ON THE ANALYSIS OF PARAMETER γIN FAST TCP

Vol.27 No.4JOURNAL OF ELECTRONICS (CHINA)July 2010ON THE ANALYSIS OF PARAMETER γ IN FAST TCPILiang Wei* " Zhang Shunyi" Ning Xiangyan' Xu Sulei*""(Nanjing University of Post and Telecommunications, Nanjing 210003, China)"(Changshu Institute of Technology, Changshu 215500,0 China)Abstract The limits of parameter γ in FAST TCP are studied in this paper. A continuous time fluidflow model of the link buffer is considered to create a linear control system related to FAST TCP.Linearing the fuid flow model and window control model, the Laplace transform version of congestioncontrol system are presented. It results in a negative feedback system with open loop transfer function.With the analysis of Nyquist curve of the system, a sufficient condition on asymptotical stability ofFAST TCP congestion window related to the parameter γ is obtained. Packet level ns-2 simulationsare used to verify the theoretical claims.Key words Computer networks; Congestion control; FAST TCP; Laplace transform; NyquistCLC indexrP393DOI 10.1007/s11767-011-0327-0I. Introductionnetwork with FAST TCP is stable and always staysFAST TCP is a TCP congestion control algo-near the equilibrium state.rithm that appeared a few years ago. It adopts theUntil recently, stability analyses have beenqueuing delays as the congestion measure, and it isbased predominantly on rate-based (rather thanused for high-speed long _latency networksl". Whenwindow based) models, almost without exception,the queuing delays are used for congestion measure,on integrator link (queue) models. In these models,the users at the sources are able to avoid packetsources control their data rates explicitly and thelosses, so as to detect congestion; hence the algo-changing rate of the queuing delay is proportionalrithm with this mechanism can be widely appliedto the difference between the aggregate incomingand is scalable in the large bandwidth delaytraffic and link capacity. Typically, these resultsproduct network.show that the system is stable when round tripIn the field of network congestion control, one ofdelays do not exceed some upper bound.the important works is the dynamics of congestionExtensive experimental and several theoreticalcontrol protocols. Stability is crucial to ensure thatefforts have been devoted to the study on the sta-the system operating point is indeed the intendedbility property of FAST TCP in Ref. [3]. Localequilibrium, with the desired efficiency and fair-stability and global stability of congestion controlnessl. To fully utilize these advantages of FASTprotocols have both attracted much attention.TCP, it is really important to ensure that theLocal stability of FAST TCP is studied for the caseof a single link without network feedback delays inRef. [4. Recently, with a new accurate link model,1 Manuscript received date: November 18, 2009; revisedlocal stability of FAST TCP is analyzed for a sin-date: March 26, 2010.Supported by National High Tech Research and Devel-gle-link network in Ref. [5]. Ref. [6] studies theopment Plan (863) of China (No. 2006AA01Z232, No.instability of FAST TCP due to Round Trip Time2009AA01Z212, No.2009012202), Natural Science Found-(RTT) heterogeneity. In Refs. [7] and [8], the localation of Jiangsu Province (No. BK207603), and High-stability condition is derived in terms of not onlyTech Research Plan of Jiangsu Province (No.the parameter γ but also a.BG2007045).Communication author: Liang Wei, born in 1971, male,n FAST TCP, the parameter γ is the sourcePh.D. candidate. Nanjing University of Post and Tele-contr中国煤化工for the networkcommunications, Nanjing 210003, China; Senior Engineerperfoa Refs. [9-11], theof Changshu Institute of Technology, Changshu 215500,China.pararHCNMHG ris the pa-Email: liangmiguel@163.com.rameter for convergence speed, which is recom-Liang et al. On the Analysis of Parameter γ in FAST TCP499mended to be 0.5121. In Ref. [13], RTT is assumed asare connected by a single bottleneck link with ca-a constant, the conclusion of r condition seems topacity C. Let w。