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T R R ECHNICAL ESEARCH EPORT Nonlinear Instabilities in TCP-RED by Priya Ranjan and Eyad H. Abed TR 2001-37 I R INSTITUTE FOR SYSTEMS RESEARCH ISR develops, applies and teaches advanced methodologies of design and analysis to solve complex, hierarchical, heterogeneous and dynamic problems of engineering technology and systems for industry and government. ISR is a permanent institute of the University of Maryland, within the Glenn L. Martin Institute of Technol- ogy/A. James Clark School of Engineering. It is a National Science Foundation Engineering Research Center. Web site http://www.isr.umd.edu Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. 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It is argued similar model has been applied to the dynamics of choke thatbysampling thestatespaceatcertaininstants, thedy- packetsinaLANtoexplainsynchronizationandsustained namicsofthesystemcanbedescribedexplicitlyasadiscrete oscillations [16]. These models indeed explain qualitative timefeedbackcontrolsystem.Thissystemisusedtoanalyze changes intheoperation ofanetworkorthatofanetwork the operating point of TCP-RED and its stability with re- component asparameters cross critical values. Incontrast specttovariouscontrollerandsystemparameters.Withthe tothedeterministicsettingof[15]and[16],multi-stability help of bifurcation diagrams, it is numerically shown that non-trivial(notduetothediscontinuityinthesystemorthe or emergence of pseudo-stable states has been reported in control law) instabilities in the system are possible due to a stochastic setting in [14]. The paper discusses the qual- the presence of a strong nonlinearity in the characteristics itative changes in the stochastic behavior of the network of TCP throughputofa sender as a functionofdropprob- due to parameter change, which may lead to degradation ability at the gateway. Some of the bifurcations observed innetworkperformance. inthe systemaretheperiod-doubling sequence andborder collisions leading to a change in the system periodicity and There have been several attempts to deal with conges- chaos. Analyticaltechniquesareprovidedtohelpintheun- tioninTCPwhichisthemostpopularnetworkmechanism derstandingofthiskindofanomalousbehavior. Anexplicit fordatatransfer. Themostimportant schemetoavoid im- stabilityconditionintermsofdifferentparametersisgiven. pending congestion was published in [2] and is known as Keywords—Congestion,computernetworks,chaos,bifur- random early drop, or RED. The basic idea of RED is to cation,control sense impending congestion before it happens and try to give feedback to the sender by dropping its packets. The I. INTRODUCTION dropping probability is the control administered by the gateways once they detect queue build-up beyond a cer- Computer networks are highly complicated systems, tain threshold. This scheme involves three parameters: 1) both in their temporal and spatial behavior [1]. Although they have traditionally been modeled and analyzed using pmax, 2)qmax, and 3) qmin that need tobe selected. (The meaningsoftheseparameterswillbeidentifiedinthenext stochastic methods, there have recently been several pa- section.) Mostoftherulesforsettingtheseparametersare pers that use deterministic nonlinear modeling and analy- empirical, and come from networking experience. These sis(e.g.,[6],[7],[8],[16],[14],[5]). rules have been evolving as the effects of controller pa- Inthispaper,westudyamodifieddeterministicdynam- rameters is not very clearly understood. There are