nonlinear instabilities in tcp-red - infocom 2002....

Nonlinear Instabilities in TCP-REDPriya Ranjan, Eyad H. Abed and Richard J. La

Abstract—This work develops a discrete time feedback system modelfor a simplified TCP (Transmission Control Protocol) network with RED(Random Early Detection [2]) control. The model involves sampling thebuffer occupancy variable at certain instants. The dynamical model is usedto analyze the TCP-RED operating point and its stability with respect tovarious RED controller and system parameters. Bifurcations are shown tooccur as various system parameters are varied. These bifurcations, whichinvolve emergence of oscillatory and/or chaotic behavior, may provide anexplanation for the parameter sensitivity observed in practice. The bifur-cations arise due to the presence of a nonlinearity in the characteristics ofTCP throughput as a function of drop probability at the gateway. Amongthe bifurcations observed in the system are the period-doubling bifurcationand border collision bifurcations. The bifurcations are studied analytically,numerically, and experimentally.

I. INTRODUCTION

Computer networks are highly complicated systems, both intheir temporal and spatial behavior [1]. Although they have tra-ditionally been modeled and analyzed using stochastic methods,there have recently been several papers that use deterministicnonlinear modeling and analysis (e.g., [6], [7], [8], [16], [14],[5]).

In this paper, we study a modified deterministic dynamicalmodel of a simple computer network with Transmission Con-trol Protocol (TCP) connections and implementing RED at therouter end. The basic model that we consider was proposed re-cently by Firoiu and Borden [5]. We modify their model with asimpler TCP throughput function [3], [4] to facilitate analysis.The calculations we give here show that the model exhibits arich variety of bifurcation behavior leading to chaotic behaviorof the computer network. The bifurcations occur as control pa-rameters are slowly varied, moving the dynamics from a stablefixed point to oscillatory behavior and finally to a chaotic state.

A glimpse at the history of network congestion control re-veals significant attempts to control congestion in the generalnetwork and telephony literature. Congestion and synchroniza-tion in tandem telephone queues have been studied in [15] us-ing a piecewise affine model. A similar model has been ap-plied to the dynamics of choke packets in a LAN to explain syn-chronization and sustained oscillations [16]. These models in-deed explain qualitative changes in the operation of a network orthat of a network component as parameters cross critical values.In contrast to the deterministic setting of [15] and [16], multi-stability or emergence of pseudo-stable states has been reportedin a stochastic setting in [14]. The paper discusses the quali-tative changes in the stochastic behavior of the network due toparameter change, which may lead to degradation in networkperformance.

There have been several attempts to address the issue of con-gestion control with TCP connections, which is the most pop-ular network mechanism for data transfer. The most important

The authors are with the Department of Electrical and Computer Engineeringand the Institute for Systems Research, University of Maryland, College Park,MD 20742 USA. Email: [email protected].

scheme to avoid impending congestion was published in [2] andis known as random early drop, or RED. The basic idea of REDis to sense impending congestion before it happens and try togive feedback to the senders by either dropping or marking theirpackets.1 The dropping probability is the control administeredby the gateways once they detect queue build-up beyond a cer-tain threshold. This scheme involves three parameters: 1) pmax,2) qmax, and 3) qmin that need to be selected. (The mean-ings of these parameters will be identified in the next section.)Most of the rules for setting these parameters are empirical, andcome from networking experience. These rules have been evolv-ing as the effects of controller parameters are not very clearlyunderstood. There are papers discouraging implementation ofRED (e.g., [9]), arguing that there is insufficient consensus onhow to select controller parameter values, and that RED doesnot provide a drastic improvement in performance.

Initially, there was very little in the way of mathematicalmodeling of TCP-RED. However, with the recent efforts to-ward modeling TCP throughput for a transmission line with apacket drop probability [5], [6], [7], [8], [4], several papers havediscussed TCP-RED in the framework of feedback control sys-tems. Most of the models used are continuous-time and the anal-ysis uses basic control theoretic results. The biggest problemwith the continuous-time models is their inability to reflect de-lay, which is prominent in networks and can be very significantfor large trunks [7], [8]. Continuous-time models with variable(state dependent) delay are hard to analyze [8]. The analysis re-ported on these models deals mostly with the stability of fixedpoints and limit cycles under different parameter settings. Forthe first time, chaotic behavior of TCP has been reported in [13].The evidence for this irregularity is mostly explored by simula-tions. Some theoretical work on flow synchronization in TCPhas been reported in [4], [17], but one of the very important is-sues which currently is not well understood is how a smoothlyoperating network transitions into chaos. To borrow dynamicalsystems terminology, the route to chaos starting from a stablefixed point is not well-studied.

