congestion control and periodic behavior

Abstract -- Feedback based congestion control —used by TCP and other transport protocols — causesthe transmission rate of a long-lived flow to oscillate.This paper is about the tendency of multiplexed flowson a bottleneck link to oscillate causing “aggregateperiodic behavior”. As a consequence, bottlenecklinks can become underutilized, wasting capacity.One simple way to prevent this is to use larger buffersin the bottleneck router. We explore how large thesebuffers need to be to absorb oscillations. Anotherproposed solution is to use a randomized active queuemanagement algorithm, such as RED. A goal of REDis to reduce “global synchronization” of TCP flows,and hence avoid oscillations. Our results suggest thatRED does not reduce aggregate periodic behaviorwhen compared to Drop Tail. Perhaps surprisingly,our results and a simple analytical model suggest thatthe tendency for aggregate traffic to oscillate is robustagainst many forms of disturbances added to the tim-ing of feedback and reaction, even if each constituentflow shows little or no sign of periodicity. Differentflows can be in different states and yet conspire tomake their sum have periodic behavior.

I. INTRODUCTION

One reason for the success of TCP and its widespreadusage in the Internet is its ability to control network con-gestion.

A TCP source uses implicit notification of networkcongestion (via packet loss) to control the rate at which itsends data. The rate is controlled by increasing ordecreasing the window of outstanding, unacknowledgeddata. In particular, TCP uses “additive increase and mul-tiplicative decrease” to increase the rate linearly duringtimes of no packet loss, and to decrease the rate geomet-rically when loss occurs. As a consequence, the rate atwhich a TCP source transmits tends to be periodic overtime. For example, Figure 1 shows the periodic behaviorof a single TCP source in an otherwise idle network.

When many TCP sources compete for the buffers in acongested router, packet loss causes each individual flowto exhibit periodic behavior. A question worth asking is:Is theaggregate behavior of all of the flows also peri-odic? For example, consider Figure 2(a) which shows a

number of periodic TCP flows. If these flows were topass through a single router over a shared link, will theaggregate flow be smooth (as in Figure 2(b)) or periodic(as in Figure 2(c))?

It has been previously observed via simulation[7][8][9] that the aggregate packet arrival rate at the bot-tleneck is periodic.

So why should we care if the aggregate rate of trafficon a link is periodic? Because the average of any peri-odic signal is lower than its peak, which means that alink with periodic traffic is under-utilized. For example,the average data rate in the output link of Figure 2(c) issignificantly lower than that of Figure 2(b).

Many experiments[7] and the intuitive explanations ofthese experiments suggest that TCP sources competingfor bandwidth on a congested link will synchronizethrough the weak coupling inherent in congestion con-trol. Intuitively speaking, a population of sources willsynchronize because they all experience loss at roughlythe same time, they all scale back their transmission ratein the presence of these losses, and then they increasetheir transmission rate until the next bout of congestionoccurs.

In order to understand if, and by how much, periodicbehavior affects the performance of individual TCPflows, we need to examine the bottleneck link. If the bot-tleneck link is under-utilized then all flows sufferreduced throughput.

Two possible ways to reduce periodicity on a bottle-neck link are: (1) Increase the size of the buffers immedi-ately prior to the link in order to absorb the oscillationsand filter them out, and (2) Use a randomized drop-pol-

Rate,cwnd

TimeFigure 1: Periodic oscillation shown by a singleTCP connection

Congestion Control and Periodic BehaviorAnna C. Gilbert, Youngmi Joo, Nick McKeown

AT&T Labs-Research, 180 Park Avenue, Florham Park, NJ 07932-0971Stanford University, Stanford, CA 94305

icy in the routers. In particular, it is a stated goal ofRED [4] to desynchronize each of the flows sharing acongested router buffer so that the periodic behavior ofeach individual flow is not synchronized with the others.RED discards packets randomly in an attempt to causedifferent TCP sources to reduce (and hence increase)their rates at different times. Intuition suggests that if theflows are not oscillating in phase, then their aggregatebehavior will be smooth (like in Figure 2(b)) and hencelink utilization will be high.

In this paper, using simulation and simple topologieswe set out to explore: (1) How much utilization is lostdue to periodic behavior, (2) How much buffering shouldbe deployed by network designers in order to filter outthe periodicity, and (3) How effective is RED in reducingor eliminating periodicity.

Our simulation results (which are based on simpletopologies) suggest that if buffers are too small, periodicbehavior occurs on bottleneck links, both with and with-out RED, even if sources are distributed at random dis-tances from the bottleneck. RED does not appear toreduce periodicity and can, under certain circumstances,increase it. Perhaps the most surprising finding of thispaper is that simulations suggest that the tendency foraggregate traffic to oscillate is quite strong, even if eachconstituent flow shows little or no sign of periodicity.