(t) denote the congestion window ofbe unreasonable. In Refs. [3] and [5], the parame-source n at time t, n∈{..*, N}. The followingter γ condition for stability is not free enough.variables are used throughout the paper. All areThis paper still studies the stability of FAST TCP.functions of timet; when written without explicitA sufficient condition on asymptotical stability oftime dependence, they denote equilibrium values.FAST TCP congestion window is obtained and ap: Queuing delay at the link;guideline on choosing this parameter γ to obtainw。: Window size of flow n;satisfactory performance is provided, as well as thex。:Arrival rate of flow n's data at the queue;ns-2 simulation results, so as to validate the theo-dn :Propagation delay for flow n;retical results.T. :RTT delay for flown, r。=d +p.The paper is organized as follows. Section IIdescribes a negative feedback system mathematicalmodel of FAST TCP. Section III focuses on sta-ODbility condition analysis. Packet level simulationssO--每are also given to verify the theorem. Section IV is .the conclusion part.sxOODwII. Analyses to F AST TCP Model1. Window -based transmission controlFig 2 The topology of snglelink multisource networkA schematic picture of the control structure forwindow- based transmission control is displayed inLet a packet sent by source n at time t, appearFig.1. The dynamics of the endpoint protocol areat the bottleneck queue at time t + Tt|. This for-represented by the three blocks: transmission con-ward delay TH models the amount of time it takestrol, window control, and congestion estimator.to travel from source n to the link, and it accountsThe system consists of an inner loop and an outerfor the constant forward latency but not queuingloop. In the outer loop, the window control adjustsdelays. The backward delay rh(t) is defined in thethe transmission window size based on the esti-same manner: it is the period from the time when amated congestion level of the network. This con-packet arrives at the link to the time when thegestion level is estimated based on the ACKs,corresponding acknowledgment is received atwhich carries implicit information in the form ofsource n. Note that the backward delay includesthe time-dependent queuing delay at the bottleneckduplicate, missing and delayed ACKs.queue. The RTT delay r。(t) seen by source n is theSanderelapsed time between the two points when a packetis sent and when the corresponding acknowledg-TrauemissioScuding. mate CNetwrork , Rociverment is received; naturally it comes T.(t)= τ(t)| Window I+r;(t). The latency of source n is denoted as d,Windowcoutroland is defined as the minimum achievable RTTEstinuateACK↓delay, ie. the RTT delay when the bottleneckCoupostionqueue is empty.ctinatorLet r,(t) be the source n sending rate. Then,the source n rate observed by the link at time tFig. 1 System view in window-based congestion controlis x(t-r), and q.(t)=p(t -) is the queuing2. Network fuid flow modelingdelay*ime t. The source中国煤化工gl4.15.In Fig. 2, the TCP connection topology of asendilFYHCNMHGsingle-link multi-source bottleneck scenario is de-= x,(t)(1)picted, in which the source and destination nodesd. +q.(t)500JOURNAL OF ELECTRONICS (CHINA), Vol.27 No.4, July 2010The bottleneck link may also carry non-win-4. Negative feedback system of FAST TCPdow-based traffic such as User Datagram ProtocolLinearing Eq. (4) around (w, 9), Combining the(UDP) traffic. Let r。(t)∈[0,c] be the rate (aver-Laplace transform version of the linear windowaged over a suitable time interval) at which non-dynamics and the communicated price (queuingwindow-based crossed-traffic is sent over the link.delay)l, it yieldsThis implies that the available bandwidth, sharedby the window based sources sending over the link,a,d,(5)u,()= -r p()-"is c-x(t). In Ref. [1], a link model is used and it isqT"based on the empirical observation. The relationFor notational simplicity, the case that the de-between the window size and the buffer size islay from the source to the link is 0 in Eq, (2) will bedescribed by Refs. [16] and [17]: .considered. Combining Eq. (5) with the frequencyw。