In this work, a discrete-time map will be used to model theTCP-RED interaction. A dynamical systems approach will beused to explain the loss of stability, bifurcation behavior, androutes to chaos in TCP-RED networks. We will use bifurcation-theoretic ideas to explain nontrivial periodic behavior of the sys-tem. The appearance of bifurcation and chaos should not besurprising, considering that the system response is nonlinear es-pecially during heavy load conditions. We will show the perfor-mance of the system as a function of various control and systemparameters in general and try to explain these irregular behav-iors with the help of bifurcation diagrams.

Our work begins by realizing that the model proposed in [5]can be viewed as a first-order (rather than third-order) discrete

1Without loss of generality we assume that packets are dropped in the rest ofthe paper.

0-7803-7476-2/02/$17.00 (c) 2002 IEEE. 249 IEEE INFOCOM 2002

nonlinear model. Our replacement of the TCP throughput func-tion of [5] with a simpler version makes the analysis feasible.However, symbolic calculations could be used to allow treat-ment of the more complex throughput function of [5]. The ad-vantage of the current work is that the calculations are simpleenough that the results are easily understood.

We borrow the model proposed by Firoiu and Borden [5] anduse the well known formula for TCP throughput proposed bymany others including [3], [4]. The motivation behind not usingFiroiu and Borden’s formula for TCP throughput is its complex-ity. Complex operations like inverse of a function in differentparameters, which are needed to connect the TCP to the controlmechanism RED, demand simplicity in the TCP throughput for-mulation. This seems to be the reason why they postponed thestudy of their proposed map [5]. Although this TCP-RED for-mulation may not be the exact representation of the complicatedmechanism, it does give a qualitative handle on its dynamics andenhances our understanding of chaos and other instabilities. Wehope that this understanding will lead to monitor the networkcongestion better and help us in formulating robust but simpli-fied control mechanisms.

This paper is organized as follows. In Section 2, we describethe TCP-RED mechanism in control system framework. Section3 contains the discrete map of TCP-RED mechanism. Section 4deals with the stability of this map which is the core of the paper.Section 5 tries to explain the different nonlinear phenomena wehave observed in our models and try to make a connection withchaotic scenario. Finally, in section 6 we discuss the results innetworking context.

II. TCP-RED: FEEDBACK SYSTEM MODELING

A computer network implementing TCP-RED is essentiallya feedback loop where senders adjust their transmission ratesbased on the feedback they receive from the routers in the formof dropped packets. Routers on the other hand implement a con-trol policy which can be either drop tail or RED [2]. There havebeen different approaches to model the dynamics of TCP-REDand various control schemes have been proposed [5], [6], [7],[8], [4] not only to control the system but to also enhance its dy-namic performance. We closely follow the approach taken in [5]with a modified TCP throughput formula.

......

..

flow(n)

flow(1)

......

..

n1 n2l,c

controlto senders

Senders

ave. queue sizeΣj

s,1

r

,qt,j _(p)

npndrop rate,

flow(2)rs,2r

Fig. 1. Simplified Network Diagram

Each flow at a router sends packets with rate rs,i. The send-ing rates of all n flows combine at the buffer of link l and gen-erate a queue of size q which is limited by its buffer size B.

The controller at the router drops packets with a probability pwhich is a function of average queue size q. For ith flow let theforwarding rate at the router be rt,i which is the same as rs,isans dropped packets. When a sender notices that its packetsare being dropped, it adjusts its sending rate based on the dropprobability p it observes.

This makes a control system with sender’s rate as control vari-able with the controller sitting at the router which issues thefeedback signal in the form of a drop probability. The aim ofthis control system is to keep the cumulative throughput belowor equal to the link’s capacity c:

Σnj=1rt,j ≤ c

We assume that TCP flows are long-lived connections and thatthe set of connections remains the same, then the throughput ofeach TCP flow follows the steady state model derived in [3], [4].

RED Module

_H

A(q )

(q )

T(p ,R)nn

Senderq

qp

n

_

n+1

n+1n+1

Averaging

Feedback Law

Fig. 2. Feedback control system corresponding to the network shown

rt,i = T (p,Ri)

T (p,R) =M

R

K√p

(1)

where

T = Throughput of a TCP flow (in bits/sec)

M = Maximum Segment Size or Packet Size

R = Round Trip Time

K = constant which varies between 1 and√

8/3 [3]

p = probability of packet loss

To simplify matters even further we assume that all flows areuniform or they all have same round trip time R, same maxi-mum segment or packet size M , and that maximum congestionwindow size advertised by TCP’s receiverWmax is large enoughto not affect T (p,R). This implies

rt,i(p,R) = rt,j(p,R), 1 ≤ i, j ≤ n and hence

≤ c

n

So this assumption enables us to reduce the n-flow system toa single flow system with feedback although it is important tokeep in mind that feedback is based on the sending rate of all


the flows since the router has no way of differentiating betweenthem, at least in this set up.