In an attempt to capture and characterize the way thatmultiple seemingly non-periodic flows might interact tocause aggregate periodic behavior, we developed a sim-ple analytical “toy” model of TCP’s congestion controlalgorithm. Although too simple to draw strong conclu-

(a)

source 2TCP

source 1TCP

source NTCP

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Figure 2: Multiple TCP connections; (a)topology, (b)flat aggregate rate, (c) periodically oscillating aggre-gate rate

sions from, the model tends to support the simulationresults that there is a strong tendency for traffic to oscil-late, and that it is likely to be a characteristic of almostany feedback congestion-control mechanism in whichthe buffers are (too) small.

II. SIMULATION RESULTS

A. Simulation Setup

We use ns-2[1][2] for our simulations, with TCP Renosources and the topology shown in Figure 3. Fifty clients(at the nodes , with ) on one side of abottlenecked router retrieve an infinite-length file from aserver, , on the other side. Hence, there are 50 TCPconnections, sharing a single oversubscribed 1.5 Mbpslink between the nodes and . Each connectionstarts at some random time between and seconds,lasting the duration of the simulation (4200 seconds).Figure 3 shows the data rate and propagation delays foreach link. In our simulations, we choose the propagationdelays between each endpoint , and thenode to be either a single value of 20 msec (which wecall “fixed RTT”), or pick them at random and uniformlyover the range 0 to 1 seconds (which we call “variableRTT”).

We compare both Drop Tail and RED queue manage-ment disciplines, with varying buffer sizes and variousthreshold values ( and ) for RED.1 Whenvarying and we use two different ways. Oneis to fix to a very small value (equal to five), whilevarying the values. We call this “RED minfix”throughout the paper. Another is to vary both and

, while keeping the ratio of to a con-stant value of three. We call this “RED ratiofix”. Thepurpose of each is to see whether the phenomenaobserved with RED have more to do with the difference

1. Other parameters for RED used in simulations arein Appendix B.

D1 D2, …DN N 50=

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between the two threshold values, or the combined effectof the difference and the value of .

B. Periodic Behavior with Small Buffers (Impact ofInsufficient Buffer Space)

Our first results show the effect of small buffers in therouter just prior to a bottleneck link. Expecting that smallbuffers will exaggerate any periodic behavior, weexplore how much utilization is lost, if any. Specifically,we keep the total buffer space, or of RED, ataround 20% of the “bandwidth-RTT product”.

Figure 4 shows the simulation results for DropTail(Figure 4(a), (b)) and RED ((c), (d)), with fixed RTT of1340 ms (Figure 4(a),(c)) and variable RTT (Figure4(b),(d)). Periodic behavior is clearly visible in all fourgraphs, which leads us to the definition of wasted utiliza-tion on the bottleneck link:

.

Drop Tail in Figure 4(a)(b), shows quite noticeableperiodicity, leading to a loss in utilization of approxi-mately 5-7%. This is to be expected — Drop Tail hasbeen previously described as leading to “global synchro-nization”.

Our simulation results for RED in Figure 4(c)(d) chal-lenge both the term “global synchronization” and theefficacy of RED in preventing it, at least when the buffersize is small. In this example, the loss in utilization withRED is approximately 15% — larger than for Drop Tail.

We shall see that “global synchronization” is not reallya good characterization of the problem for Drop Tail.

We start here by testing the intuition that periodicbehavior comes from synchronized sources. Figure 5shows that it does not. (We’ll be spending quite a bit oftime in the remainder of this paper trying to understandhow periodic aggregate behavior occurs even when eachindividual flow shows little sign of oscillation.) The fig-ure shows the totalwnd2 values plotted against time,along with cwnd values for 3 randomly chosen TCPflows. cwnd represents an individual TCP flow’s con-gestion ‘state’, while the ‘totalwnd’ represents theaggregate state of flows on the link.3 The top half of Fig-ure 5 shows totalwnd oscillating in a sawtooth pattern,in agreement with the oscillatory behavior of the aggre-gate packet arrival rate. However, individualcwnd val-

2. Sum ofwnd=min(cwnd, rwnd ) values for allflows.3. It has been also checked from simulation resultsthat the totalwnd values are roughly the same as thepacket arrival rate on the time granularity ofRTT+queueing delay on the path.

minth

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-----------------------------------------------------– ⋅=

Figure 4:Aggregate Packet Arrival, Departure rates andqueue lengths for (a) DropTail, fixed RTT, (b) DropTail,variable RTT, (c) RED, fixed RTT, (d) RED, variable RTT

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ues are not necessarily in sync with one another, asshown in the bottom half of Figure 5. Instead, at eachcongestion epoch somecwnd values continue toincrease while others do not. Note that, although Figure 5(a) is consistent with the results in [10] which indicatesthat the individualcwnd behavior of the sources appearschaotic, simulation results show that the aggregatebehavior is decidedly deterministic.

Surprisingly, introducing randomness in packet dropsdoesn’t change the aggregate periodic behavior, neitherat the input nor at the output of the buffer. The outgoingline utilization is lower in RED in Figure 2, when com-pared with Drop Tail.

