Author name used in this publication: Chi K. TseRefereed conference paper2007-2008 > Academic research: refereed > Refereed conference paperVersion of RecordPublishe

Chen, X

Trajković, L

Tse, CKM

Wong, SC

Author name used in this publication: Chi K. TseRefereed conference paper2007-2008 > Academic research: refereed > Refereed conference paperVersion of RecordPublishedVoR allowe

PolyU Institutional Repository

Stability Study of the TCP-RED System UsingDetrended Fluctuation AnalysisXi Chen∗, Siu-Chung Wong∗, Chi K. Tse∗, and Ljiljana Trajkovic´†∗The Hong Kong Polytechnic University, Hong Kong†Simon Fraser University, Vancouver, CanadaAbstract— It has been observed that the TCP-RED systemmay exhibit instability and oscillatory behavior. Control methodsproposed in the past have been based on the analytical modelsthat rely on statistical measurements of network parameters. Inthis paper, we apply the detrended fluctuation analysis (DFA)method to analyze stability of the TCP-RED system. The DFA hasbeen used for detecting long-range correlations in seemingly non-stationary noisy signals. The key indicator emanating from DFAis known as the scaling exponent. By examining the variationsof the DFA scaling exponent when varying system parameters,we quantify the stability of the TCP-RED system in terms ofsystem’s characteristics.I. INTRODUCTIONInternet applications, such as World Wide Web, ﬁle transfer,and email, are transmitted using the Transmission Control Pro-tocol (TCP) [1]–[3]. TCP allows multiple information sourcesto compete for a fair share of physical bandwidth allocatedbetween two end-points of a connection. Active Queue Man-agement (AQM) in bottleneck gateways plays an importantrole in controlling the trafﬁc ﬂow when the number of sourcesincreases. The Random Early Detection (RED) [4] gatewaywas introduced to provide early feedback to the TCP sendersby marking (or dropping) packets before buffers become fullthus preventing large number of packets from being lost.However, it has been reported that TCP gateways employingRED may exhibit instability and oscillatory behavior [5]–[8]that degrade the transmission performance.The TCP-RED algorithms have been implemented in thens-2 network simulator [9]. The ns-2 simulator enables analy-sis and veriﬁcation of communication networks in a controlledenvironment. A number of analytical models have been de-veloped to characterize the TCP-RED system and analyze itsoscillatory behavior [10]–[14]. The stability conditions derivedusing these analytical models often depend on statistical mea-surements of various parameters and their upper and/or lowerbounds. These measured parameters may be non-stationary.Hence, a universal and reliable method, robust and simple forimplementation, would facilitate characterizing the TCP-REDsystem for various scenarios of network connections.Detrended ﬂuctuation analysis (DFA) is a method for de-tecting long-range power-law correlations in seemingly non-stationary noisy/randomized signals. It has been widely usedin computer science, biodynamics, bioinformatics, economics,and meteorology [15]–[18]. The outcome is usually givenin terms of a parameter called power-law scaling exponent.Given a time series (waveform), a chosen statistical variableis observed for various time scales. When plotted over varioustime scales, a typically seen straight line on a log-log scaleimplies a long-range power-law correlation for the observedvariable. Its slope is equal to the scaling exponent. Thisline may display a deﬂection (crossover) at a certain timescale where the slope abruptly changes. The interpretations ofscaling exponents and the crossovers are system dependent.In this paper, we apply the DFA method to the individualand aggregated trafﬁc ﬂows in a bottleneck RED gateway ofa TCP-RED system which may exhibit a change of stabilityunder variation of system parameters. We analyze the changein scaling exponents and the locations of the crossovers.The paper is organized as follows. In Section II, we describethe TCP-RED system. In Section III, we illustrate the DFAmethod and analyze the long-range power-law correlationproperty of the queue length. A physical interpretation of theobtained scaling exponent values in relation to the waveformsof the queue length is provided in Section IV. We concludewith Section V.II. OVERVIEW OF TCP-RED ALGORITHMSA. Congestion Control AlgorithmWe assume a window based ﬂow control between the twoend-points of a TCP connection. The TCP sender organizesa stream of data into a sequence of windows of packets.Each window contains w packets. When the TCP receives apacket from the sender, it returns an acknowledgment packet.Upon receiving the ﬁrst acknowledgment packet from the lastwindow of data sent, the TCP sender begins sending packetswithin the next window. The duration between sending apacket and receiving the acknowledgment is called the round-trip-time (RTT). Therefore, a TCP sender may send w packetsfor each RTT. If