To define this control system mathematically, we model thequeue as a function of control variable q = G(p), which acts as aplant in control system literature. To analyze this control systemwe also need the control function p = H(q) implemented atthe gateways. This control function H is given by the policyimplemented at the queue, such as Drop-Tail or RED [2]. Nowfollowing the procedure suggested in [5] we can define the plantfunction G(p) as follows:

G(p) ={

min(B, cM (T−1

R (p, cn ) −R0)) : p ≤ p0

0 : otherwise(2)

where

Ro = round-trip propagation and transmission time and

po = T−1p (c/n,R0)

Here, T−1R (c/n,R0) denotes the inverse of T (p,R) in R,

T−1p (c/n,R0) denotes the inverse of T (p,R) in p, and p0 is

maximum probability for which the system is fully utilized i.e.,for p ≥ p0 senders will have their rates too small to keep thelink fully utilized. For T (p,R) defined by eq. 1.

p0 =(MK

R0cn

)2

(3)

G(p) =

{min

(B, c

M ( MKcn

√p −R0)

), if p ≤ p0

0 , otherwise(4)

RED control law can be expressed as follows:

p = H(qe)

=

0 , 0 ≤ qe < qminqe−qmin

qmax−qminpmax , qmin ≤ qe < qmax

1 , qmax ≤ qe ≤ B

(5)

where qe is the exponential weighted moving average of queuesize, qmin, qmax, pmax are configurable RED parameters, andB is buffer size.

III. DISCRETE MODEL FOR TCP-RED

It is argued in [5] that TCP adjusts its sending rate dependingon whether it has detected a packet drop or not. Hence, thisprocess can be modeled as a stroboscopic map where the instantof observation is one round trip time or RTT. This techniquehas been utilized before for different clocked systems in powerelectronics for modeling the dynamics of power converters [12].Following similar arguments it seems reasonable to model TCP-RED dynamics as a discrete map. Although one would preferthat the sampling interval be regular, there are models where thedynamics is sampled at irregular intervals and the resulting mapsare known as “impact maps” [11].

Let pk be the packet drop probability at tk. At time tk+1 =tk + RTT the senders observe drop rate pk and in an averagesense, adjust their transmission rates. This in turn forces thebuffer to its new state qk+1 = G(pk) following the queue law

in eq. 4. The RED mechanism now computes a new estimate ofqueue size qe,k+1 = A(qe,k, qk+1), following the exponentialweighted moving average:

A(qe,k, qk+1) = (1 − w)qe,k + w · qk+1 (6)

where w is the weight used for averaging. After computingqe,k+1, the RED module adjusts it dropping rate to pk+1 =H(qe,k+1) given by its “feedback control law” in eq. 5. Thisresults in the following mathematical relationships:

qk+1 = G(pk)qe,k+1 = A(qe,k, qk+1)pk+1 = H(qe,k+1) (7)

From eq. 7, we derive a simple one dimensional discrete timedynamical system representation. Since the maps G(.) andH(.) are static, the only dynamics that appear are from the mapA(., .). Using substitution, we can easily derive the followingequation for the exponential weighted moving average for thequeue length at time tk+1:

qe,k+1 = (1 − w)qe,k + w ·G(H(qe,k)) (8)

Eq. 8 also provides a formula to compute the instantaneousqueue occupancy qk+1 at time tk+1 from the exponentially av-eraged queue occupancy qe,k+1. This will be utilized later toplot both averaged and instantaneous queue occupancies.

qk+1 =qe,k+1 − (1 − w)qe,k

w(9)

Below, we illustrate some interesting dynamical behavior ofeq. 8. This equation is rather simple in most of its domain ofdefinition.

We know that G(.) is identically 0 for all p ≥ p0. So we canfind a corresponding value b1 of qe,k such that for any qe,k ≥ b1,G(.) is identically 0 if we assume a monotone feedback law.

b1 =

{p0(qmax−qmin)

pmax+ qmin , if pmax ≥ p0

qmax , otherwise(10)

This gives an explicit formula for the map in eq. 8 for all qe,k ≥b1:

qe,k+1 = (1 − w)qe,k

Now consider the other boundary value b2 of qe,k such thatfor all qe,k ≤ b2 we have G(.) = B or buffer is always full.This value can be computed from eq. 4 and eq. 5. b2, and isgiven by:

b2 =

(nK

B+R0cM

)2

pmax(qmax − qmin) + qmin (11)

This gives an explicit formula for map in eq. 8 for all qe,k ≤b2:

qe,k+1 = (1 − w)qe,k + wB


0 0.01 0.02 0.03 0.04 0.05 0.060

100

200

300

400

500

600

Drop prob.(p)−>

Qu

eu

e s

ize

(q)−

>

Fig. 3. Feedback law H(qe,k) in red and queue law G(p) in blue

It is clear that most of the interesting dynamics happens for b2 ≤qe,k ≤ b1. Map in eq. 8 can be written for this region as follows:

qe,k+1 = (1 − w)qe,k + w(nK√

pmax(qe,k−qmin)

(qmax−qmin)

− R0c

M)

:= f(qe,k, ρ) (12)

where ρ summarizes the parameters in the system.