papers ical model of a simple computer network running TCP at discouraging implementation of RED (e.g., [9]), arguing the sender end and implementing RED at the router end. that there is insufficient consensus on how to select con- The basic model that we consider was proposed recently troller parameter values, and that REDdoes not provide a by Firoiu and Borden [5]. We modify their model with a drasticimprovementinperformance. simplerTCPthroughputfunction[3],[4]tofacilitateanal- ysis. The calculations we give here show that the model Initially, there was very little in the way of mathemat- exhibits a rich variety of bifurcation behavior leading to ical modeling of TCP-RED. However, with the recent ef- chaotic behavior of the computer network. The bifurca- forts toward modeling TCP throughput for a transmission tions occur as control parameters are slowly varied, mov- line with a packet drop probability [5], [6], [7], [8], [4], ing the dynamics from a stable fixed point to oscillatory severalpapershavediscussedTCP-REDintheframework behavior andfinallytoachaoticstate. offeedback control systems. Mostofthemodels used are A glimpse at the history of network congestion con- continuous-time and the analysis uses basic control the- trol reveals significant attempts to control congestion in oretic results. The biggest problem with the continuous- the general network and telephony literature. Conges- time models is their inability to reflect delay, which is tionandsynchronization intandemtelephonequeueshave prominentinnetworksandcanbeverysignificantforlarge trunks [7], [8]. Continuous-time models with variable TheauthorsarewiththeDepartmentofElectricalandComputerEn- (statedependent)delayarehardtoanalyze[8]. Theanaly- gineeringandtheInstituteforSystemsResearch,UniversityofMary- land,CollegePark,MD20742USA.Email:[email protected]. sisreportedonthesemodelsdealsmostlywiththestability ISRTECHNICALREPORT,2002 2 of fixed points and limit cycles under different parameter whichisthecoreofthepaper. Section5triestoexplainthe settings. For the first time, chaotic behavior of TCP has different nonlinear phenomena we have observed in our been reported in[13]. Theevidence forthis irregularity is modelsandtrytomakeaconnectionwithchaoticscenario. mostlyexploredbysimulations. Sometheoreticalworkon Finally, in section 6 we discuss the results in networking flowsynchronizationinTCPhasbeenreportedin[4],[17], context. but one of the very important issues which currently isn’t wellunderstoodishowdoesasmoothlyoperatingnetwork II. TCP-RED: FEEDBACK SYSTEM MODELING transitionintochaos. Toborrowdynamicalsystemstermi- Acomputer network implementing TCP-REDisessen- nology,theroutetochaosstartingfromastablefixedpoint tially a feedback loop where senders adjust their sending isnotwell-studied. rate based on the feedback they receive from their near- Inthis work, adiscrete-time mapwillbeused tomodel est routers in the form of dropped packets. Routers on theTCP-REDinteraction. Adynamical systemsapproach the other hand implement a control policy which can be willbeusedtoexplain thelossofstability, bifurcation be- either drop tail or RED[2]. There have been different ap- havior,androutestochaosinTCP-REDnetworks. Wewill proaches tomodelthedynamicsofTCP-REDandvarious use bifurcation-theoretic ideas to explain nontrivial peri- control schemes have been proposed [5], [6], [7], [8], [4] odic behavior of the system. The appearance of bifurca- not only to control the system but to also enhance its dy- tion and chaos should not be surprising, considering that namicperformance. Wecloselyfollowtheapproachtaken the system response is nonlinear especially during heavy in[5]withamodifiedTCPthroughput formula. loadconditions. Wewillshowtheperformanceofthesys- Senders temasafunctionofvariouscontrolandsystemparameters ingeneralandtrytoexplaintheseirregularbehaviors with flow(1) r s,1 thehelpofbifurcation diagrams. flow(2) r inO[5u]rcwaonrkbebevgiienwsebdyarseaalizfiirnsgt-othrdaterth(eramthoedretlhparnopthoisredd- ........ s,2 n1 S jrlt,,jc(p) _ n2 ........ order) discrete nonlinear model. Our replacement of the ave. queue size,q flow(n) n TCP throughput function of [5] with a simpler version drop rate, pn makes the analysis feasible. However, symbolic calcula- control tionscouldbeusedtoallowtreatmentofthemorecomplex to senders throughput function of [5]. The advantage of the current Fig.1. SimplifiedNetworkDiagram work is that the calculations are simple enough that the resultsareeasilyunderstood. Each flow at a router sends packets with rate rs;i. The We borrow the model proposed by Firiou et al. [5] and sending rates of all n flows combine at the buffer of link usethewellknownformulaforTCPthroughput proposed l and generate a queue of size q which is limited by its by many others including [3], [4]. The motivation behind buffer size B. The controller at the router drops packets not using Firiou’s forumla for TCPthroughput is itscom- with probability p which is a function of average queue plexity. Complex operations like inverse of a function in sizeq. Forithflowlettheforwarding rateattherouter be differentparameters,whichareneededtoconnecttheTCP rt;iwhichisthesameasrs;isansdroppedpackets. Whena to the control mechanism RED, demand simplicity in the sendernoticesthatitspacketsarebeingdropped,itadjusts TCPthroughput formulation. Thisseems to bethe reason itssendingratebasedonthedropprobabilitypitobserves. why Firiou et al. postponed the study of their proposed This makes a control system with sender’s rate as con- map [5]. Although this TCP-RED formulation may not trol variable with the controller sitting at the router which betheexactrepresentationofthecomplicatedmechanism, issuesthefeedbacksignalintheformofadropprobability. it does give a qualitative handle on its dynamics and en- The aim of this control system is to keep the cumulative hances our understanding of chaos and other instabilities. throughput beloworequaltothelink’scapacityc: We hope that this understanding will lead to monitor the network congestion better and help us in formulating ro- (cid:6)n r (cid:20) c j=1 t;j bustbutsimplifiedcontrolmechanisms. Thispaperisorganized asfollows. InSection2,wede- Let’s assume for simplicity that TCP flows have a long scribe the TCP-REDmechanism incontrol system frame- duration and that their number n does not change, then work. Section 3 contains the discrete map of TCP-RED the throughput of each TCP flow follows the steady state mechanism. Section 4 deals with the stability of this map modelderivedin[3],[4]. ISRTECHNICALREPORT,2002 3 Sender Gateway q ( T(p ,R) n A(q ) max(B; c (T−1(p; c)−R )) : p (cid:20) p G(p) = M R n 0 0 n n 0 : otherwise _ (2) _ q Where H(q ) n p n n Ro = propagation andtransmission timeand RED Module p = T−1(c=n;R ) o p 0 Fig.2. Feedbackcontrolsystemcorrespondingtothenetwork shown TR−1(c=n;R0) denotes the inverse of T(p;R) in R, Tp−1(c=n;R0)denotestheinverseofT(p;R)inpandp0is maximumprobabilityforwhichthesystemisfullyutilized i.e. for p (cid:21) p0 senders will have their rates too small to keepthelinkfullyutilized. ForT(p;R)definedbyeq.1. r = T(p;;R ) t;i i ! 