We note that RED has little room to exercise random-ness when the buffers are too small. Since issmall, the majority of the packet drops occur in burststhat are not much different from Drop Tail. Moreover,since RED drops packets until thetime-averaged queuelength falls below both of its thresholds, RED tends topunish flows more severely than Drop Tail. Thisbecomes clear when we look at how many flows experi-enced multiple packet drops4 within a burst, or the num-ber of drops when packet drops come in well-separatedbursts. Figure 6 shows histograms of those counts forsimulation results shown in Figure 4 (a) and (c),observed at each congestion epoch. Note that REDforces more flows to reduce theircwnd (and hence trans-mission rate) compared to Drop Tail. Consequently the

4. We regard this as a congestion notification with ahigher magnitude, since TCP Reno source is likely toreducecwnd to 1 when this happens [3]. Therefore,this particular interpretation is closely tied to thedetails of the TCP implementation.

(a)

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Figure 5: totalwnd values (top) andcwnd values ofthree randomly chosen flows, (a) fixed RTT (1340ms),(b) variable RTT.

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aggregate arrival rate decreases more with RED, makingthe buffer empty for longer periods of time. Also, sincethe maximum buffer occupancy is smaller for RED thanDrop Tail, the queues will go empty quicker for REDonce the aggregate arrival rate falls below the outgoingline rate.

C. Periodic Behavior with Large Buffers

We can expect that when the router buffer is largeenough, the bottleneck link will no longer exhibit peri-odic behavior, due to the smoothing effect of the buffer.Simulations indeed show this to be the case. While thelinks immediately prior to the buffer continue to oscil-late, the output link from the buffer does not regardlessof the queue management algorithm.

It is of practical interest to know how large the bufferneeds to be to prevent periodic behavior from occurringfor both Drop Tail and RED. Armed with some data, net-work operators may be able to prevent oscillations fromoccurring and hence increase link utilization. In what fol-lows, we provide quantitative answers on buffer sizingfor the network topology considered in our simulations.

1) Preventing Periodic Behavior

First we’ll try and prevent periodic behavior with DropTail. We tested the well-known rule-of-thumb that thebuffers at the bottleneck router should be at least equal insize to the bandwidth-RTT product. Note, however, thatthis value comes from assuming that the aggregatearrival pattern is a ‘pulse train’, i.e. it consists of burstsof packet arriving (exactly) every RTT, which we knowfrom our simulation results is not true. Despite the differ-ence in behavior, we found that with Drop Tail and onebandwidth-RTT of buffering, the bottleneck link was100% utilized for both fixed and variable RTT (for whichwe substituted the average RTT of 2300ms). The loss ofutilization as a function of buffer size is shown in Figure7 for both incoming5 and outgoing links.

For RED, it is less clear how to interpret the band-

(a) (b)

Figure 6: Histogram of number of flows with multiplepacket drops within a burst, (a) DropTail(Figure 4 (a)),(b) RED (Figure 4 (c)).

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width-RTT guideline: Is it only the value that mat-ters, or should the bandwidth-RTT product liesomewhere between and ?

So we considered various values of or(RED minFix, RED ratiofix). In each case, the loss ofutilization eventually falls to zero, as increases. A

5. We define the loss of utilization on the ingress linkas .Since the average arrival rate can be larger than theoutgoing line rate, this value can be negative, whichindicates that packets are being buffered. A positivevalue indicates that the arrival process, even withoutconsidering the buffer becoming empty, is ‘ineffi-cient’ due to its periodic behavior.

100 1 Avg Arrival Rate Egress Line Rate⁄( )–( )

Figure 7: Loss of utilization on ingress and egresslinks, (a)long fixed RTT, (b) short fixed RTT, (c)variable RTT.

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few things to note are:• Comparing RED minfix and RED ratiofix, it appears

that has more impact on the loss of utilization.In Figures 5(a)-(c), RED minfix needs smallerthan ratiofix to achieve 100% utilization, when thetwo have roughly the same .

• When RTT is short compared to the transmissiontime of onewnd of data, as in Figure 7 (b), REDrequires much larger buffer space than DropTail toachieve 100% utilization.

2) Changes in Aggregate Behavior with IncreasingBuffer Space

In this subsection, we examine how thecollectivebehavior of the sources changes as buffer size andthresholds increase.

Figure 8 shows totalwnd values as a function of timefor different buffer sizes. For Drop Tail, as buffer sizeincreases, totalwnd stays constant. This corresponds toall 50 flows transmitting with theirwnd values set to the

maxthmaxth

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Figure 8: totalwnd values vs. time for different bufferspaces, (a)DropTail, fixed RTT, (b) DropTail, variableRTT, (c) RED minfix, fixed RTT, (d) RED minfix,variable RTT, (e) RED ratiofix, fixed RTT, (f) REDratiofix, variable RTT

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maximum possible value, and the buffer size appears tobe large enough to accommodate this. We can verify thisthrough a simple calculation. If the buffer size is largeenough to satisfy the equality

where, for Drop Tail, the maximum queueing delay isroughly (buffer size)/(bottleneck bandwidth), then thenetwork can accommodate all the sources transmitting attheir peak rates without dropping any packets. In onesimulation shown in Figure 7(a) we had RTT(withoutqueueing delay) of 1340 ms, bottleneck bandwidth 1.5Mbps, number of flows 50, maximumwnd(=rwnd ) 20packets, and the packet size of 1000 bytes. This yieldsthe buffer size of 748 packets, which is in agreementwith the buffer size of 750 packets that was just able tocompletely eliminate any aggregate periodic fluctuationfor Drop Tail.