48 49 50 51 52 53 54 5548

50

52

54

56

58

60

qe,n−>

qe,n

+1, q

e,n

+2−>

MapSecond Return

First Return Map

45o

Line

Fig. 4. 1. Red curve shows first return map, 2. blue curve shows second returnmap, and 3. green line is 45o line whose intersection denotes the fixed pointsof the map

We remark that solving eq. 12 leads to a third degree poly-nomial in fixed point q∗e which interestingly does not depend onw as should be expected since, both the “queue law” and the“feedback control law” are not functions of w. The polynomial

is given below.

(q∗e − qmin)(q∗e +R0c

M)2 =

(nK)2

pmax(qmax − qmin) (13)

IV. STABILITY

Stability of this fixed point q∗e can be assessed by computingits eigenvalue:

df(qe,k, ρ)dqe,k

= 1 − w − wnK

2(q∗e − qmin)32

√qmax − qmin

pmax

:= λ(ρ) (14)

Although λ(q∗e, ρ) is a function of the fixed point q∗e itself, weknow that the fixed point will always be bounded from aboveby f(b2, ρ) and from below by f(b1, ρ). It is also clear thatf(b2, ρ) > f(b1, ρ) since control mechanism kicks in once thequeue length at the router grows beyond b2 decreasing the aver-age queue length. In fact f(qe,k, ρ) decreases monotonically inthe interval b2 to b1 [18], but the slope decreases in the magni-tude. Hence, an approximate stability condition for fixed pointin terms of parameters can be derived by taking the upper boundof f(q∗e, ρ) which is f(b2, ρ), where f(qe,k, ρ) has its eigenvaluenegative and largest in magnitude. Thus, this stability conditioncan be formulated as:

|λ(q∗e, ρ)| < 1, or by substituting b2 by q∗e∣∣∣∣1 − w − wnK

2(b2 − qmin)32

√qmax − qmin

pmax

∣∣∣∣ < 1 (15)

where b2 is given by eq. 11. Please note that stability conditiongiven by eq. 15 involves the buffer size B in spite of the factthat fixed point of the map does not depend on the buffer size.Inclusion of the buffer size makes the result conservative but itcan be argued that a conservative design is good for the sys-tem’s convergence since it has finite capacity and hence, even amarginally stable system may not be acceptable in practice.

V. NUMERICAL RESULTS

The behavior of the map can be explored numerically in pa-rameter space to look for interesting dynamical phenomena. Asthe eigenvalue moves towards the unit circle, the fixed point willbecome unstable and depending on the nature of the ensuing bi-furcation there can be new fixed points or chaos. There is alsoa possibility of the fixed point colliding with either border b1 orb2, leading to a rich set of possible bifurcations.

A whole range of different dynamical scenarios is presentedhere. Consider first the effect of varying qmin on the fixed pointof the map with different values of exponential averaging weightw. We analyze this effect with the help of numerical bifurcationdiagrams.

A. Bifurcation Diagrams

A bifurcation diagram shows the qualitative changes in thenature or the number of steady-state solutions of a dynamicalsystem as a parameter varies. On the horizontal axis we plotthe parameter value (qmin or w in this case). The vertical axis


displays a measure of the corresponding fixed points or periodicorbits, which coincides with queue build-ups in the present con-text. We have normalized the actual queue buildup by dividingit by qmin for ease of visualization. This is why the legend onthe vertical axis reads Norm. queueing at the router. The wayto read a bifurcation diagram is to fix a point on the horizontalaxis and draw a vertical line through it. The number of placesthe bifurcation curve intersect that vertical line is the number ofequilibrium points of the system. If there is only one point thenit is a stable fixed point for that parameter whereas the presenceof more than one point indicates that system has a stable peri-odic orbit as we have plotted only stable solutions correspondingto that particular parameter value. The intersection of the verti-cal line and the bifurcation curve only indicates the number ofstable solutions. Disappearance of a branch implies that the so-lution corresponding to that branch becomes unstable and viceversa. All the bifurcation diagrams use three types of symbols.Red star, green triangle and blue dot denote the normalized bor-ders b2, b1 and the system solution (assuming that solutions canbe chaotic) respectively.

B. Effect of Exponential Averaging Weight w

The following parameters are common to the next three bifur-cation plots [5].

qmax=100, qmin=50, c=1500kbps,K=√

8/3B=300 packets, R0 =0.1sec, M=0.5kb

n=20, w=bifurcation parameter

The first three bifurcation plots (Figs. 5,7,8) show the effect ofvarying the exponential weight w for different values of pmax.For small w, these plots have a fixed point which looks like astraight line but after some critical value of w this straight linesplits into two and this map exhibits period-doubling bifurca-tion. This is the first indication of oscillatory behavior appearingin the system due to its inherent nonlinearity, as opposed to dis-continuity in “queue or control law” which has been proposedearlier. This period two oscillation starts batching load at therouter as shown in the plots. Increasing w further shows thatone of the branches collides with the upper border of the mapgiving a chaos type phenomenon. This is basically a bifurcationsequence expressed briefly as 1 → 2 → chaos. This is a caseof border collision bifurcation [12]. Border collision bifurcationis a well understood phenomenon in piecewise smooth systemsand has been shown responsible for chaos in different electricalcircuits and economic system models. A technical proof for theborder collision type bifurcation phenomena is reported in [18].

