Re: [tcpm] alpha_cubic

[Bob Briscoe <[e-mail]>]

Yoshi,

On 11/09/2021 22:02, Yoshifumi Nishida wrote:
> 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.

[BB] The paper gets the same answer as I got under assumption #3 (all 
sawtooth vary around a set point half-way between max and min).
I didn't assume shallow queues, so it can't be assuming shallow queues.

> 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.

[BB] I didn't use Equation-based CC at all and, at least under one of my 
three assumptions, I got the same result as the [FHP00] paper (with one 
further assumptions about how many drops there are at the top of 
different sawteeth).

I just wanted to check the outcome wouldn't be different if I used a 
model of varying RTT, 'cos I'd already recently done this maths for 
another reason - to derive the recovery time after a loss. This proved 
that, given [FHP00] got the same result as one of my cases just using 
basic AIMD sawtooth geometry, at least in this one case, the varying RTT 
doesn't make any difference.

Nonetheless, I think you're right that deriving the correct formula 
would involve deeper research. I don't think publication of the RFC 
should block on that research.

> So, I think it would be better for this doc to use the model as it is 
> and leave its enhancements for future discussions.

[BB] That's up to the WG. But I would not refer to that paper, at least 
not alone. And I don't think the draft should claim that the formula for 
Reno-friendly Cubic that it uses from that paper is correct, when it 
clearly doesn't produce the desired result, even in the paper that 
proposed it.

>
> The second point is α in the draft might be too big.

[BB] How so? The paper's own simulations (exploring a very limited 
space) conclude that α from the model is more than 2x too small.

> 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.

[BB] Correct

> So, it is obvious that this model does not aim for the cases where 
> there's no other traffic.

[BB] It is certainly obvious that comparing flow rates is not about 
single flows.

But, by the same token, does it mean anything to say a flow is 'more 
aggressive when there's no other traffic'?
You can have different cwnd for the same throughput, if the avg queue 
delay is higher and therefore RTT is higher.  But I don't think that 
says anything about aggression. If anything, aggression is about shorter 
recovery time.

Yes, the analysis I used was for each flow on its own. When they are not 
on their own, but competing in the same queue, (obviously) there cannot 
be two different average queue delays. But you can think of them as each 
progressing in parallel, each taking losses and each contributing to the 
queue. I'm not saying my analysis is the final word by any stretch of 
the imagination. However, if on their own they have different recovery 
times, then when together A will be pulling B away from its natural 
recovery time and towards A's recovery time, and vice versa.

>
> 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 α_3  = 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.

[BB] This is the same equation as in [FHP00], but with numbers not 
variables. it's also the same as the formula in the current Cubic RFC. 
And it gives the same answer you've got when you plug in the numbers. 
Because you've used the same technique to compare the geometry of the 
sawteeth.

But is there empirical evidence to support this formula, given the 
simulations in the paper itself don't support it? That's the question.

Also, I'd like to hear Markku's response to your original question to 
him. I was only trying to second-guess what he was trying to say.