Number of flows Maximum cwnd×( )Bottleneck Bandwidth( )

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RTT Maximum Queueing Delay+=

Figure 9: Aggregate packet arrival (a) DropTail withbuffer size of 50, (b) DropTail with buffer size 750,(c) RED with maxth=15, (d) RED with maxth=1500.

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RED shows different trends in totalwnd values as value increases for both ‘minfix’ and ‘ratiofix’

cases; the magnitude of oscillation (of totalwnd)decreases, eventually staying constant. But the overallvalues of totalwnd do not increase as rapidly as in DropTail, although the consequences of this “convergence”leads to the elimination of periodicity in the aggregatepacket arrival rate. This is verified in Figure 9, whichshows the packet arrival and departure rates for bothDrop Tail and RED ‘minfix’ with varying buffer size (forDrop Tail) and threshold values of and

(for RED). Although the queue length stillfluctuates with , there isn’t any prominentperiodic fluctuation in the aggregate arrival process.

Disappearing fluctuations of totalwnd values asincreases can be explained by changes in timing dynam-ics of packet drops. As increases past the band-width-RTT product and beyond, RED begins to haveenough opportunities to introduce randomness in packetdrops before the time-averaged queue length reaches

maxth

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Figure 10: Distribution of inter-drop times ofpacket, (a) max_th=15, aggregate, (b) max_th=15,each flow, (c) max_th=1500, aggregate, (d)max_th=1500, each flow.

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, beyond which point it drops packets deterministi-cally. As a result, packet drops at the bottleneck nolonger occur in short, clearly delineated bursts, occurringinstead at random intervals. The intervals are typicallylonger than the back-to-back inter-drop times, and gener-ally increase with , as shown on the left half of Fig-ure 10. Figure 10 (a) and (b) are for and (c)and (d) for . For the larger value of ,the distribution of inter-drop time is centered around 2.0second, whereas the distribution for the smallerhas two distinctive ‘islands’, one for the back-to-backpacket drops, the other for the packet drops belonging toneighboring ‘bursts’.

The same trend is observed when we look at inter-droptimes seen by each individual flow, as shown in the rightside of Figure 10, at the top and bottom, respectively. Animportant consequence is that the flows are no longer alldriven in to the same state (namely,cwnd=1, as a resultof a burst of packet drops), hindering or eliminating theaggregate periodic behavior.

III. A “T OY” A NALYTICAL MODEL

Simulation has only limited value — it can only tell usdefinitive things about very specific topologies. Whilewe may convince ourselves that our experiments are rep-resentative, we cannot, in any scientific way, concludethat our results extend to more general network or situa-tions.

So, in an attempt to capture and characterize the waythat a feedback-based congestion control algorithmmight cause aggregate periodic behavior, we created avery simple analytical model. The model attempts tocrudely capture the feedback behavior of TCP with a ran-dom drop policy such as RED. Our analysis suggests thatalthough sources may be placed at different distancesfrom the bottleneck, and that they may initially startasynchronously to each other, the feedback congestioncontrol algorithm will eventually cause the aggregatebehavior to oscillate. Perhaps surprisingly, the modelsuggests that the oscillation is quite stable in that itremains prominent in spite of various irregularities andperturbations. It appears that it would take quite a drasticchange to the feedback or drop mechanism to prevent theperiodic behavior.

The conclusions derived from this model are valid onlyin the regime where buffering is inconsequential, whenthe buffer is suitably small. The analytical results showthat the aggregate periodic behavior is an extremely sta-ble feature of a small buffer network configuration.However, the model does not accurately predict thedeparture process from a large buffer, a situation inwhich we in simulations can eliminate oscillatory behav-

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ior. A more complete model would have to take suchbuffering into account for a more accurate picture.

A. Description of model

We model a TCP connection as a source with an infi-nitely long file to transmit to a receiver across a bottle-neck link. There are sources or connections sharingthe bottleneck link which has capacity . In the absenceof congestion or feedback from the bottleneck, eachsource increases its sending rate linearly over time (i.e.,

where is the rate of source at time ).

