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CONGESTION CONTROL FOR SELF-SIMILAR NETWORK TRAFFIC Network Systems Lab, Department of Computer Sciences, Purdue University, West Lafayette, IN 47907 18.1 INTRODUCTION Recent measurement

CONGESTION CONTROL FOR

SELF-SIMILAR NETWORK TRAFFIC

Network Systems Lab, Department of Computer Sciences, Purdue University,

West Lafayette, IN 47907

18.1 INTRODUCTION

Recent measurements of local-area and wide-area traf®c [8, 28, 42] have shown thatnetwork traf®c exhibits variability at a wide range of scales What is striking is theubiquitousness of the phenomenon, which has been observed in diverse networkingcontexts, from Ethernet to ATM, LAN and WAN, compressed video, and HTTP-based WWW traf®c [8, 15, 23, 42] Such scale-invariant variability is in strongcontrast to traditional models of network traf®c, which show burstiness at short timescales but are essentially smooth at large time scales; that is, they lack long-rangedependence Since scale-invariant burstiness can exert a signi®cant impact onnetwork performance, understanding the causes and effects of traf®c self-similarity

is an important problem

In previous work [33, 34], we have investigated the causal and performanceaspects of traf®c self-similarity, and we have shown that self-similar traf®c ¯ow is anintrinsic property of networked client=server systems with heavy-tailed ®le sizedistributions, and conjoint provision of low delay and high throughput is adverselyaffected by scale-invariant burstiness From a queueing theory perspective, theprincipal distinguishing characteristic of long-range-dependent (LRD) traf®c is thatthe queue length distribution decays much more slowlyÐthat is, polynomiallyÐvis-aÁ-vis short-range-dependent (SRD) traf®c sources such as Poisson sources, whichexhibit exponential decay A number of performance studies [1, 2, 11, 29, 32, 34]

Self-Similar Network Traf®c and Performance Evaluation, Edited by Kihong Park and Walter Willinger ISBN 0-471-31974-0 Copyright # 2000 by John Wiley & Sons, Inc.

447

Self-Similar Network Traf®c and Performance Evaluation, Edited by Kihong Park and Walter Willinger

Copyright # 2000 by John Wiley & Sons, Inc Print ISBN 0-471-31974-0 Electronic ISBN 0-471-20644-X

have shown that self-similarity has a detrimental effect on network performance,leading to increased delay and packet loss rate In Grossglauser and Bolot [18] andRyu and Elwalid [37], the point is advanced that for small buffer sizes or short timescales, long-range dependence has only a marginal impact This is, in part, due to asaturation effect that arises when resources are overextended, whereby the burstinessassociated with short-range-dependent traf®c is suf®cientÐand, in many cases,dominantÐto cause signi®cant buffer over¯ow.

What is still in its infancy, however, is the problem of controlling self-similarnetwork traf®c By the control of self-similar traf®c, we mean the problem ofmodulating traf®c ¯ow such that network performance including throughput isoptimized Scale-invariant burstiness introduces new complexities into the picture,which make the task of providing quality of service (QoS) while achieving highutilization signi®cantly more dif®cult First and foremost, scale-invariant burstinessimplies the existence of concentrated periods of high activity at a wide range of timescales which adversely affects congestion control Burstiness at ®ne time scales iscommensurate with burstiness observed for traditional short-range dependent traf®cmodels The distinguishing feature is burstiness at coarser time scales, whichinduces extended periods of either overload or underutilization and degrades overallperformance However, on the ¯ip side, long-range dependence, by de®nition,implies the existence of nontrivial correlation structure, which may be exploitablefor congestion control purposes, information to which current algorithms areimpervious

In this chapter, we show the feasibility of ``predicting the future'' under similar traf®c conditions with suf®cient reliability such that the information can beeffectively utilized for congestion control purposes First, we show that long-rangedependence can be on-line detected to predict future traf®c levels and contention attime scales above and beyond the time scale of the feedback congestion control.Second, we present a traf®c modulation mechanism based on multiple time scalecongestion control framework (MTSC) [46] and show that it is able to effectivelyexploit this information to improve network performance, in particular, throughput.The congestion control mechanism works by selectively applying aggressivenessusing the predicted future when it is warranted, throttling the data rate upward if thepredicted future contention level is low, being more aggressive the lower thepredicted contention level We show that the selective agressiveness mechanism is

self-of bene®t even for short-range-dependent traf®c; however, being signi®cantly moreeffective for long-range dependent traf®c, leading to comparatively large perfor-mance gains We also show that as the number of connections engaging in selectiveaggressiveness control (SAC) increases, both fairness and ef®ciency are preserved.The latter refers to the total throughput achieved across all SAC-controlled connec-tions