0. pmax = 0.1,Both exponentially averaged queue length qe,k and instanta-

neous queue length qk derived from qe,k according to eq. 9 havebeen plotted here to present the implication of border collision.It can be seen that when bifurcation diagram of qe,k collideswith the border b1, qk bifurcation diagram touches the emptybuffer level. The implication of relatively small oscillation inaverage queue length is rather grave for instantaneous queuelength since the buffer starts getting empty and overly filled in

0.04 0.042 0.044 0.046 0.048 0.05 0.052 0.054 0.05651

52

53

54

55

56

57

58

Weight for exp. avg.(w)−>

Avg

. q

ue

uin

g (

qe

,k)−

>

0.04 0.042 0.044 0.046 0.048 0.05 0.052 0.054 0.0560

20

40

60

80

100

120

140


Actu

al queuin

g (

qk)−

>

b1

b2

Period Doubling

Border Collision

Fig. 5. Bifurcation diagram of average and instantaneous queue length w.r.t. w,pmax = 0.1

every alternate cycle. This dynamical phenomenon is commonto all the plots.

0.3

−0.3

0

Lyp.

Exp.−

>

−>w0.04 0.056

PeriodDoubling

Border Collision

Fig. 6. Lyapunov exponent computed for average queue length w.r.t. w,pmax = 0.1

1. pmax = 0.3,2. pmax = 1We also plot the Lyapunov exponents for the bifurcation sce-

nario in Fig. 5 where pmax = 0.1. Fig. 6 shows that in the be-ginning the exponent is negative which corresponds to the fixedpoint. It slowly increases to zero near period doubling bifurca-tion and then goes negative again. Finally, it jumps to a posi-tive value when the border collides with the periodic solution.A positive Lyapunov exponent confirms the presence of chaoticbehavior.

Lyapunov exponents for the other two scenarios also exhibitsimilar behavior.

C. Effect of RED Control Parameter qmin

The following parameters are common to the next four bifur-cation plots [5].

pmax=0.3, qmax=100, c=1500kbps, K=√

8/3B=300 packets, R0 =0.1sec, M=0.5kb,

n=20, qmin=bifurcation parameter


0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.019550

50.5

51

51.5

52

52.5

53


Avg. queuin

g (

qe

,k)−

>

0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.01950

20

40

60

80

100

120

140


Actu

al q

ue

uin

g (

qk)−

>

b1

b2

Period Doubling

Border Collision

Fig. 7. Bifurcation diagram of average and instantaneous queue length w.r.t.w,pmax = 0.3

4.6 4.8 5 5.2 5.4 5.6 5.8

x 10−3

50

50.2

50.4

50.6

50.8


Avg. queuin

g (

q e,k

)−>

4.6 4.8 5 5.2 5.4 5.6 5.8

x 10−3

0

20

40

60

80

100

120


Actu

al queuin

g (

qk)−

>

b1

b2

Period Doubling

Border Collision

Fig. 8. Bifurcation diagram of average and instantaneous queue length w.r.t. w,pmax = 1

0. w = 2−5

1. w = 2−6

Similar phenomena are exhibited in these four scenarios.Here also there is bifurcation sequence like 1 → 2 → chaosin figs. 9, 10 and 11, but the scenario of Fig. 12 shows 1 → 2 →4 → chaos. It should be noted that the transition 2 → 4 is not asmooth bifurcation like period doubling but, rather, it is a bordercollision bifurcation.

2. w = 2−7

3. w = 2−8

Finally, we plot the Lyapunov exponent corresponding to thethe bifurcation scenario in Fig. 10. Lyapunov exponent shownin Fig. 13 also stays negative in the beginning like the other one

21 21.5 22 22.5 23 23.5 24 24.5 25 25.5 2621

22

23

24

25

26

27

28

29

30

qmin

−>

Avg.

que

uing

(qe,

k)−>

21 21.5 22 22.5 23 23.5 24 24.5 25 25.5 260

10

20

30

40

50

60

70

qmin

−>

Actu

al qu

euing

(qk)−

>

Fig. 9. Bifurcation diagram of average and instantaneous queue length w.r.t.qmin, w = 2−5

48 49 50 51 52 53 54 55 56 57 5848

50

52

54

56

58

60

62

qmin

−>

Avg.

que

uing

(qe,

k)−>

48 49 50 51 52 53 54 55 56 57 580

20

40

60

80

100

120

140

160

qmin

−>

Actu

al q

ueui

ng (q

k)−>

Fig. 10. Bifurcation diagram of average and instantaneous queue length w.r.t.qmin, w = 2−6

plotted in Fig. 6. In a similar fashion it increases to zero whenthe system goes through a period doubling bifurcation and againdecreases when the system has a stable period two trajectory.Finally, it jumps to a positive value after border collision bifur-cation and stays there.