The system of sources experiences congestion when the

total rate reaches the link capacity . At

each congestion epoch when , a fraction of the sources (chosen uniformly at random) experiences

losses and these sources receive a notification of con-gestion. Each source upon congestion notification, resetsits transmission rate to zero. We assume that these eventssignaling congestion and resetting the transmission rateoccur instantaneously when the aggregate rate reachesthe link capacity. After reset, the rates of the sources con-tinue to grow linearly until the system reaches the nextcongestion epoch, at which point the process repeatswith another random fraction of sources experiencinglosses and adjusting their rates. This model assumes thatthere is an entity at the bottleneck link that can measurethe instantaneous aggregate rate, signal, and reset thesources instantaneously.

B. Conclusions drawn

Rather than following the possible states of allsources, we assume that is large enough for us toemploy a mean-field approximation for the system. Wedescribe the state of the system at time by a density

that represents the percentage of sources withtransmission rate at time . We let take on anyreal positive value, not necessarily an integer multiple of

. Furthermore, we approximate the result of eachreset upon as removing a uniform percentage ofthe sources at rate and time (i.e., that the approxi-mate effect upon the population of sources is the mean orexpected effect) and resetting the rates of that percentage

by placing a point mass of weight at rate . If represents the distribution of sources immediately

before the th congestion epoch, then the mean-fieldapproximation gives us the representation

= +

N

C

tdd

xi t( ) 1= xi t( ) i t

A t( ) xi t( )i 1=

N

∑= C

A t( ) C= r

N

r

r

N

N

t

ρ x t,( )

x t ρ x t,( )

1 N⁄ρ x t,( ) r

x t

r r x 0=

ρ x t-,( )

n

ρ x tn+,( ) r δ0 x( )⋅ 1 r–( ) ρ x tn

-,( )⋅

for the distribution of source rates immediately follow-ing the reset of the sources. In addition, the linearincrease of the sources’ rates between congestion epochstells us that the configuration of the system attime is simply the translation of the distribution attime by , .

We demonstrate that:• there is a configuration of the source rates that is

invariant under the reset operation at each conges-tion epoch; i.e., the distribution of source ratesimmediately following a reset is the same as the dis-tribution after the preceding congestion epoch,

• there is a constant time intervalbetween congestion epochs,

• the aggregate rate of the sources is a -periodicfunction that oscillates between a maximum valueand a minimum value with an average of

, and• any initial configuration of the sources will tend to

this invariant exponentially fast in .The reader interested in technical details will find them

in Appendix A. See Figure 11 for a picture of this config-uration, where the mean field approximation is plotted inblack for connections and .

The invariant distribution of source rates has twoimportant properties. First, it consists of distinct popula-tions of sources, each population with one rate, ratherthan constraining all sources to transmit at the same

ρ x t tδ+,( )

t tδ+

t tδ ρ x t tδ+,( ) ρ x tδ–( )=

ρ*

t∆ r C⋅( ) N⁄=

A t( ) t∆C

1 r–( ) C⋅C r C⋅ 2⁄–

r

N 50= r 1 4⁄=

Figure 11: The invariant configuration of sources forthe mean-field approximation and a simulatedinstance of the model with N = 50 sources, the frac-tion r = 1/4 of the sources reset randomly at eachepoch, the bottleneck link capacity C=1, and theepoch duration . The distribution of50 sources after 1000 cycles in a realization of thetoy model is also plotted in grey.

t∆ r N⁄ 0.002= =

rate

no. c

onne

ctio

ns

0.0 0.02 0.04 0.06

02

46

810

12

mean-field approx.N = 50 sources

N

rate. These distinct populations point out a slight misno-mer in the term “synchronization” commonly used todescribe this phenomenon. The sources do not synchro-nize their rates with respect to one another, they conspirein groups to produce an aggregate periodic rate, and sothe term “aggregate periodic behavior” is a more accu-rate description of this phenomenon. The sizes of the ratepopulations do decay exponentially, showing that manyof the sources transmit at the lowest rate while very fewsources transmit at a high rate.

The second important property the invariant configura-tion possesses is stability. To explore the stability of theinvariant distribution, we define a perturbation of thesystem at the th congestion epoch to be a random devia-tion from the time the th reset occurs. We choose uni-formly at random in and jitter the duration of theth epoch . The time between the

st reset and when the sources next reach linkcapacity is still , we simply randomly perturbwhen this reset occurs as we do in a queueing policy thatattempts to inject a small amount of randomness into theloss process. At the new time , we reset a uniformlychosen random fraction of the sources and let the sys-tem evolve again until the total rate exceeds capacity .We repeat this process for each congestion epoch (draw-ing an independent each time) and show that the per-turbed system remains close to the unperturbed system(which, we emphasize, is converging exponentially fastto the invariant distribution independent of the initialconditions). The perturbed system differs from its origi-nal trajectory with respect to its rates by a maximumvalue of and with respect to its congestion epoch dura-tions by . On average, the perturbed system does notstray from its original course, as the perturbation is arandom variable with mean 0.

C. Extensions and generalizations

Several aspects of this mathematical model are onlysimple abstractions of more complex realistic behavior.To generate a less artificial picture of the behavior of asystem of sources, we must change both the manner inwhich the sources increase their rates and the rate resetpolicy in the face of congestion. However, neither modi-fication changes the gross behavior of the system and sowe feel reasonably confident drawing conclusions fromthe behavior of the simple abstract model.