The rest of the chapter is organized as follows In Section 18.2, we give a briefoverview of self-similar network traf®c and the speci®c setup employed in thischapter In Section 18.3, we describe the predictability mechanism and its ef®cacy atextracting the correlation structure present in long-range dependent traf®c This isfollowed by Section 18.4, where we describe the SAC protocol and a re®nement of

448 CONGESTION CONTROL FOR SELF-SIMILAR NETWORK TRAFFIC

the predictability mechanism for on-line, per-connection estimation In Section 18.5

we show performance results of SAC and show its ef®cacy under different range dependence conditions and when the number of SAC connections is varied

long-We conclude with a discussion of current results and future work

18.2 PRELIMINARIES

18.2.1 Self-SimilarTraf®c: Basic De®nitions

Let …Xt†t2Z‡be a time series, which, for example, represents the trace of data ¯ow at

a bottleneck link measured at some ®xed time granularity We de®ne the aggregatedseries Xi…m† as

Xi…m†ˆ1

m…Xim m‡1‡


‡ Xim†:

That is, Xtis partitioned into blocks of size m, their values are averaged, and i is used

to index these blocks

Let r…k† and r…m†…k† denote the autocorrelation functions of Xt and Xi…m†,respectively Xt is self-similarÐmore precisely, asymptotically second-order self-similarÐif the following conditions hold:

for k and m large, where 0 < b < 1 That is, Xtis ``self-similar'' in the sense that thecorrelation structure is preserved with respect to time aggregationÐrelation (18.2)Ðand r…k† behaves hyperbolically withP1kˆ0r…k† ˆ 1 as implied by Eq (18.1) Thelatter property is referred to as long-range dependence

Let H ˆ 1 b=2 H is called the Hurst parameter, and by the range of b,1

2< H < 1 It follows from Eq (18.1) that the farther H is away from1

2 the morelong-range dependent Xt is, and vice versa Thus, the Hurst parameter acts as anindicator of the degree of self-similarity

A test for long-range dependence can be obtained by checking whether Hsigni®cantly deviates from1

2or not We use two methods for testing this condition.The ®rst method, the variance±time plot, is based on the slowly decaying variance of

a self-similar time series The second method, the R=S plot, use the fact that for aself-similar time series, the rescaled range or R=S statistic grows according to apower law with exponent H as a function of the number of points included Thus, theplot of R=S against this number on a log±log scale has a slope that is an estimate of

H A comprehensive discussion of the estimation methods can be found in Beran [4]and Taqqu et al [39]

18.2 PRELIMINARIES 449

A random variable X has a heavy-tailed distribution if

PrfX > xg  x a

as x ! 1, where 0 < a < 2 That is, the asymptotic shape of the tail of thedistribution obeys a power law The Pareto distribution,

p…x† ˆ akax a 1;with parameters a > 0; k > 0, x  k, has the distribution function

PrfX  xg ˆ 1 …k=x†a;and hence is clearly heavy tailed

It is not dif®cult to check that for a  2 heavy-tailed distributions have in®nitevariance, and for a  1, they also have in®nite mean Thus, as a decreases, a largeportion of the probability mass is located in the tail of the distribution In practicalterms, a random variable that follows a heavy-tailed distribution can take onextremely large values with nonnegligible probability

18.2.2 Structural Causality

In Park et al [33], we show that aggregate traf®c self-similarity is an intrinsicproperty of networked client=server systems where the size of the objects (e.g., ®les)being accessed is heavy-tailed In particular, there exists a linear relationshipbetween the heavy-tailedness measure of ®le size distributions as captured by aÐthe shape parameter of the Pareto distributionÐand the Hurst parameter of theresultant multiplexed traf®c streams That is, the aggregate network traf®c that isinduced by hosts exchanging ®les with heavy-tailed sizes over a generic networkenvironment running ``regular'' protocol stacks (e.g., TCP, ¯ow-controlled UDP) isself-similar, being more burstyÐin the scale-invariant senseÐthe more heavy-tailedthe ®le size distribution are This relationship is shown in Fig 18.1 The relationship

is robust with respect to changes in network resources (bandwidth, buffer capacity),topology, the in¯uence of cross-traf®c, and the distribution of interarrival times Wecall this relationship between the traf®c pattern observed at the network layer and thestructural property of a distributed, networked system in terms of its high-levelobject sizes structural causality [33] H ˆ …3 a†=2 is the theoretical valuepredicted by the on=off model [42]Ða 0=1 renewal process with heavy-tailed on

or off periodsÐassuming independent traf®c sources with no interactions due tosharing of network resources