(t-r)domain Laplace transform version of the linear=c- x。(t)(2)d + p(t)queue dynamics Eq, (2) results in a negativefeedback system with open loop transfer func-In Eq, (2), a change in the source congestiontionl20.2]windows w(t) results in a proportional change inthe queuing delay p one forward delay t' later.uo)=SH, dyne(6)3. Window model of FAST TCP8+Y%qFAST TCP is an algorithm which aims at im-This is the static link model; it will now be usedproving TCP Reno's performance especially forto analyze the stability of FAST TCP.networks with large bandwidth delay productsl@l.II，Stability AnalysisFAST sets the congestion window based on theFor given parameters, the stability degrades asqueuing delay p(t - r)u), seen by the packets. Thesending rate of FAST TCP is implicitly adjustedq→0, and L(S) will have large gain and lose morevia the congestion window mechanism. Each senderphases. In the rest of the paper, q→0 will be theupdates its window size in discrete time accordingonly case that is discussed. Then the open lootransfer function tends to betw,(k+1)=(1-r.)w,(k)(7)d。Tn8+γ,-w()+ 7.0。(3)"d, +9。(k)Now we can analyze the stability of the systemof Eq. (7). The fllwing results are validated withwhere a∈Z*，and γ are protocol parameters.packet level data generated by using the ns-2 FASTThis update is performed once every RTT. TherCP modul2.23.queuing delay q(k) is estimated by the source and1. N=1the kth estimate is denoted by q。(k)u0. The win-There is a single flow in a bottleneck linkdow algorithm operates in time scale of RTTs whileIf there is a single flow, μ= 1 in Eq. (7) and thethe estimator is such that it operates on a timestatic link model tends to bescale of packets; for the case of high bandwidth andlatency, the estimator operates at a much faster[(8)=re~"(8)T8time scale than the window algorithm. This workwill therefore ignore estimator dynamics and usewhere 8 is the operator of Laplace transform, T is(k)=q,(k) in Eq. (3).RTT delay. Next, we analyze how the protocolThe continuous time window update isl8!:parameter ry will affect the stability of Eq, (8).p.(t-τ)aTheo中国煤化工TCP model dein()=-r(d, +p.(t-1)5.()+7.+p.t-)scribefYH_link single-sourcelink,CNMHGγ>π/2, FAST(4) TCP model described by Eq. (8) over a single-linkLiang et al. On the Analysis of Parameter r in FAST TCP501single source link, is unstable.this case, γ> π/2 and its Nyquist plot is given inProof The frequency property of Eq. (8) is: .Fig.4. We can see that the Nyquist curve doesindeed encircle -1+ j0 and it predicts that theL(jw)=eresulting system is unstable. The correspondingjwTns-2 simulation results are given in Fig. 5(b) andcos8(ur)- jsin(wr)Fig. 5(c), which have shown that the congestionwindow is instable. Fig. 5(b) shows that there are a=-rsin(wT)_ cos(wT)(9)little oscillations and Fig. 5(c) show that there arewTJY-wrsustained obvious oscillations, indicating instabil-ity.Re(u(u)=-siml(o)(10a)0.8cos(wr)Im([(jw))=-r(10b)0.6If w→0*, L(jw→0叶) tends to-γ- joo;)2 t(-1,70)If w→+∞, L(jw-→+∞) tends to 0.Let m([(jw)= 0, that gives uT = (2k + 1)(π/2),-0.21/u+0+where k= 0,12...-0Then Re([(jw))= <<(-1)* /(2k + 1)(π/2))-0.6When k=0, namely T=π/2, the most left-1.5-0.5).5intersection of the Nyquist curve of L(jw) and thefig.3 Nyquist plot of the system of γ<π/2.real axis is Re(L(jw))= ->(2/π). .This crossing must be clockwise. By the Nyquiststability criterion, the system is stable if - >(2/才)>--1.Namely if γ< π/2, the model Eq. (8) is stable.If-r(2/)<-1, that is γ> π/2, the model(-1,j0)Eq. (8) is unstable.The ns-2 simulations are provided to show how-0.4+/w-0+the value of parameter r can be chosen in theobtained stability condition in various networkenvironments.-2 -1.5 -1 -0.5 0 0.5Example 1 Consider a single-link single-sourceFig. 4 Nyquist plot of the system of γ>π/2network, with the link capacity being 40 Mbit/sand each packet size being 1000 bytes. The2. N=2propagation delay is 20 ms and the bottleneckConsider two FAST flows with从=从=1/2propagation delay is 4 ms，a = 10 packets. .