D. Effect of system parameters on stability

To understand the effect of number of connections on thesystem stability, we have plotted the bifurcation diagram w.r.t.number of connections (n). Other parameters for this bifurca-tions diagram are as follows:


8/3


70 71 72 73 74 75 7670

71

72

73

74

75

76

77

78

qmin

−>

Avg.

que

uing

(qe,

k)−>

70 71 72 73 74 75 760

20

40

60

80

100

120

140

160

180

200

qmin

−>

Actu

al q

ueui

ng (q

k)−>

Fig. 11. Bifurcation diagram of average and instantaneous queue length w.r.t.qmin,w = 2−7

83 84 85 86 87 88 8983

84

85

86

87

88

89

90

qmin

−>

Avg.

que

uing

(qe,

k)−>

83 84 85 86 87 88 890

50

100

150

200

250

300

qmin

−>

Actu

al q

ueui

ng (q

k)−>

Fig. 12. Bifurcation diagram of average and instantaneous queue length w.r.t.qmin,w = 2−8

B=300 packets, R0 =0.1sec, M=0.5kbw=2−7, n=bifurcation parameter

Bifurcation diagram in Fig. 14 shows that system stabilizesas the number of connections (n) increases. In general, there isagreement [7] that more number of users will stabilize the sys-tem. To further characterize the stability with respect to n, westudy the dependence of critical value of averaging parameterw as a function of other system parameters. A particular valueof w will be called critical if corresponding eigenvalue given byeq. 14 is −1. It can be expressed in a closed form as follows:

wcrit =2

1 + nK

2(q∗e−qmin)32

√qmax−qmin

pmax

(16)

48 58

0.3

−0.3

qmin

0

Lyp

. Exp

.−>

−>

Period Doubling

Border Collision Type

Fig. 13. Lyapunov exponent computed for average queue length w.r.t. qmin,w = 2−6

6 6.5 7 7.5 8 8.550

50.2

50.4

50.6

50.8

51

51.2

51.4

51.6

51.8

52

number of connections(n)−>

Avg.

que

uing

(qe,

k)−>

6 6.5 7 7.5 8 8.50

50

100

150

200

250

300

number of connections(n)−>

Actu

al q

ueui

ng (q

k)−>

Fig. 14. Bifurcation diagram of average and instantaneous queue length w.r.t.number of connections (n)

where q∗e is a fixed point of the system and is given by the so-lution of eq. 13. This expression as function of the number ofactive TCP-sessions is plotted in Fig. 15. It shows that criticalvalue of wcrit increases with the increasing number of activeTCP sessions. Its implication for the stability is that increasingthe number of active TCP-session renders the queue length sta-ble since bigger value ofw is needed to destabilize it (first perioddoubling bifurcation). This result is in agreement with the resultshown in [7] where it is shown that under certain conditions alarger number of active TCP sessions will stabilize the system.

Similarly, we also plot a bifurcation diagram w.r.t. round-trippropagation delay (R0). Plot in Fig. 16 is in agreement with theresult in [7] that larger delays cause instability. Other parametersfor this bifurcations diagram are as follows:


8/3


50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Number of users

weig

ht

Number of users vs. bifurcation point

Fig. 15. wcrit as a function of number of active TCP-sessions.

0.19 0.195 0.2 0.205 0.21 0.215 0.2250.2

50.4

50.6

50.8

51

51.2

51.4

51.6

51.8

52

round−trip prop. delay (R0)−>

Avg.

que

uing

(qe,

k)−>

0.19 0.195 0.2 0.205 0.21 0.215 0.220

50

100

150

round−trip prop. delay (R0)−>

Actu

al q

ueui

ng (q

k)−>

Fig. 16. Bifurcation diagram of average and instantaneous queue length w.r.t.round-trip propagation delay (R0)

B=300 packets, n=20, M=0.5kbw=2−7, R0 =bifurcation parameter

Variation of wcrit as a function of round-trip propagation de-lay (R0) is plotted in Fig. 17. It shows that system is more stablefor smaller values of round-trip propagation delay as larger aver-aging weight w is needed to make it oscillate. This result againis in agreement with the general result of [7] that smaller delaystend to keep the system stable.

