth have the same Rmin
> From Eqn (3):
>     Tr = r.Rmin2 (1-β2) / (2β2α)
> Total packets sent over period Tr is r.Tr during which time assume 1
> packet is lost (or marked) at the sawtooth peak {Note 1}. So loss
> probability:
>     p = 1 / r.Tr
> Substituting for Tr and rearranging :
>     r2 = 2β2α / Rmin2 (1-β2)p
>     r = (β/Rmin) sqrt(2α / (1-β2)p)
> Equating the Reno packet rate, rReno, from eqn (6) to the C-Reno packet
> rate, r, for all α and β:
>     r = rReno
>        = (1/Rmin) sqrt(2 / 3p).
> Therefore,
>     (1/Rmin) sqrt(2 / 3p) ~= (β/Rmin) sqrt(2α / (1-β2)p)
> Rearranging
>     2/3 ~= 2αβ2 / (1-β2)
>     α ~= (1-β2)/3β2
>        ~= (1/β2 - 1) / 3
>                                 (8)
> #2) All sawteeth have the same Rmax = Rmin / β
> From Eqn (4):
>     Tr = r.Rmax2 (1-β2) / (2α)
> As before, after substituting for Tr and rearranging:
>     p = 1 / r.Tr
>     r = (1/Rmax) sqrt(2α / (1-β2)p)
> Equating to rReno from eqn (7) for all α and β,
>       ~= (1/Rmax) sqrt(8 / 3p).
> Rearranging
>     8/3 ~= 2α / (1-β2)
>     α ~= 4(1-β2) / 3
>                             (9)
>
> #3) All sawteeth have the same Ravg Ravg = (Rmax + βRmax )/2
>          = Rmax (1+β)/2
> From Eqn (4):
>     Tr = r.Rmax2 (1-β2) / (2α)
>          = 2r.Ravg2 (1-β) / α(1+β)
> As before, after substituting for Tr and rearranging:
>     p = 1 / r.Tr
>     r = (1/Ravg) sqrt(α (1+β) / 2(1-β)p)
> Equating to rReno from eqn (5) for all α and β,
>       ~= (1/Ravg) sqrt(3 / 2p).
> Rearranging
>     3/2 ~= α(1+β) / 2(1-β)
>     α ~= 3(1-β) / (1+β)
>                         (10)
>
> This last formula is same as the equation used for C-Reno in RFC8312 and
> in the RFC8312bis draft.
>
>
> {Note 1}: The assumption in [FHP00] of deterministic marking is suspect
> for the Internet (meaning that the spacing between drops or marks is even
> and there is always 1 packet dropped or marked at the top of each sawtooth).
> The actual average dropped (or marked) per sawtooth will depend on whether
> the buffer uses tail-drop or an AQM, and if so which AQM. It is perhaps
> better not to assume the actual average number dropped (or marked) is
> known, but instead assume the ratio of the average drops or marks between C-Reno
> and Reno will be roughly proportional to their AI factors. This would
> modify the formula for C-Reno's drop probability to:
>     p = αReno / αC-Reno.r.Tr
> Substituting αReno = 1:
>     p = 1 / αC-Reno.r.Tr
> This would modify all the results to:
>
> #1) All sawteeth have the same Rmin: α ~= sqrt( (1/β2 - 1) / 3 );
> Example: β = 0.7; α ~= 0.59
> #2) All sawteeth have the same Rmax: α ~= sqrt( 4(1-β2) / 3 );
> Example: β = 0.7; α ~= 0.82
> #3) All sawteeth have the same Ravg: α ~= sqrt( 3(1-β) / (1+β) );
> Example: β = 0.7; α ~= 0.73
>
> However, these would then be wrong where there really is deterministic
> marking.
>
> References
> =========
> [FHP00]    Floyd, S., Handley, M., and J. Padhye, "A Comparison of
> Equation-Based and AIMD Congestion Control", May 2000,  <
> https://www.icir.org/tfrc/aimd.pdf>
>
> [MSJ+19] Ayush Mishra, Xiangpeng Sun, Atishya Jain, Sameer Pande, Raj
> Joshi, and Ben Leong. The Great Internet TCP Congestion Control Census. Proc.
> ACM on Measurement and Analysis of Computing Systems, 3(3), December 2019
>
> [FF99] Sally Floyd and Kevin Fall, "Promoting the Use of End-to-End
> Congestion Control in the Internet", IEEE/ACM ToN (1999)
>
>
> Bob
>
> --
> ________________________________________________________________
> Bob Briscoe                               http://bobbriscoe.net/
>
>