the window size w is adaptively selected toreach its maximum share of throughput according to the net-work conditions, the TCP communication eventually reachesits steady-state as a self-regulated RTT clocked system. SeveralTCP algorithms have been employed for adjusting the windowsize according to network conditions (Tahoe [1], Reno [2],Vegas [3], and their variants). Both TCP Tahoe and TCP Renoemploy an Additive Increase Multiplicative Decrease (AIMD)algorithm for congestion avoidance. TCP Vegas uses a moreproactive approach for controlling window size. In this study,978-1-4244-1684-4/08/$25.00 ©2008 IEEE 324we consider TCP Reno, being the most widely implementedversion of TCP.The TCP congestion control mechanism employs four algo-rithms: slow start, congestion avoidance, fast retransmit, andfast recovery. TCP uses slow start to detect network bandwidthimmediately following connection establishment or timeout.The algorithm starts with a window size w of one or twopackets that doubles for each RTT when acknowledgments arereceived. When w exceeds a threshold, the algorithm enters thecongestion avoidance phase. In a typical congestion avoidancephase, the window size w increases linearly by 1/w for eachreturned acknowledgment (additive increase). Thus, after awindow of w packets is transmitted, w will be increased by 1within one RTT. Window size w is reduced to half its currentvalue (multiplicative decrease) if there is an indication ofcongestion (the receipt of three duplicated acknowledgmentsof a packet or the presence of an Explicit Congestion Notiﬁ-cation (ECN) header information received from a gateway andforwarded in the acknowledgment sent by the receiver). Thewindow then follows the additive increase algorithm. Whenthree duplicated acknowledgments of a packet are received,the TCP sender enters the fast retransmit phase where lostpackets are retransmitted without waiting for timeout. Thefast recovery algorithm then retransmits all lost packets untila non-duplicate acknowledgment is received. The congestionavoidance phase follows, where the TCP sender attempts tooptimize its fair share of the link bandwidth connecting thetwo TCP end-points. If at least one link on this communicationpath becomes the bottleneck, an AQM gateway regulates thetrafﬁc ﬂow in this bottleneck link at its maximum capacity.B. The RED AlgorithmThe RED mechanism calculates a weighted moving averagexk of the current queue size qk as:xk = (1− α)xk−1 + αqk, (1)where α ∈ (0, 1) is the ﬁlter resolution. A probability pk iscalculated based on the current value of xk:pb =⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩0 0 ≤ xk < Xminxk −XminXmax −Xminpmax Xmin ≤ xk ≤ Xmaxpmax+ 1−pmaxXmax(xk− Xmax) Xmax < xk ≤ 2Xmax1 2Xmax ≤ xk ≤ B(2)pk =pb1− cmpb, (3)where Xmin and Xmax are minimum and maximum thresholdsof xk, respectively; B is the buffer size; pmax is the maximumthreshold of pk; and cm is the number of packets that arrivedafter the last marking. Each incoming packet is marked (ordropped) with probability pk. The target-queue-length q0can, therefore, be maintained by properly choosing systemparameters, such as Xmin, Xmax, and pmax.S1S2SiSNG1 G2D1D2DiDNFig. 1. A system of N TCP ﬂows, from Si to Di, where i = 1, 2, · · · , N ,passing through a common bottleneck link between G1 and G2.TABLE Ins-2 SIMULATION PARAMETERSns-2 parameter valueN 170C/N 1.5 MbpsRo 12 msro 19.46 msXmin 256 packetsq0 384 packetsXmax 512 packetsBottleneck delay 6 msRED redwait falseTCP ECN 1C. The TCP-RED SystemWe consider a system of N TCP ﬂows passing througha bottleneck RED gateway shown in Fig. 1 [6], [8]. Weassume that the system operates in its desired stable condition,satisfying a designed objective [4]. Without loss of generality,the system is assumed ECN capable. The N long-lived TCPconnections are controlled by the ECN markings of the REDgateway, which aims to maintain an even distribution of thebandwidth in the bottleneck link among the N TCP ﬂows.The average queue size can be maintained at the targetqueue level (Xmin + Xmax)/2. The system can be analyzedusing small-signal linearization and perturbation for the cal-culation of instability boundaries [10].The nonlinear dynamics of the aggregated ﬂow at the REDgateway was analyzed using the ns-2 network simulator andthe parameters shown in Table I. As the RED ﬁlter resolutionα varies, the system crosses the boundary of stability. Stableand unstable RED queue length waveforms of the TCP-REDsystem for two values of α are shown in Fig. 2 [10]. Thewaveforms observed over various time scales exhibit self-similarity. In this paper, we analyze its effect on the queuestability.III. LONG-RANGE POWER LAW CORRELATIONS IN TCPFLOWSA. Self-Similarity and DFA MethodDetrended fluctuation analysis (DFA) method [17], [18] isa popular approach to analyze self-similarity and long-rangedependence (LRD) properties of signals.325600 650 700200300400500600(a)600 650 7000200400600(b)600 601 602 603 604 605300350400450(c)Intantaneous