VI. EXPERIMENTAL RESULTS

In this section, we present ns-2 simulation results to provideempirical evidence to our claim of rather continuous change inthe dynamics of the system through bifurcations. The networktopology is as shown in Fig. 1. All TCP connections are Renoconnections. The propagation delays of the links that connectthe sources to node n1 are random variables selected from [10

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

we

igh

t

Round−trip propagation delay

Propagation delay vs. bifurcation point

Fig. 17. wcrit as a function of round trip propagation delay.

ms, 50 ms] and those of the links that connect node n2 to thedestinations are uniform random variables in [5ms, 15ms]. Thecapacity of these edge links are set to 30 Mbps. We set n = 100,c = 30 Mbps, and the propagation delay of the bottleneck linkto 5 ms. Given these parameters the average round-trip propaga-tion delay (without any queuing delay) is 92 ms. The packet sizefor both UDP and TCP connections is 500 bytes, and the buffersize at node n1 and n2 is 1,500 packets or 750 kbytes. Thevalue of pmax of the RED mechanism at n1 is set to 1

3 . In thefirst part threshold values qmin and qmax are set to 250 packetsand 1,000 packets, respectively. The first part of the simulationstudies the instability introduced by increasing exponential av-eraging weight (Fig. 18 and 19), and the second part (Fig. 21)illustrates the effects of qmin on the system stability, while fix-ing the weight and qmax constant. Each simulation is run for400 seconds for each run.

220 240 260 280 300 320 340 360 380 400250

260

270

280

290

300

310

320

330

340

time in sec

avg

queue s

ize (

pkt

s)

Fig. 18. Average queuing as a function of time for w = 0.00013.


220 240 260 280 300 320 340 360 380 400200

220

240

260

280

300

320

340

360

380

time in sec

avg

queue s

ize (

pkt

s)

Fig. 19. Average queuing as a function of time for w = 0.00019

0 50 100 150 200 250 300 350 4000

100

200

300

400

500

600

700

time in sec

act

ual q

ueue s

ize (

pkt

s)

w = 0.00019

Fig. 20. Instantaneous queuing as a function of time for w = 0.00019. Notethat buffer indeed gets empty every once in a while which is the hallmarkof a border collision bifurcation.

A. Effects of the Exponential Averaging Weight

In the first scenario for w = 0.00013, average queue occu-pancy qe,k has been plotted as a function of time in Fig. 18.Similarly, in the second scenario for w = 0.00019, averagequeue occupancy qe,k has been plotted as a function of time inFig. 19, and instantaneous queue occupancy qk has been plot-ted as a function of time in Fig. 20. In Fig. 18, though systemshows bursts but their length and amplitude is relatively shorterthan that shown in Fig. 19 and most of the time it stays close toan equilibrium point near 300 packets. On the other hand, plotof average queue occupancy in Fig. 19 exhibits sustained andirregular oscillations. Corresponding instantaneous queue oc-cupancy qk plot in Fig. 20 shows the occupancy levels touchingthe zero line which confirms that during these oscillations bufferactually gets empty every once in a while. It is also shown thatbuffer does not get empty in any regular manner which indicatesthe existence of dynamical phenomenon certainly more compli-cated than fixed points or periodic orbits. The amplitude of vari-ation has also noticeably increased. As we have predicted in

our theoretical modeling, that chaos is possible if average queueoccupancy qe,k hits certain border or instantaneous queue occu-pancy qk becomes zero, here we see the same phenomena hap-pening in the simulation.

B. Effects of the Lower Threshold Value qmin

200 220 240 260 280 300 320 340 360 380 400170

180

190

200

210

220

230

240

250

qmin

= 150 pkts, w = 0.00019

Time (sec)

Ave

rag

e q

ue

ue

siz

e (

pkt

s)

(a)

200 220 240 260 280 300 320 340 360 380 400250

300

350

400

450

500A

vera

ge

qu

eu

e s

ize

(p

kts)

Time (sec)

qmin

= 340 pkts, w = 0.00019

(b)

Fig. 21. Average queuing as a function of time for w = 0.00019. (a) qmin =150 packets, (b) qmin = 340 packets.

In Fig. 21, we have plotted the experimental output for twodifferent values of qmin while keeping all other parametersfixed. Here again, we observe that system shows much moresustained oscillations as qmin increases, which is in a goodagreement with our results from the bifurcations diagram.

VII. DISCUSSION

We have demonstrated in this paper that instability in TCP-RED can be induced by the inherent nonlinear behavior of thenetwork, rather than by discontinuity in the “queue or the con-trol law” as has been believed so far [5]. The subharmonic loadbatching very clearly indicates that the system can oscillate ifthe parameters are not properly tuned. We have also given a


conservative criterion for stable parameter settings based on lin-earized stability analysis. This paper further reports the effectof the number of users, n, and the round trip propagation delay,R0, on system stability and the corresponding critical bifurca-tion value of the exponential averaging parameter w.

The results are based on a simple discrete time dynamical sys-tem model for the average queue length. This model is muchmore tractable than the system of delay differential equationsgiven in [7]. The model is useful for both local and global anal-ysis, and has the advantage of allowing a finite buffer size.