2 Mk M k p0 = R c (3) T(p;R) = R pp (1) ( 0n G(p) = max(B; Mc (ncMpkp −R0) ; ifp (cid:20) p0 (4) where 0 ; otherwise REDcontrol lawcanbeexpressed asfollows: T = Throughput ofaTCPflow(inbits/sec) M = MaximumSegmentSizeorPacketSize p = H(qe) 8 R = RoundTripTime >< 0 ; 0 (cid:20)qe < qmin p k = constant whichvariesbetween1and 8=3[3] = >: qmqea−x−qmqmininpmax ; qmin (cid:20) qe < qmax (5) 1 ; q (cid:20) q (cid:20) B p = probability ofpacketloss max e where qe is the exponential weighted moving average of Tosimplifymattersevenfurtherlet’sassumethatallflows queue size, qmin, qmax, pmax are configurable RED pa- areuniform ortheyallhavesameroundtriptimeR,same rameters,andB isbuffersize. maximum segment or packet size M and maximum congestion window size advertised by TCP’s receiver Wmax III. DISCRETE MODEL FOR TCP-RED islargeenough tonotaffectT(p;R). Thisimplies It is argued in [5] that TCP adjusts its sending rate de- pending on whether it has sensed that packets are dis- rt;i(p;R) = rt;j(p;R); 1(cid:20) i;j (cid:20) nandhence c carded. Hence, this process can be modeled as a strobo- (cid:20) scopic map where the instant of observation is one RTT n orreturntriptime. Thistechniquehasbeenutilizedbefore So this assumption enables us to reduce the n-flow sys- fordifferentclockedsystemsinpowerelectronicsformod- tem to a single flow system with feedback although it is eling the dynamics of power converters. Following simi- important to keep in mind that feedback is based on the lararguments itseemsreasonable tomodelTCP-REDdy- sending rateofalltheflowssincetherouterhasnowayto namicsasadiscrete map. Although onewould prefer that differentiate betweenthem,atleastinthissetup. the sampling interval be regular, there are models where Todefinethiscontrolsystemmathematically, wemodel the dynamics is sampled at irregular intervals and the re- the queue as a function of control variable q = G(p), sultingmapsareknownas“impactmaps”[11]. which actslike aplant incontrol system literature. Toan- Let pk be the drop probability at tk. At time tk+1 = alyze this control system we also need the control func- tk +RTT the sender observes drop rate pk and in an av- tionp = H(q)implemented atthegateways. Thiscontrol eragesense, adjust itssending rate. Thisinturnforcesthe functionH isgivenbythepolicyimplementedatthedrop buffer to its new state qk+1 = G(pk) following the queue module,suchasDrop-TailorRED[2]. Nowfollowingthe law in eq. 4. The REDmodule now computes a new esti- proceduresuggestedin[5]wecandefinetheplantfunction mate of queue size qe;k+1 = A(qe;k;qk+1), following the G(p)asfollows: exponential weightedmovingaverage: ISRTECHNICALREPORT,2002 4 600 A(qe;k;qk+1)= (1−w)qe;k +wqk+1 (6) where w is the weight used for averaging. After comput- 500 ing q , the RED module adjusts it dropping rate to e;k+1 pk+1 = H(qe;k+1) given by its “feedback control law” 400 in eq. 5. Thiscompletes the description ofa discrete time > − dynamical systemmodeledasfollows: q) e( e siz300 u e u Q q = G(p ) k+1 k 200 q = A(q ;q ) e;k+1 e;k k+1 pk+1 = H(qe;k+1) (7) 100 From eq. 7 one can derive a simple one dimensional re- currence equation since map G(:) and map H(:) has no 0 0 0.01 0.02 0.03 0.04 0.05 0.06 dynamics. The only dynamics that comes into the picture Drop prob.