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

Bob Briscoe <ietf@bobbriscoe.net> Tue, 07 September 2021 12:17 UTC

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To: Markku Kojo <kojo=40cs.helsinki.fi@dmarc.ietf.org>, Yoshifumi Nishida <nsd.ietf@gmail.com>

Cc: "tcpm@ietf.org Extensions" <tcpm@ietf.org>, tcpm-chairs <tcpm-chairs@ietf.org>

References: <CAAK044SjMmBnO8xdn2ogWMZTcecXoET1dmZqd6Dt3WzOUi359A@mail.gmail.com> <alpine.DEB.2.21.2108300740560.5845@hp8x-60.cs.helsinki.fi>

From: Bob Briscoe <ietf@bobbriscoe.net>

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

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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 This paper has been cited 250 times but it is not peer reviewed.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> [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 Briscoe http://bobbriscoe.net/

- [tcpm] Concluding WGLC for draft-ietf-tcpm-rfc831… Yoshifumi Nishida
- Re: [tcpm] Concluding WGLC for draft-ietf-tcpm-rf… Yoshifumi Nishida
- Re: [tcpm] Concluding WGLC for draft-ietf-tcpm-rf… Lars Eggert
- Re: [tcpm] Concluding WGLC for draft-ietf-tcpm-rf… Vidhi Goel
- Re: [tcpm] Concluding WGLC for draft-ietf-tcpm-rf… Vidhi Goel
- Re: [tcpm] Concluding WGLC for draft-ietf-tcpm-rf… Lisong Xu
- Re: [tcpm] Concluding WGLC for draft-ietf-tcpm-rf… Sangtae Ha
- Re: [tcpm] Concluding WGLC for draft-ietf-tcpm-rf… Markku Kojo
- Re: [tcpm] Concluding WGLC for draft-ietf-tcpm-rf… Lars Eggert
- Re: [tcpm] Concluding WGLC for draft-ietf-tcpm-rf… Lars Eggert
- Re: [tcpm] Concluding WGLC for draft-ietf-tcpm-rf… Markku Kojo
- Re: [tcpm] Concluding WGLC for draft-ietf-tcpm-rf… Yoshifumi Nishida
- Re: [tcpm] Concluding WGLC for draft-ietf-tcpm-rf… Lars Eggert
- [tcpm] alpha_cubic (was: Concluding WGLC for draf… Bob Briscoe
- Re: [tcpm] alpha_cubic (was: Concluding WGLC for … Yoshifumi Nishida
- Re: [tcpm] Concluding WGLC for draft-ietf-tcpm-rf… Markku Kojo
- Re: [tcpm] alpha_cubic (was: Concluding WGLC for … Yoshifumi Nishida
- Re: [tcpm] alpha_cubic (was: Concluding WGLC for … Jonathan Morton
- Re: [tcpm] alpha_cubic Bob Briscoe
- Re: [tcpm] Concluding WGLC for draft-ietf-tcpm-rf… Bob Briscoe
- Re: [tcpm] alpha_cubic Yoshifumi Nishida
- Re: [tcpm] alpha_cubic Bob Briscoe
- Re: [tcpm] alpha_cubic Neal Cardwell
- Re: [tcpm] alpha_cubic Bob Briscoe
- Re: [tcpm] alpha_cubic Bob Briscoe
- Re: [tcpm] alpha_cubic Markku Kojo
- Re: [tcpm] alpha_cubic Yoshifumi Nishida
- Re: [tcpm] alpha_cubic (Issue 1) Bob Briscoe
- Re: [tcpm] alpha_cubic (Issue 1) Bob Briscoe