The rate increase profile in this simple mathematicalmodel does not reflect the complex behavior of the rateincrease in TCP with its slow-start and congestion avoid-ance phases. However, if we impose a more complicatedrate increase , we do not change the overall behav-ior of the sources although we do change the details. The

i

i εi

ε– ε,[ ]i t∆ i t∆ εi N⁄+= t∆

i 1–( )C r C⋅( ) N⁄

tir

C

ε

εε N⁄

ε

N

tdd

xi t( )

congestion epoch duration is different and is given by

but with the same reset policy, the system evolves asbefore. The source rate distribution moves according to

the expression . Similarly, if

we choose a different reset policy, perhaps setting therates to zero of those sources with higher rates, wechange the congestion epoch duration slightly and thepercentage of sources at a given rate no longer decaysexponentially. In fact, if we adopt a “deterministic” resetpolicy where we reset the sources with the highestrates, we will get distinct rate populations of equal sizeand a constant epoch duration depending on .

Finally, the mean-field approximation is just that, onlyan approximation of the true behavior of the mathemati-cal model (itself an approximation). However, a morecareful analysis of the behavior of sources under theaction of this random dynamical system,not assumingthat is large, reveals that the mean-field approximationis an excellent approximation of the system in Figure 11.

IV. CONCLUSION

In our work we set out to explore and quantify by howmuch RED reduces “global synchronization”. Intuitively,it is a small step from understanding how sources maymove in lock-step to conclude that a randomized droppolicy will break the synchronization and hence elimi-nate aggregate periodic behavior.

Our first conclusion is that the intuition is wrong —while RED may prevent sources moving in lock-step,this is not enough to prevent the traffic rate on a linkfrom oscillating, and hence wasting throughput for allflows.

Our second conclusion — aided by a simple analyticalmodel — is that there is evidence that when buffers aresmall, a feedback-based congestion control algorithmhas a strong propensity to cause aggregate periodicbehavior, regardless of the queue management policy. Inother words, randomized drop policies are unlikely toreduce aggregate periodic behavior. Even when the sys-tem is perturbed in a variety of ways, such as placingsources as very different distances from the bottleneck,traffic on the bottleneck link can still oscillate. Particu-larly with small buffers, flows in apparently very differ-ent states interact — and appear to conspire — to createaggregate periodicity. However, our results are based ona simple topology and a toy analytical model — it is pre-mature to make any strong generalizations from our

A t-n 1+( ) C 1 r–( ) C⋅ N

tdd

xi s( ) sdtn

tn 1+

∫⋅+= =

ρ x t,( ) ρ xtd

dxi s( ) sd

0

t

∫– 0,( )=

N q⁄q

q

N

N

results. Further work is needed to determine how broadlythese results apply.

Fortunately, although traffic on the link prior to thebottleneck router may continue to oscillate, with largeenough buffers (equal to at least the bandwidth-RTTproduct), neither Drop Tail and RED lead to significantperiodic behavior on the bottleneck link itself.

REFERENCES

[1] S. Bajaj et. al., “Virtual InterNetwork Testbed: Sta-tus and Research Agenda,” USC/ISI Technical Re-port 98-678, July 1998.

[2] S. McCanne and S. Floyd, “UCB/LBNL/VINT Net-work Simulator - ns (version 2),” http://www-mash.cs.berkeley.edu/ns/

[3] K. Fall, S. Floyd, “Simulation-based comparisonsof TCP Tahoe, Reno, and SACK TCP,”ComputerCommunication Review, Vo. 26, No. 3, pp.5-21,1996

[4] S. Floyd and V. Jacobson, “Random Early Detec-tion Gateways for Congestion Avoidance,”IEEE/ACM Transactions on Networking, Vol. 1, No. 4,pp.397-413, 1993

[5] M. Allman and V. Paxson and W. Stevens, “TCPCongestion Control,” Request for Comments 2581,April 1999

[6] B. Braden et. al., “Recommendations on QueueManagement and Congestion Avoidance in the In-ternet,” Request for Comments 2309, April 1998

[7] S. Shenker, L. Zhang, D. Clark, “Some Observa-tions on the Dynamics of a Congestion Control Al-gorithm,” ACM Computer Communication Review,Vol. 20, No. 4, pp.30-39, October 1990.

[8] L. Zhang, S. Shenker and D. Clark, “Observationson the Dynamics of a Congestion Control Algo-rithm: The Effect of Two-Way Traffic,”Proc. ofACM/SIGCOMM’91, pp. 133-149, 1991.

[9] Y. Joo, V. Ribeiro, A. Feldmann, A. Gilbert, W.Willinger, “On the impact of variability on the buff-er dynamics in IP networks,” Proc. of 37th AnnualAllerton conference on Communication, Control,and Computing, Sept. 1999

[10] A. Veres, M. Boda, “The Chaotic Nature of TCPCongestion Control,”Proc. of Infocom 2000, TelAviv, Israel, March 2000.