(p)−> is in map A(:;:). After substitution one can easily derive Fig.3. FeedbacklawH(q )inredandqueuelawG(p)inblue e;k the following equation for exponential weighted moving averageforqueuelengthattimetk+1: qe;k+1 = (1−w)qe;k +wG(H(qe;k)) (8) nk R c q = (1−w)q +w(r − 0 ) e;k+1 e;k M Below, we illustrate some interesting dynamical behavior pmax(qe;k−qmin) (qmax−qmin) of eq. 8. This equation is rather simple in most of its do- := f(q ;(cid:26)) (11) mainofdefinition. e;k We know that G(:) is identically 0, 8 p (cid:21) p0. So we where(cid:26)summarizestheparameters inthesystem. can find corresponding value b1 of qe;k such that for any qe;k (cid:21) b1, G(:) is identically 0 if we assume a monotone 60 feedback law. ( 58 Second Return b1 = p0(qmpamxa−xqmin) +qmin ; ifpmax (cid:21) p0 (9) Map qmax ; otherwise 56 First Return Map > Thisgivesanexplicitformulaformapineq.88qe;k (cid:21) b1: −+2 n q54 q = (1−w)q , 1 e;k+1 e;k + n q Now consider the other boundary value b2 of qe;k such 52 that 8 qe;k (cid:20) b2 we have G(:) = B or buffer is always full. This value can becomputed from eq. 4 and eq. 5. b2 50 isgivenby: 45oLine (cid:18) (cid:19) 2 48 nk 48 49 50 51 q−> 52 53 54 55 B+R0c n b2 = p M (qmax −qmin)+qmin (10) max Fig.4. 1.Redcurveshowsfirstreturnmap,2.bluecurveshows Thisgivesanexplicitformulaformapineq.88q (cid:20) secondreturnmap, and 3. greenline is 45o line whose in- e;k b2: tersectiondenotesthefixedpointsofthemap q = (1−w)q +wB e;k+1 e;k We remark that solving eq. 11 leads to a third degree It is clear that most of the interesting dynamics happens polynomial in fixed point q(cid:3) which interestingly does not e for b2 (cid:20) qe;k (cid:20) b1. Map in eq. 8 can be written for this dependonwasshouldbeexpectedsince, boththe“queue regionasfollows: law” and the “feedback control law” are not functions of ISRTECHNICALREPORT,2002 5 w. Thepolynomial isgivenbelow. on fixed point of the map with different values of exponential averaging weight w. We analyze this effect with R c (nk)2 (q(cid:3)e −qmin)(q(cid:3)e + M0 )2 = p (qmax−qmin) (12) thehelpofnumericalbifurcation diagrams. max A. HowtoReadaBifurcation Diagram IV. STABILITY A bifurcation diagram shows the qualitative changes in Stability of this fixed point q(cid:3) can be assessed by com- e the nature or the number of solutions of a dynamical sys- putingitseigenvalue: tem as a parameter varies. On the horizontal axis we plot s the different value of parameter (qmin or w in this case). df(q ;(cid:26)) wnk q −q e;k = 1−w− max min The vertical axis shows the corresponding value of fixed dqe;k 2(q(cid:3)e −qmin)32 pmax points or periodic orbits, which is different queue build- := (cid:21)((cid:26)) (13) ups in this context. We have normalized the actual queue buildup by dividing it with qmin for the easy of visual- Although (cid:21)(q(cid:3)e;(cid:26)) is a function of the fixed point q(cid:3)e it- ization. Thats why the legend on the vertical axis reads self, weknow that thefixedpoint willalways bebounded Norm. queueing at the router. The way to read a bifur- from above by f(b2;(cid:26)) and from below by f(b1;(cid:26)). It is cation diagram is fix a point on the horizontal axis and also clear that f(b2;(cid:26)) > f(b1;(cid:26)) since control mecha- draw avertical line. Thenumber of places the bifurcation nism kicks in once the queue length at the router grows curve intersect that vertical line is the number of equilib- beyond b2 decreasing the average queue length. In fact riumpointsofthesystem. Ifthereisonlyonepointthenit f(qe;k;(cid:26))decreases monotonically intheintervalb2 tob1, isastable fixedpointforthatparameter whereas thepres- but the slope decreases in the magnitude. Hence an ap- ence of more than one point indicates that system has a proximate stability condition for fixed point in