Structural causality is of import to self-similar traf®c control since (1) it provides

an environment where self-similar traf®c conditions are easily facilitatedÐjustsimulate a client=server networkÐ(2) the degree of self-similar burstiness can beintimately controlled by the application layer parameter a, and (3) the self-similarnetwork traf®c induced already incorporates the actions and modulating in¯uence of

450 CONGESTION CONTROL FOR SELF-SIMILAR NETWORK TRAFFIC

the protocol stack since the observed traf®c pattern is a direct consequence of hostsexchanging ®les whose transport was mediated through protocols (e.g., TCP, ¯ow-controlled UDP) in the protocol stack This provides us with a natural environmentwhere the impact of control actions by a congestion control protocol can bediscerned and evaluated under self-similar traf®c conditions.

18.3 PREDICTABILITY OF SELF-SIMILAR TRAFFIC

18.3.1 Predictability Setup

In this section, we show that the correlation structure present in long-rangedependent (LRD) traf®c can be detected and used to predict the future over timescales relevant to congestion control Time series analysis and prediction theory havelong histories with techniques spanning a number of domains from estimation theory

to regression theory to neural network based techniques to mention a few [3, 17, 22,

40, 44, 45, 49] In many senses, it is an ``art form'' with different methods givingvariable performance depending on the context and modeling assumptions Our goal

is not to perform optimal time series prediction but rather to choose a simple, to-implement scheme, and use it as a reference for studying congestion controltechniques and their ef®cacy at exploiting the correlation structure present in LRDtraf®c for improving network performance Our prediction method, which isdescribed next, is a time domain technique and can be viewed as an instance ofconditional expectation estimation

easy-Fig 18.1 Hurst parameter estimates (R=S and variance±time) for a varying from 1.05 to1.95

18.3 PREDICTABILITY OF SELF-SIMILAR TRAFFIC 451

Assume we are given a wide-sense stationary stochastic process …xt†t2Z and twonumbers T1; T2> 0 At time t, we have at our disposal

i2‰t T 1 ;t†qi;where qiis a sample path of xtover time interval ‰t T1; t† For notational clarity, let

i2‰t T1;t†xi; V2ˆ P

i2‰t;t‡T2†xi:

a may be thought of as the aggregate traf®c observed over the ``recent past''

‰t T1; t† and V1, V2 are composite random variables denoting the recent past andnear future We are interested in computing the conditional probability

for b in the range of V2 For example, if a represented a ``high'' traf®c volume, then

we may be interested in knowing what the probability of encountering yet anotherhigh traf®c volume in the near future would be Let

In our case, eight levels …h ˆ 8† were found to be suf®ciently granular forprediction purposes In practice, Vt

18.3.2 Estimation of Conditional Probability Density

To explore and quantify the potential predictability of self-similar network traf®c, weuse TCP traf®c traces used in Park et al [33] whose Hurst parameter estimates areshown in Fig 18.1 as the main reference point First, we use off-line estimation ofaggregate throughput traf®c, which is then re®ned to on-line estimation of aggregatetraf®c using per-connection traf®c when performing predictive congestion control.Other traces including those collected from ¯ow-controlled UDP runs yield similarresults The traces used are each 10,000 seconds long at 10 ms granularity Theyrepresent the aggregate traf®c of 32 concurrent TCP Reno connections recorded at abottleneck router

We observe that the aggregate throughput series exhibit correlation structure atseveral time scales from 250 ms to 20 s and higher To estimate PrfL2jL1ˆ lg fromthe aggregate throughput series Xt, we segment Xt into

N ˆT10;000 …seconds†

1‡ T2…seconds†

contiguous nonoverlapping blocks of length T1‡ T2 (except possibly for the lastblock), and for each block j 2 ‰1; NŠ compute the aggregate traf®c V1, V2 over thesubintervals of length T1, T2

For l; l02 ‰1; 8Š, let hl2 ‰0; NŠ denote the total number of blocks such that

L1…V1† ˆ l and let hl0 2 ‰0; hlŠ denote the size of the subset of those blocks such that

L2…V2† ˆ l0 Then

PrfL2ˆ l0jL1ˆ lg ˆhhl0

l:Figure 18.2 shows the estimated conditional probability densities for a ˆ 1:05, 1.95traf®c for time scales 500 ms, 1 s, and 5 s In the following, T1ˆ T2