(corresponding to the current practice that allFirstly, it is assumed that the source controlFAST flows share the same a). Then Eq. (7) givesgain r=1.54. Then the condition of Theorem is2eP e-2satisfed, namely γ< π /2. Its Nyquist plot is given[(8)= 7(11)2[ Tj8° T28 )in Fig.3, the Nyquist curve crosses the real axis inthe right of -1+ j0. It indicates the resultingLet T2= η, where η measures the heteroge-system is stable. The corresponding ns-2 simulationneity推化r3e of value ηisresults are given in Fig. 5(a), which predicts that(1,+中国炒ator of Laplacethe congestion window is stable, and they convergetransfMHCN M H G the protocol pa-to their equilibrium points very fast.rameter γ will affect the stability of Eq. (11). TheNext, change the value ofγ to 1.65 and 1.75, infrequency characteristic of Eq. (11) is502JOURNAL OF ELECTRONICS (CHINA), Vol.27 No.4, July 2010Re(u))=-:2( sin(uT.) + sin(muT.)(12)By the Nyquist stability criterion, the systemdepicted by Eq. (11) is unstable if and only if2[ 叮ηwTL(jw)∈(-∞,-1) or -(r/2)中()>-1 for someIm(uw)=- (0(工)+ c(0)] (13)w; the system depicted by Eq. (11) is stable if2(町L(jw)4(-∞o,-1) or-(r/2)(8)<-1 for all w.We use Matlab to acquire the value of γ whichLet (u)= si() + sin(旭)(14)satisfies - (r/2)(0)<-1. For example， whenη= 5.5, the curveof中(0) and φ(0) are shown inp(E)- : co() + cos(7)(15)Fig. 6. When θ* = 1.50 gives q(0°")= 0 and中(0")reaches its maxima. In this case, the stabilityηθcondition of Eq. (11) is-(r/2)(0)<-1. We can。3.acquire stability bound r < 2.575.10当2.55(0)1.5003. 07766)虽1.50昌120.5Fig6 Curveof 中(用) and中间) when η= 5.5416(-1.0)(月) 7=1.54-1喜1614良12-2告10享8Fig.7 r= 2.575 is the stability bound when T2 = 5.5T,Meanwhile, let η vary at the range of (1, +∞),with small variable step of n, we acquire the sta-bility bound r curve by Matlab as in Fig. 8. The3121620system is stable when the valueof r is on top of theTime (间)curve. The system is unstable when the value of γ(b) 1=1.65is at the bottom of the curve. When the heteroge-neity η of two flows increases enough (over 25), thestability bound tends to be 3.s 2.51.51(5.6, 2.577)B 0.55202530Fig.8中国煤化IntRrTinthecaseof.Time (sN=2(c) r=1.80MHCNMHGFig.5 Congetion windows of FASTExample 2 Consider a network with a single bot-Liang et al. On the Analysis of Parameter γ in FAST TCP503tleneck link c = 500 m/s, carrying two flows andfsu(co(ur)dr =0(16)each pack size being 1000 bytes. The maximalwTqueuing length is 1000 packets, T2 = 5.5T.We have drawn a conclusion that the stabilitythe transfer function should satisfybound of r is 2.575. The Nyquist curve of Eq. (9)Re(L(ju))>-1(17)exactly passes (-1,0) as shown in Fig. 7, which isthe bound between stability and instability. Ns-2Most existing literature on window based con-packet level simulations in Fig. 9 and Fig. 10 showgestion controll, assumes that the sending rate isthe congestion window stability with r = 2.37 andproportional to the window size divided by thethe congestion window instability with γ= 2.65,round trip delay and may further model the queueas a simple integrator, integrating the excess rate atrespectively.the link, ie. integrator link model:o.(t-)薯1p(t)=d。+ p(t)+x.(t)-c|(18)12Combining Eq. (5) with the frequency domain10量8Laplace transform version of the linear queue dy-6tInamics Eq. (18) results in a negative feedbacksystem with open loop transfer functionTime (a)u=立n-e(19)! r8+ 7n9Fig. 9 Congestion window is stable when r=2.37 with T, =5.5T, r=2.37 in caseof N=2Consider the caser% = 7, gives哥1L(s)=-+1台8T2(20)1Slope 1/(wu2))5 t8121620(-1,j0)Time (间)Fig 10 Osillations of the congestion window when γ= 2.65 with-0.5 tτ:=5.5T; in the case of N=23. N=∞--0.50.5In reality, the link is likely to be shared by manyFig. 11 Half plane under the line that pases -1+ j0 with theflows. It is interesting to find the statistical meanslope 1/(wf)value of the stability bound for those scenarios. Wewill now consider the case of many flows with con-Define H(w) as the half plane under the line-1十i∩with olnne. 