terms of stableperiodicorbit. Sincewehaveonlyplottedstableso- parameters can be derived by taking the upper bound of lutions corresponding tothedifferent valueofparameters, f(q(cid:3)e;(cid:26))whichisf(b2;(cid:26))andthatswhenf(qe;k;(cid:26))hasits the intersection of vertical line and the bifurcation curve eigenvalue negative and largest in magnitude. Hence, this only indicated the number of stable solutions. Disappear- stability condition canbeformulatedas: ance of a branch implies that, solution corresponding to that branch became unstable and vice versa. All the bifurcation diagrams use three types of symbols. Red star, (cid:12) j(cid:21)(q(cid:3)e;(cid:26))j < 1,orbyssubstituting b2 b(cid:12)yq(cid:3)e green triangle and blue dotdenote thenormalized borders (cid:12) (cid:12) (cid:12)(cid:12)1−w− wnk qmax−qmin(cid:12)(cid:12) < 1 (14) b2,b1 andthesystemsolution respectively. (cid:12) 2(b2−qmin)32 pmax (cid:12) B. EffectofExponential Averaging Weightw whereb2 isgivenbyeq.10. Pleasenotethatstability con- Thefollowing parameters arecommontonextthree bi- dition given by eq. 14 involves the buffer size B despite furcation plots[5]. of the fact that fixed point of the map does not depend on the the buffer size. Theinclusion ofbuffer size makes the q q =100; q =50; c=1500kbps;k= 8=3 resultconservative butitcanbearguedthataconservative max min design is good for the system’s convergence since it has B=300packets; R0=0:1sec; M=0:5kb finitecapacity andhence, evenamarginally stable system n=20; w=bifurcation parameter maynotbeacceptable inpractise. The first three bifurcation plots(Figs. 5,6,7) show the V. NUMERICAL RESULTS effect of varying the exponential weight w with differ- The behavior of this map can be explored numerically ent values of pmax. For the small value of w these plots in parameter space to look for interesting dynamical phe- haveafixedpointwhichlookslikeastraight linebutafter nomena. As eigenvalue moves towards unit circle, fixed some critical value of w this straight line splits into two pointwillbecomeunstableanddependingonthenatureof and this map exhibits period-doubling bifurcation. This bifurcationtherecanbenewfixedpointsorchaos. Thereis is first indication of an oscillatory behavior appearing in alsoapossibilityoffixedpointcollidingwitheitherborder the system due to its inherent nonlinearity as opposed to b1 orb2 whichhasitsownrichworldofbifurcations. discontinuity in “queue or control law” which has been A whole range of different dynamical scenario is pre- proposed earlier. This period two oscillation starts batch- sented here. Lets first consider the effect of varying qmin ing load at the router as shown in the plots. Increasing ISRTECHNICALREPORT,2002 6 w further shows that one of the branches collide, with the 2. pmax = 1 upper border of the map showing a chaos type phenomena. This is basically abifurcation sequence expressed as 1 ! 2 ! chaos. We suspect this to be a case of border 1.016 collision bifurcation [12]. Bordercollision bifurcation isa well understood phenomenon in piecewise linear systems 1.014 andhasbeenshownresponsibleforchaosindifferentelec- > −1.012 trical circuits and economic system models. A technical er ut proof for the border collision bifurcation will be reported he ro 1.01 somewhereelse. at t 0. pmax = 0:1; euing 1.008 u q 1.18 Norm. 1.006 1.16 1.004 >1.14 − uter 1.00120−2.35 10−2.33 10−2.31 10−2.29 10−2.27 10−2.25 e ro1.12 Weight for exp. avg.