APPENDIX A: Analysis of Toy Model

In this section we sketch the proofs of the results dis-cussed in Section III.

1) Invariant configurations

Theorem 1:There is a unique density thatdescribes the state of sources (for large) immedi-ately after resetting the rates of sources (chosenuniformly at random) to zero such that after the nextreset, reached at time after the first reset,

is restored. This density is given by

In addition, the aggregate rate is a periodic functionwith period , oscillating between a maxi-mum value of and .

Proof: The aggregate rate at time is where the density

describes the state the system is in at time and thecapacity constraint gives us the time at which the ratereaches capacity

We obtain an iterated map on the density andaggregate rate . Assume that is the time immedi-ately after the th reset, then the iterated map is

where . The last two conditions on theaggregate rate tell us that if a density exists, thensatisfies or that .

The global attractor for the iterated map onmust therefore satisfy

(1)

Observe that we can translate condition specified in (1)into a condition on , the Laplace transform of , andthen we can expand in a uniformly convergent powerseries

Next we compute the inverse Laplace transform of by inverting each term in the power series term-by-

term, giving us

ρ*N N

r N⋅

t∆ r C⋅( ) N⁄=

ρ*

ρ* x( ) r 1 r–( ) l δ0 x l t∆⋅–( )⋅ ⋅l 0=

∞

∑⋅ x 0≥,

0 x 0<,

=

A t( )

t∆ r C⋅( ) N⁄=

C 1 r–( ) C⋅

t

A t( ) N x ρ x t,( )⋅( ) xd∫⋅= ρ x t,( )t

tn-

C

C A tn-

N x ρ x tn,( )⋅( ) xd∫⋅= =

ρ x t,( )

A t( ) tn+

n

ρ x tn+,

r δ0 x( )⋅ 1 r–( ) ρ x tn 1+∆– tn 1+

-,( )⋅+=

A tn 1+-

( ) C A tn+

( ) N tn 1+∆⋅+= =

A tn+

( ) A tn 1++

( ) 1 r–( ) C⋅= =

tn 1+∆ tn 1+ tn–=

ρ* x( ) t∆1 r–( ) C⋅ N t∆⋅+ C= t∆ r C⋅( ) N⁄=

ρ x t,( )

ρ* x( )r δ0 x( )⋅ 1 r–( ) ρ* x t∆–( )⋅+ x 0≥,

0 x 0<,

=

ρ* ρ*

ρ* s( ) r

1 1 r–( ) es t∆⋅( )–⋅–

--------------------------------------------------- r 1 r–( ) le

s l t∆⋅ ⋅( )–⋅l 0=

∞

∑⋅= =

ρ* s( )

ρ* x( ) r 1 r–( ) l δ0 x l t∆⋅–( )⋅ ⋅l 0=

∞

∑⋅=

when and for .Note that if we take an iterative approach, we find the

following theorem:

Theorem 2:If a fraction of a system of sourceswhose states are initially described by the densityfollows the dynamics outlined above, then the density

converges weakly to the global attractor ,independent of the initial conditions of the system. Thedensity at time after the -th reset is

and the congestion epoch length is , afterthe first congestion epoch (whose length depends on theinitial state of the system).

The careful reader might argue that allowing even aninfinitesimal percentage of the sources’ rates to growarbitrarily large is not a physically relevant result. Let usnow assume that the sources’ rates cannot grow largerthan a maximum rate . That is, once a source increasesits transmission rate to , its rate remains fixed thereuntil it experiences a loss. We have to assume that

to avoid trivial and physically meaninglesssituations. We express this condition upon the source ratedistribution mathematically by placing a point mass atrate whose weight is the total percentage of sourcerates that at time are larger than and leavingunchanged the distribution for smaller rates. Let be thelargest integer such that for

. Then we may conclude using argumentssimilar to the above theorems that there is an invariantdistribution for -bounded sources.

Corollary 1:The invariant distribution for -bounded sources has the form

(2)

where is the per-

centage of sources at rate .

2) Stability of invariant configurations

We work with the more physically meaningful rate-bounded model as in Corollary 1 and assume

for ease of calculation (if not, then thesummation in (2) ends at ).

Lemma 1:If we jitter the duration of the th conges-tion epoch by for a uniformly distributed value in

x 0≥ ρ* x( ) 0= x 0<

r N

ρ x 0,( )

ρ x t,( ) ρ* x( )

rn n

ρ x tn,( ) r 1 r–( ) l δ0 x l t∆⋅–( )⋅ ⋅l 0=

∞

∑⋅ 1 r–( ) n ρ x tn– 0,( )⋅+=

t∆ r C⋅( ) N⁄=

D

D

C N⁄ D C< <

x D=

t x D=

L

L t∆⋅ D≤ L 1+( ) t∆⋅<t∆ r C⋅( ) N⁄=

ρ̃* D

D

ρ̃* x( ) r 1 r–( ) l δ0 x l t∆⋅–( )⋅ ⋅l 0=

L

∑⋅ m r L,( ) δ0 x D–( )⋅+=

m r L,( ) r 1 r–( ) l

l L 1+=

∞

∑⋅ 1 r–( ) L 1+= =

D

ε N⁄ D L t∆⋅–<l L 1+=

i

εi N⁄ εi

, then only the duration of the st epoch isaffected and , where

.