Simulations of the average queue length (qe,k) indicated anamplitude for the period-two oscillation within five percent ofthe nominal amplitude. However, simulations of the correspond-ing instantaneous queue length (qk) clearly showed large ampli-tude oscillations. The appearance of bifurcations is of signifi-cance for several reasons. It provides some insight into the ac-tual system parametric sensitivity. Understanding the specificbifurcation sequence and how it leads to chaotic behavior pro-vides a basis for the design of control schemes to yield a desiredform of dynamical behavior of the network.

Extension of the results to router cascades is an interestingopen problem. If one router begins to oscillate, it may impartthis instability on the routers up or downstream depending on themajor traffic direction. Also, depending on traffic conditions thisinstability may propagate throughout the network with differenttime scales and a systemic oscillation can develop.

ACKNOWLEDGMENTS

This research has been supported in part by the Institute forSystems Research, University of Maryland and by the Officeof Naval Research under Multidisciplinary University ResearchInitiative (MURI) Grant N00014-96-1-1123. Authors also ap-preciate the valuable suggestions from the anonymous refereeswhich have contributed towards significant improvement of thispaper.

REFERENCES

[1] Y. Korilis and A. Lazar, “Why is Flow Control Hard: Optimality, Fairness,Partial and Delayed Information,” Proc. 2nd ORSA TelecommunicationsConference, March 1992.

[2] S. Floyd and V. Jacobson, “ Random Early Detection Gateways for Con-gestion Avoidance,” IEEE Trans. on Networking, Vol.1, no. 7, pp. 397-413,1993.

[3] M. Mathis, J. Semke, J. Mahdavi, and T. Ott, “The Macroscopic Behav-ior of the TCP Congestion Avoidance Algorithm,” Computer Communica-tions Review, Vol. 27, no. 3, 1997.

[4] J. P. Hespanha, S. Bohacek, K. Obraczka and J. Lee, “Hybrid Modelingof TCP Congestion Control,” Lecture notes in Computer Science no. 2034pp. 291-304, 2001.

[5] V. Firoiu and M. Borden, “A Study of Active Queue Management for Con-gestion Control,” Proc. of Infocom 2000.

[6] V. Misra, W. Gong and D. Towsley, “A Study of Active Queue Manage-ment For Congestion Control,” Proc. of Sigcomm 2000.

[7] C. Hollot, V. Misra, D. Towsley and W. Gong, “A Control Theoretic Anal-ysis of RED,” CMPSCI Technical Report TR 00-41, 2000 and in Proc. ofIEEE Infocom, 2001.

[8] P. Kuusela, P. Lassilaand and J. Virtamo, “Stability of TCP-RED Congestion Control,” Submitted for publication, available athttp://www.tct.hut.fi/tutkimus/com2/publ/, 2000.

[9] M. May, J. Bolot, C. Diot, B. Lyles , “Reasons Not to Deploy RED,” Proc.IWQoS’97, 1997.

[10] D. Lin and R. Morris, “Dynamics of Random Early Detection,” Proc. ofSIGCOMM, 1997.

[11] M. di Bernardo, F. Garofalo, L. Glielmo and F. Vasca, “Analysis of ChaoticBuck, Boost and Buck-Boost Converters through Impact Maps,” Proceed-

ings PESC97 (IEEE Power Electronics Specialist Conf.), St. Louis, USA,1997.

[12] S. Banerjee, P. Ranjan and C. Grebogi, “Bifurcations in two-dimensionalpiecewise smooth maps - Theory and applications in switching circuits,”IEEE Trans. on Circuits and Systems–I: Fundamental Theory and Appli-cations, Vol. 47, no. 5, pp. 633-643, 2000.

[13] S. Grishechkin, M. Devetsikiotis,I. Lambadaris and C. Hobbs, “On Catas-trophic Behavior of Queueing Networks,” Workshop on Analysis and Sim-ulation of Communication Networks, The Fields Institute for Research inMathematical Sciences, Toronto, 1998.

[14] A. Veres and M. Boda, “The Chaotic Nature of TCP Congestion Control,”Proc. of Infocom,2000.

[15] A. Erramilli, L. J. Forys, “Oscillations and Chaos in a Flow Model of ASwitching System,” IEEE Journal on Selected Areas in Communication,Vol. 9, No. 2, pp. 171-178, Feb. 1991.

[16] A. Erramilli, L. J. Forys, “Traffic Synchronization Effects in TeletrafficSystems,” Proc. ITC-13, Copenhagen, 1991.

[17] R. Johari and D. Tan, “End-to-End Congestion Con-trol for the Internet: Delays and Stability,” available athttp://www.statslab.cam.ac.uk/Reports/2000/2000-2.pdf, 2000.

[18] P. Ranjan and E. Abed, “Nonlinear Analysis and Control of TCP-RED ina Simple Network Model,” Accepted for presentation at American ControlConference (ACC), Alaska, 2002.


nonlinear instabilities in tcp-red - infocom 2002....

Documents