(w)−> h g at t 1.1 Fig.p7. B=ifu1rcationdiagram of averagequeue length w.r.t. w, uin max e u q1.08 m. We also plot the Lyapunov exponents for the bifurca- Nor1.06 tionscenarioinfig.5wherepmax = 0:1. Fig.8showsthat inthebeginning expoent isnegativewhichcorresponds to 1.04 thefixedpoint. Itslowlyincreasestozeronearperioddou- bling bifurcation and then goes negative again. Finally, it 1.02 10−1.35 10−1.33 10−1.31 10−1.29 10−1.27 10−1.25 jumpstoapositivevaluewhenthebordercollideswiththe Weight for exp. avg.(w)−> periodic solution. PositiveLyapunov exponent conmfirms Fig. 5. Bifurcation diagramof average queuelength w.r.t. w, theexistenceofchaos. p =0:1 max 0.3 1. pmax = 0:3; Border Coillision 1.06 > − 1.05 p. x E0 > Period − outer1.04 yp. Doubling e r L h g at t1.03 n ui e u q m. 1.02 or −0.3 N 1.01 0.04 w−> 0.056 Fig.8. Lyapunovexponentcomputedforaveragequeuelength 1 10−1.8 10−1.7 w.r.t.w,pmax =0:1 Weight for exp. avg.(w)−> Fig. 6. Bifurcation diagram of average queue length w.r.t.w, Lyapunovexponentsforothertwoscenariosalsoexhibit p =0:3 max similarbehavior. ISRTECHNICALREPORT,2002 7 C. EffectofREDControlParameterqmin Similar phenomena are exhibited in these four scenar- ios. Herealso thereisbifurcation sequence like1 ! 2 ! The following parameters are common to the next four chaos in figs. 9, 10 and 11 but scenario of fig. 12 shows bifurcation plots[5]. 1 ! 2 ! 4 ! chaos. It should be noted that 2 ! 4 is notasmoothbifurcation likeperioddoublingrather,itisa q p =0:3; q =100; c=1500kbps; k= 8=3 bordercollision bifurcation. max max B=300packets; R0=0:1sec; M=0:5kb; 2. w = 2−7 n=20; qmin=bifurcation parameter 0. w = 2−5 1.02 1.018 1.18 1.016 1.16 −> uter1.014 o >1.14 e r ng at the router−1.11.21 orm. queuing at th11..100.010821 ui N e u 1.006 q1.08 m. or 1.004 N 1.06 1.002 70 71 72 73 74 75 76 Q −> 1.04 min Fig. 11. Bifurcation diagram of average queue length w.r.t. 1.0221 21.5 22 22.5 23 Q23.5−> 24 24.5 25 25.5 26 qmin,w=2−7 min Fig.9. Bifurcationdiagramofaveragequeuelengthw.r.t.qmin, 3. w = 2−8 w=2−5 1. w = 2−6 1.05 1.045 > − 1.01 er ut uter−>1.10.3054 g at the ro o n e r eui ng at th 1.03 orm. qu1.005 ui1.025 N e u q m. 1.02 or N 1.015 1 83 84 85 86 87 88 89 Q −> 1.01 min Fig. 12. Bifurcation diagram of average queue length w.r.t. 1.00548 49 50 51 52 53 54 55 56 57 58 qmin,w=2−8 Q −> min Fig. 10. Bifurcation diagram of average queue length w.r.t. Finally, we plot the Lyapunov exponent corresponding qmin,w=2−6 to the the bifurcation scenario in fig. 10. Lyapunov expo- ISRTECHNICALREPORT,2002 8 nent shown in fig. 13 also stays negative in the beginning ysis has been studied here, it would be interesting to con- like the other one plotted in fig. 8. In a similar fashion it sider implications oftheapproach forcontrol andforpro- increases to zero when the system goes through a period tocoldesign. doublingbifurcation andagaindecreases whenthesystem has a stable period two trajectory. Finally, it jumps to a ACKNOWLEDGMENTS positive value after border coliision bifurcation and stays Thisresearchhasbeensupported inpartbytheInstitute there. for Systems Research, University of Maryland and by the Officeof Naval Research under Multidisciplinary Univer- 0.3 sityResearchInitiative(MURI)GrantN00014-96-1-1123. REFERENCES Border Collision [1] Y. Korilisand A. Lazar, “WhyisFlow Control Hard: Optimal- > ity,Fairness,PartialandDelayedInformation,”Proc.2ndORSA − Period Doubling p. TelecommunicationsConference,March1992. Ex0 [2] S.FloydandV.Jacobson, “RandomEarlyDetectionGateways p. for Congestion Avoidance,” IEEE Trans. on Networking, Vol.1, y no.7,pp.397-413,1993. L [3] M.Mathis,J.Semke,J.Mahdavi,andT.Ott,“TheMacroscopic Behavior