Bob


>
> 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] 
> <mailto:[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 R_min : α ~= (1/β^2 - 1) / 3;   
>         Example: β = 0.7; α ~= 0.35
>     #2) All sawteeth have the same R_max : α ~= 4(1-β^2 ) / 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:
>     α_1 = 0.10
>     α_2 = 0.31
>     α_3 = 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}, α_3 = 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}, α_2 = 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, T_r , 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.
>         T_r = Σ^J-1 _j=0     (R_min + 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.R_min + J(J-1)α/2r
>            ~= J.R_min + J^2 α/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:
>         (R_max - R_min ) = Jα/r,                      (2a)
>     Or by the multiplicative relationship between the max and min
>     RTTs, by the definition of β: R_min = β.R_max .
>         (R_max - R_min ) = (R_min / β) - R_min
>                                    = R_min (1-β)/β              (2b)
>     Equating (2a) and (2b):
>         J = r.R_min (1-β) / (αβ)               (2)
>
>     Substituting for J from (2) in (1):
>         T_r = r.R_min ^2 (1-β) / (βα) + r.R_min ^2 (1-β)^2 /(2β^2 α)
>              = r.R_min ^2 (1-β) / (βα) * (1 + (1-β)/2β )
>              = r.R_min ^2 (1-β)(1+β) / (2β^2 α)
>              = r.R_min ^2 (1-β^2 ) / (2β^2 α) (3)
>     And, because R_min = β.R_max .
>         T_r = r.R_max ^2 (1-β^2 ) / (2α)                           
>                  (4)
>
>     The well-known steady-state Reno formula from [FF99] is:
>         r_Reno ~= (1/R_avg ) 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,
>         R_avg ~= (R_max + R_min )/2
>     and
>         R_min = R_max /2
>     Therefore
>         r_Reno ~= (1/R_min ) sqrt(2 / 3p)                           
>                             (6)
>         r_Reno ~= (1/R_max ) 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 R_min
>
>     From Eqn (3):
>         T_r = r.R_min ^2 (1-β^2 ) / (2β^2 α)
>     Total packets sent over period T_r is r.T_r during which time
>     assume 1 packet is lost (or marked) at the sawtooth peak {Note 1}.
>     So loss probability:
>         p = 1 / r.T_r
>     Substituting for T_r and rearranging :
>         r^2 = 2β^2 α / R_min ^2 (1-β^2 )p
>         r = (β/R_min ) sqrt(2α / (1-β^2 )p)
>     Equating the Reno packet rate, r_Reno , from eqn (6) to the C-Reno
>     packet rate, r, for all α and β:
>         r = r_Reno
>            = (1/R_min ) sqrt(2 / 3p).
>     Therefore,
>         (1/R_min ) sqrt(2 / 3p) ~= (β/R_min ) sqrt(2α / (1-β^2 )p)
>     Rearranging
>         2/3 ~= 2αβ^2 / (1-β^2 )
>         α ~= (1-β^2 )/3β^2
>            ~= (1/β^2 - 1) / 3 (8)
>
>
>           #2) All sawteeth have the same R_max = R_min / β
>
>     From Eqn (4):
>         T_r = r.R_max ^2 (1-β^2 ) / (2α)
>     As before, after substituting for Tr and rearranging:
>         p = 1 / r.T_r
>         r = (1/R_max ) sqrt(2α / (1-β^2 )p)
>     Equating to r_Reno from eqn (7) for all α and β,
>           ~= (1/R_max ) sqrt(8 / 3p).
>     Rearranging
>         8/3 ~= 2α / (1-β^2 )
>         α ~= 4(1-β^2 ) / 3                                            
>             (9)
>
>
>           #3) All sawteeth have the same R_avg
>
>     R_avg = (R_max + βR_max )/2
>              = R_max (1+β)/2
>     From Eqn (4):
>         T_r = r.R_max ^2 (1-β^2 ) / (2α)
>              = 2r.R_avg ^2 (1-β) / α(1+β)
>     As before, after substituting for Tr and rearranging:
>         p = 1 / r.T_r
>         r = (1/R_avg ) sqrt(α (1+β) / 2(1-β)p)
>     Equating to r_Reno from eqn (5) for all α and β,
>           ~= (1/R_avg ) 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 = α_Reno / α_C-Reno .r.T_r
>     Substituting α_Reno = 1:
>     p = 1 / α_C-Reno .r.T_r
>     This would modify all the results to:
>
>     #1) All sawteeth have the same R_min : α ~= sqrt( (1/β^2 - 1) / 3
>     );        Example: β = 0.7; α ~= 0.59
>     #2) All sawteeth have the same R_max : α ~= sqrt( 4(1-β^2 ) / 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
>     <https://www.icir.org/tfrc/aimd.pdf>>
>
>     [MSJ+19] AyushMishra,XiangpengSun,AtishyaJain,
>     SameerPande,RajJoshi,andBenLeong.The
>     GreatInternetTCPCongestionControlCensus. 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 Briscoehttp://bobbriscoe.net/  <http://bobbriscoe.net/>
>

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