18.3 PREDICTABILITY OF SELF-SIMILAR TRAFFIC 453

For the aggregate throughput traces with a ˆ 1:05ÐFigure 18.2 (top row)Ðthethree-dimensional (3D) conditional probability densities can be seen to be skeweddiagonally from the lower left side toward the upper right side This indicates that ifthe current traf®c level L1is low, say, L1ˆ 1, chances are that L2will be low as well.That is, the probability mass of PrfL2jL1ˆ 1g is concentrated toward 1 Conversely,the plots show that PrfL2jL1ˆ 8g is concentrated toward 8 This is more clearly seen

in Fig 18.3(a), which shows two cross sections, that is, 2D projections, re¯ectingPrfL2jL1ˆ 1g and PrfL2jL1ˆ 8g

For the aggregate throughput traces with a ˆ 1:95 (Fig 18.2 (bottom-row)), onthe other hand, the shape of the distribution does not change as the conditioningvariable L1 is varied This is more clearly seen in the projections of PrfL2jL1ˆ 1gand PrfL2jL1ˆ 8g shown in Fig 18.3(b) This indicates that for a ˆ 1:95 traf®cobserving the past (over the time scales considered) does not help much in predictingthe future beyond the information conveyed by the ®xed a priori distribution Giventhe de®nition of Lk, the Gaussian shape of the marginal densities is consistent withshort-range correlations, making the central limit theorem approximately applicableover larger time scales In both cases …a ˆ 1:05, 1.95), the shape of the distributionstays relatively constant across a wide range of time scales 500 ms to 20 s For

a ˆ 1:35, 1.65 the predictability structure lies ``in-between'' (not shown here)

18.3.3 Predictability and Time Scale

An important issue is how time scale affects predictability when traf®c is long-rangedependent Going back to Fig 18.2 (top row), one subtle effect that is not easilydiscernible is that as time scale is increased the conditional probability densitiesPrfL2jL1ˆ lg become more concentrated Given that PrfL2jL1ˆ lg is a function of

T1, T2, we would like to determine at what time scale predictability is maximized.One way to measure the ``information content''Ðthat is, in the sense ofrandomness or unstructurednessÐin a probability distribution is to compute its

18.3 PREDICTABILITY OF SELF-SIMILAR TRAFFIC 455

entropy For a discrete probability density pi, its entropy S…pi† is de®ned as

S…pi† ˆPipilog…1=pi† In the case of our conditional density PrfL2jL1 ˆ lg,

Sl ˆ P8

l 0 ˆ1PrfL2ˆ l0jL1 ˆ lg log PrfL2ˆ l0jL1ˆ lg:

Thus, entropy is maximal when the distribution is uniform and it is minimal if thedistribution is concentrated at a single point Since we are given a set of eightconditional probability densities, one for each L1ˆ 1; 2; ; 8, we de®ne theaverage entropy S as

S ˆ P8lˆ1Sl=8:

The average entropy remains a function of T1; T2: that is, S ˆ S…T1; T2†

Figure 18.4 plots S…T1; T2† ˆ S…T1† (recall that T1ˆ T2) for the a ˆ 1:05throughput series as a function of time scale or aggregation level T1 Entropy ishighest for small time scales in the range 250 ms, and it drops monotonically as T1

is increased Eventually, S…T1† begins to ¯atten out near the 3±5 second mark,reaching saturation, and stays so as time scale is further increased From our analysis

of various long-range dependent traf®c traces, we ®nd that the ``knee'' of the entropycurve is in the range of 1±5 seconds Note that increasing T1 further and further togain small decreases in entropy brings forth with it an important problem, namely, ifprediction is done over a ``too long'' time interval, then the information may not beeffectively exploitable by various congestion control strategies In the next section,

Aggregation Level (seconds)

456 CONGESTION CONTROL FOR SELF-SIMILAR NETWORK TRAFFIC

we show that one strategyÐselective aggressivenessÐis effective at exploiting thepredictability structure found in the 1±5 second range.

18.4 SAC AND PREDICTIVE CONGESTION CONTROL

In this section we present a congestion control strategy called selective ness control (SAC) and show its ef®cacy at exploiting the predictability structurepresent in long-range dependent traf®c for improving network performance Ourcontrol scheme is a form of predictive congestion control based on the multiple timescale congestion control framework (MTSC) [46] Explicit prediction of the longterm network state l is performed at SAC's time scale (1±5 seconds) A certaincontrol action e…l† is made by SAC based on this information about the future and isincorporated into the underlying congestion control to affect traf®c control deci-sions The overall structure is shown in Figure 18.5 SAC is aimed to be robust,ef®cient, and portable such that it can easily be incorporated into existing congestioncontrol schemes

aggressive-SAC's modus operandi is to complement and help improve the performance ofexisting reactive congestion controls Toward this end, we set up a simple, genericrate-based feedback congestion control as a reference and let our control module