1/wa8 as showntinuously distributed RTTs. Let all T。be in thein Fig中国煤化工be writen 88.range of Tn∈sA(Tg,7), where干possibly infinite.The stability condition of Eq. (7) can be written asHCNMH Gin the case ofH(w) = {xlarg(x + 1) - arctan|))(-.o(21)504JOURNAL OF ELECTRONICS (CHINA), Vol.27 No.4, July 2010Lemma If γ<-台T。(22)4ld0六sin(wT.)γ<-(29)wT,0u-sin6Then the L(s) depicted by Eq. (20) satifes02-dθL(jw)∈H(w)(23)Note thatProof [(jw) ∈H(w) equalsa( sincosθsinθ0=0-θ一d6(30)arg(L(jw)+1)- arctan(一)∈(-π,0) (24)Eq. (29) becomesLet 8= jw, substitute Eq. (20) and note thatarg(jw + =) + arctan((25)Condition Eq. (24) can be further rewritten asγ<一-(0(31)d0argn=icos(wT,)- jsin(wr.),+ (jw+-1、jur2千](26)Consider the condition of Eq. (16), Eq. (31)becomese-2:2)which is equivalent to!dRel之cos(wT.)- jsin(wT.)γ<( sin0)dθ(32)jwr'(0+1)>0“了喝(27)T。Dividing by r> 0, gives the hypothesis of theLemma.Q.E.D.下盟。6Theorem2 if(33),日、sinθl; θ、sinθ( sin(w元)_ sin(uTo)]'9 hrowrWT.As an example, assume RTTs follow a uniformmaxlog.distribution. As units of time are arbitrary, this canbe modeled without loss of generality as1in the case of Eq. (16), the system depicted by .μ(r)={F-To,r∈(ro,F)(34)Eq. (20) is stable.Proof If there are many flows N-→o, Eq. (22)|0, otherwise :givesIn this case, Eq. (33) becomes" r) sin(wr)(28)(35)[sin0r*"Considering RTTs of those flows drawn from a中国煤化工continuous distribution, set θ= wr, and r= θ/w,MHCNMHGdT= d0/w, Eq. (28) givesLiang et al. On the Analysis of Paramneter γ in FAST TCP505It is alsopresented. It results in a negative feedback systemwith loop transfer function. Applying the Nyquist7<-(36)stable criterion to the Nyquist curve of the system,( sin(w元)_ sin(To)\the stability conditions of γ has been acquired. WewiwT。maxhave drawn Theorem 1 in the case of a single-linkw>0og -single-source. The stability bound is γ= 1.57.Next, the stability bound of r is analyzed in theAccording to Lemma, L(jw)∈H(w).cases of N=2. Finally, a sufficient condition Theo-Then the system depicted by Eq. (20) is stable,rem 2 on asymptotical stability of FAST TCPTheorem 2 is proved.Q.E.D.congestion window related to the parameter γ isAccording to Theorem 2, the stability boundobtained. While the FAST flows increase, the sta-curve of γ with many FAST flows shared a singlebility bound of parameter r increases, the stabilitybottleneck link can be drawn. By settingT/To asof the system becomes loose and the stability valuethe horizontal coordinate, and γ as the verticalis far higher than 1.57. In other words, however,coordinate, the stability bound curve of r deter-many FAST flows are in the link, the congestionmined by Theorem 2 is shown in Fig. 12. We can seecontrol system under γ< 1.57 is always stable. Asthe stability change range of γ when many flowsthe conclusion of this paper is based on the controlwith maximum ratio of i/ T。reaches 200 sharing atheory, it has been well verified by the simulationbottleneck link. With the increase of the differenceresults. The achievement of the paper will be helpfulbetween the maximum RTT and the minimumin applying FAST TCP and giving a definite guideRTT, the curve monotonically increases and itin the setting of parameter 7. In future, furtherresearch can be carried on in terms of the suggestedseems that the increase of the value of 7 becomes aparameter r when FAST TCP is widely applied ingeneral trend. When 干/To>50, the value of γreal networks.goes beyond 3. Meanwhile, Theorem 2 also indicatesthat in the case of many FAST flows shared in aReferencesbottleneck link, when the ratio of于/T。is low, the[1] M. Chen and X. Fan. Normalized queueing delay:stability upper bound ofγ will be low. When theCongestion control jointly utilizing delay and marking,ratio of彳/T。is high, the stability upper bound ofIEEE/ ACM Transactions on Networking, 17(2009)2,γ will be high.618 631.2] Salem Belhaj. VFAST TCP: An improvement ofFAST TCP. 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