of the TCP Congestion Avoidance Algorithm,” Com- puterCommunicationsReview,Vol.27,no.3,1997. [4] J. P. Hespanha, S. Bohacek, K. Obraczka and J. Lee, “Hybrid −0.3 Modeling of TCP Congestion Control,” Lecture notes in Com- 48 q −> 58 puterScienceno.2034pp.291-304,2001. [5] V.FiroiuandM.Borden,“AStudyofActiveQueueManagement min forCongestionControl,”Proc.ofInfocom2000. Fig.13. Lyapunovexponentcomputedforaveragequeuelength [6] V. Misra, W. Gong and D. Towsley, “A Study of Active Queue w.r.t. qmin,w =2−6 ManagementForCongestionControl,”Proc.ofSigcomm2000. [7] C.Hollot, V.Misra, D.TowsleyandW.Gong, “AControlThe- oretic Analysis of RED,” CMPSCI Technical Report TR 00-41, VI. DISCUSSION 2000andinProc.ofIEEEInfocom,2001. [8] P. Kuusela, P. Lassilaand and J. Virtamo, “Stability of TCP- We have demonstrated in this paper that instability in REDCongestionControl,”Sumittedforpublication,availableat TCP-RED can be induced by the inherent nonlinear be- http://www.tct.hut.fi/tutkimus/com2/publ/,2000. [9] M. May, J. Bolot, C. Diot, B. Lyles , “Reasons Not to Deploy havior of the network, rather than by discontinuity in RED,”Proc.IWQoS’97,1997. the “queue or the control law” as has been believed so [10] D.Linand R.Morris, “Dynamicsof Random EarlyDetection,” farciteaqm1. Thesubharmonic loadbatching veryclearly Proc.ofSIGCOMM,1997. indicatesthatthesystemcanoscillateiftheparametersare [11] M.diBernardo,F.Garofalo,L.GlielmoandF.Vasca,“Analysis ofChaoticBuck,BoostandBuck-BoostConvertersthroughIm- notproperly tuned. Wehavealsogivenaconservative cri- pactMaps,”ProceedingsPESC97(IEEEPowerElectronicsSpe- terionforstableparametersettingsbasedonlinearizedsta- cialistConf.),St.Louis,USA,1997. bilityanalysis. [12] S. Banerjee, P. Ranjan and C. Grebogi, “Bifurcations in two- Although the simulations have indicated a period-two dimensionalpiecewisesmoothmaps-Theoryandapplicationsin switchingcircuits,”IEEETrans.onCircuitsandSystems–I:Fun- oscillation withamplitude within fivepercent ofthe nom- damental Theory andApplications, Vol. 47, no. 5, pp.633-643, inal amplitude, the appearance of period doubling behav- 2000. ior is of significance. The importance of period doubling [13] S. Grishechkin, M. Devetsikiotis,I. Lambadaris and C. Hobbs, bifurcation is not in predicting the amplitude of the os- “OnCatastrophicBehaviorofQueueingNetworks,”Workshopon cillation, but rather in explaining the routes to larger and AnalysisandSimulationofCommunicationNetworks,TheFields InstituteforResearchinMathematicalSciences,Toronto,1998. more pronounced oscillations including chaos. Also, the [14] A.VeresandM.Boda,“TheChaoticNatureofTCPCongestion results become stillmoresignificant whenwethink ofthe Control,”Proc.ofInfocom,2000. network as a cascade involving multiple routers. If one [15] A.Erramilli,L.J.Forys,“OscillationsandChaosinaFlowModel router starts oscillating, it may transfer this instability on ofASwitchingSystem,”IEEEJournalonSelectedAreasinCom- munication,Vol.9,No.2,pp.171-178,Feb.1991. theroutersupordownstreamdependingonthemajortraf- [16] A.Erramilli,L.J.Forys,“TrafficSynchronizationEffectsinTele- ficdirection. Alsodepending on trafficconditions this in- trafficSystems,”Proc.ITC-13,Copenhagen,1991. stability may propagate throughout the network with dif- [17] R. Johari and D. Tan, “End-to-End Congestion Con- ferent time scales and a systemic oscillation can develop. trol for the Internet: Delays and Stability,” available at http://www.statslab.cam.ac.uk/Reports/2000/2000-2.pdf,2000. This needs further study. In addition, although only anal-