Proof: Because we reset the source rates after an epochduration , the total rate immediately beforereset is allowed to grow to and immedi-ately following reset is . We letthe system of sources evolve as before until their totalrate reaches capacity . The duration of the st epochis the time interval given by

or. At this time (with no jit-

ter), we reset a fraction of the source rates and proceedas usual. Because at the st epoch, the total rate is

and we use the same evolution procedure and reset pol-icy at the nd epoch, the durationremains unaffected by the perturbation at the th epoch.

Note that if we jitter the st epoch duration by, then the total rate before reset is

(and this rate is not affected by the jit-ter at the th epoch). We may then apply the lemma tothe st epoch. Applying this lemma inductively, wemay conclude:

Theorem 3:If we jitter each congestion epoch dura-tion independently and let the system evolve, these per-turbations do not propagate and force the system awayfrom the invariant configuration with respect to the dura-tion of the epochs. That is, the duration of each epochvaries only by , the amount by which we perturb theduration, and on average .

These calculations do not tell us if the rates of the per-turbed system remain close to those of the original sys-tem however. We return to the setting of Lemma 1 andassume that the system has evolved long enough in timeto have the form immediately after a reset

where the mass at the rate bound is

Lemma 2:If we jitter the duration of the th conges-tion epoch by for a uniformly distributed value in

, then the source rates change by.

Proof: The reader is asked to verify that if and , then before

the st reset, the first rate population is at rate

ε– ε[ , ] i 1+( )t∆ i 1+ t∆ 1 r–( ) εi⋅ N⁄–=

t∆ r C⋅( ) N⁄=

t∆ i t∆ εi N⁄+=

A ti-

( ) C εi+=

A t+

i( ) 1 r–( ) C εi+( )⋅=

C i 1+

C A t+

i

N ti 1+∆⋅+ 1 r–( ) C εi+( )⋅ N ti 1+∆⋅+= =

ti 1+∆ rC 1 r–( ) εi⋅( )–( ) N⁄=

r

i 1+

C

i 2+ ti 2+∆ rC( ) N⁄=

i

i 1+

εi 1+ N⁄A t

-i 1+( ) C εi 1++=

i

i 1+( )

ε N⁄ti∆ t∆ rC( ) N⁄= =

ρ x t,( ) r 1 r–( ) l δ0 x l t∆⋅–( )⋅ ⋅l 0=

L

∑⋅ m̃n r L,( ) δ0 x D–( )⋅+=

m̃ r L,( ) x D=

m̃ r L,( ) r 1 r–( ) l

l L 1+=

∞

∑⋅ 1 r–( ) L 1+ ρ x 0,( ) xd0

∞∫⋅+=

i

εi N⁄ εi

ε– ε[ , ]

max 1 r–( ) εi⋅( ) N⁄ rεi( ) N⁄,( )

ti∆ t∆ εi N⁄+= ti 1+∆ t∆ 1 r–( ) εi( ) N⁄–=

i 1+

while all the others are at rate. In addition, for

iterations after the perturbation at the th epoch,the source rates have wandered from their original trajec-tory by at most . Afteriterations proceeding the perturbation, the system returnsto its initial trajectory.

By applying this lemma inductively, we may conclude:

Theorem 4:If we perturb each congestion epochduration independently and let the system evolve, thenthe maximum deviation of the source rates from integermultiples of is , for i.i.d. uni-form random variables on .

We observe that if the maximum source rate is onthe order of , then is approximately . Hence themaximum deviation is roughly and on aver-age this deviation is zero since is a uniformly distrib-uted random variable with mean zero.

APPENDIX B: RED parameters used in simulations

We use the same notation for the parameters as [4].• Queue occupancy is measured in number of packets.• Dropping probability is NOT increased slowly when

average queue length exceeds maxthresh.• When the algorithm decides to drop a packet, it

drops from the tail.• The algorithm doesn’t favor any packet based on its

type(control or data) or size.• (queue weight given to current queue size

sample).• (Maximum probability of dropping a

packet).

t∆ 1 r–( ) εi( ) N⁄–

k t∆ εi N⁄ 1 r–( ) εi( ) N⁄–+ k t∆ rεi( ) N⁄+=

k L 1–< i

max 1 r–( ) εi⋅( ) N⁄ rεi( ) N⁄,( ) L 1–

t∆ rC( ) N⁄= r N⁄ εkk 0=

L

∑⋅ εk

ε– ε[ , ]

D

C L N r⁄rLε N⁄ ε=

ε

qw 0.002=

pb 0.1=

congestion control and periodic behavior

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