queue length (packet)Time (second)600 601 602 603 604 6050200400600(d)600 600.05 600.1 600.15 600.2300350400(e)600 600.05 600.1 600.15 600.20200400600(f)Fig. 2. Simulated RED queue length waveforms using ns-2 simulator forﬁlter resolutions α = 0.0001 and α = 0.0008 illustrate stable waveforms (a),(c), and (e) and unstable waveforms (b), (d), and (f), respectively. Figures (c)and (e) are enlarged views of (a) while (d) and (f) are enlarged views of (b).There are 170 TCP connections. Each connection shares a ﬁxed bandwidthof 1.5 Mbps in the bottleneck link.Consider a signal s(i), where i = 1, · · · , L, and L is thelength of the signal. An integrated time series y(i) of s(i) isobtained as:y(i) =i∑j=1(s(j)− s) , (4)where s is the mean of s(j) given as:s =1LL∑j=1s(j). (5)The integrated time series y(i) is divided into m bins of equallength l, where ml = L. A least-squares ﬁtted line ykl(i)of y(i), also called the local trend of y(i), is calculated forthe kth bin of length l, where k = 1, · · · ,m. The series y(i)is detrended by subtracting the local trend ykl(i), for i =(k − 1)l + 1, · · · , kl, as:Yl(i) = y(i)− ykl(i). (6)For the kth bin of length l, the root mean square (rms)ﬂuctuation for the integrated and detrended time series iscalculated as:fk(l) =√√√√1lkl∑j=(k−1)l+1[Yl(j)]2. (7)Thus, the averaged rms ﬂuctuation for the entire time series0 1 2 3 4 5 6 70123456log10(l)log 10(F(l))α=1e?005α=0.0001α=0.0004α=0.0005α=0.0006α=0.0008α= 0.001α=0.0015α= 0.002α=  0.01α=   0.1β1 = 2.0β1 = 1.5β2 = 0.1β2 = 0.2β3 = 0.5β3 = 0.5region 1 region 2 region 3Fig. 3. DFA scaling functions obtained from RED instantaneous queue lengthfor various choices of α.is given as:F (l) =√√√√ 1mm∑k=1f2k (l). (8)This computation is repeated for all time scales l to obtainthe behavior of F (l). Scale invariant signals with power-lawcorrelations possess a power-law relationship between the rmsﬂuctuation function F (l) and the scale l, given by::F (l) ∼ lβ . (9)If logF (l) increases linearly with log l, the slope of the linelogF (l) versus log l gives the scaling exponent β. F (l) is alsocalled the scaling function. We analyze the scale exponent βfor the TCP-RED system.B. Scaling Exponent and StabilityFig. 3 illustrates application of the described DFA methodto the waveforms of the queue length in the TCP-RED systemwith parameters shown in Table I. The instantaneous queuelength in the RED gateway is recorded every 1 millisecondover the 5,000 seconds interval. The scaling exponent (slopeof the line) changes with varying α. For each DFA curve, weobserve two deﬂections (crossovers) and three linear regions.We thus obtain three scaling exponents for each DFA curveover the entire range of bin size l, labeled β1, β2, and β3 forregions 1, 2, and 3, respectively. A plot of exponent β1 vs.the RED ﬁlter resolution α is shown in Fig. 4. Also shownare the stability boundaries for the TCP-RED system [10]. Thevalue of β1, which varies between 1.5 and 2.0, provides a clearindication of the stability of the TCP-RED system. Values ofβ1 below ∼1.85 correspond to stable region and above ∼1.85correspond to unstable region of operation.32610?5 10?4 10?3 10?2 10?11.51.61.71.81.922.12.2αβunstable stablestableFig. 4. DFA exponents in region 1 of Fig. 3 for varying α.0.5 1 1.5 2 2.5 3 3.5 4?1.5?1?0.500.511.522.533.5log10(l)log 10(F(l))DFA of sawtoothDFA of sinusoidDFA of sinusoid+sawtoothβ = 1.5β = 2.0β = 1.75Fig. 5. DFA method applied to stationary signals.IV. INTERPRETATION FROM A WAVEFORM VIEWPOINTIn the analysis of time series, the DFA exponent value al-lows a consistency characterization of seemingly nonstationarytime series [18]. Two time series are DFA similar if theyhave nearly equal DFA exponent values. Hence, if a stationarytime series that is DFA similar to the RED queue lengthtime series could be identiﬁed, the two would have consistentcharacteristics.The unstable RED queue length waveform shownin Fig. 2(e) resembles a sinusoidal wave. The DFA curvefor a sinusoidal waveform with scaling exponent 2 is shownin Fig. 5. The crossover depends on the frequency of thesinusoidal wave. Also shown in Fig. 5 is the DFA curvefor a sawtooth waveform with a scaling exponent 1.5. Sucha sawtooth waveform is the native waveform of the stableAIMD congestion algorithm. DFA exponent values between1.5 and 2.0 may be generated by combining “sinusoid” and“sawtooth”. For example, a DFA exponent value 1.75 maybe obtained by combining “sinusoid” and “sawtooth” with anamplitude ratio of 2:1, as shown in Fig. 5.V. CONCLUSIONBased on the data collected from the ns-2 simulations, long-range power-law correlations of the queue length waveforms inthe RED gateway have been studied using the DFA method.The results show that the scaling exponent varies with thestability of the RED gateway. 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Stability study of the TCP-RED system using detrended fluctuation analysis

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