``run on top'' of it SAC always respects the decision made by the underlyingcongestion control with respect to the directional change of the traf®c rateÐup ordown; however, it may choose to adjust the magnitude of change That is, if, at anytime, the underlying congestion control decides to increase its sending rate, SACwill never take the opposite action and decrease the sending rate Instead, what SAC

Fig 18.5 The overall structure of predictive congestion control SAC module is active attime scale (1±5 sec) exceeding the time scale of the underlying congestion control of itsfeedback loop

18.4 SAC AND PREDICTIVE CONGESTION CONTROL 457

will do is amplify or diminish the magnitude of the directional change based on itspredicted future network state.

In a nutshell, SAC will try to aggresively soak up bandwidth if it predicts thefuture network state to be ``idle,'' adjusting the level of aggressiveness as a function

of the predicted idleness We will show that the performance gain due to SAC ishigher the more long-range dependent the network traf®c is

18.4.1 Underlying Congestion Control

18.4.1.1 Generic Feedback Congestion Control Congestion control has been anactive area of networking research spanning over two decades with a ¯urry ofconcentrated work carried out in the late 1980s and early 1990s [5, 6, 16, 19, 24, 25,

27, 30, 31, 35, 36, 38] Gerla and Kleinrock [16] laid down much of the earlygroundwork and Jacobson [24] has been instrumental in in¯uencing the practicalmechanisms that have survived until today A central part of the investigation hasbeen the study of stability and optimality issues [5, 13, 24, 25, 30, 31, 35, 38]associated with feedback congestion control A taxonomy for classifying the variousprotocols can be found in Yang and Reddy [43]

More recently, the delay-bandwidth product problem arising out of width networks and quality of service issues stemming from support of real-timemultimedia communication [7, 10, 12, 20, 21, 41] have added further complexities

high-band-to the problem with QoS reigning as a unifying key theme One of the lessonslearned from congestion control research is that end-to-end rate-based feedbackcontrol using various forms of linear increase=exponential decrease can be effective,and asymmetry in the control law needs to be preserved to achieve stability

We employ a simple, generic instance of rate-based feedback congestion control

as a reference to help demonstrate the ef®cacy of selective aggressiveness controlunder self-similar traf®c conditions SAC is motivated, in part, by the simple yetimportant point put forth in Kim [26], which shows that the conservative nature ofasymmetric controls can, in some situations, lead to throughput smaller than thatachieved by a ``nearly blind'' aggressive control By applying aggressivenessselectivelyÐbased on the prediction of future network contentionÐwe seek tooffset some of the cost incurred for stability

Let l denote packet arrival rate and let g denote throughput Our generic linearincrease=exponential decrease feedback congestion control has a control law of theform1

1 We use continuous notation for expositional clarity.

458 CONGESTION CONTROL FOR SELF-SIMILAR NETWORK TRAFFIC

sely, if increasing the data rate results in a decrease in throughput (i.e., dg=dl < 0),then exponentially decrease the data rate In general, condition dg=dl < 0 can bereplaced by various measures of congestion.

Of course, dif®culties arise because Eq (18.4) is, in reality, a delay differentialequation (the feedback loop incurs a time lag) and the sign of dg=dl needs to bereliably estimated The latter can be implemented using standard techniques

18.4.1.2 Unimodal Load-Throughput Relation One item that needs furtherexplanation is throughput g ``Throughput'' (in the sense of goodput) can be de®ned

in a number of ways depending on the context, from reliable throughput (number ofbits reliably transferred per unit time when taking into account reliability mechanismoverhead), to raw throughput (number of bits transferred per unit time), to power(one of the throughput measures divided by delay) Raw throughput, denoted n, isboth easy to measure ( just monitor the number of packets, in bytes, arriving at thereceiver per unit time) and to attain (in most contexts n ˆ n…l† is a monotoneincreasing function of l, e.g., M=M=1=n), but it does not adequately discriminatebetween congestion controls that achieve a certain raw throughput without incurringhigh packet loss or delay and those that do

For example, achieving reliability using automatic repeat request (ARQ) with

®nite receiver and sender side buffers requires intricate control and coordination, andhigh packet loss can have a severe impact on the ef®cient functioning of suchcontrols (e.g., TCP's window control) In particular, for a given raw throughput, ifthe packet loss rate is high, this may mean that a signi®cant fraction of the rawthroughput is taken up by duplicate packets (due to early retransmissions) or bypackets that will be dropped at the receiver side due to ``fragmentation'' and bufferover¯ow Thus, the reliable throughput associated with this raw throughput=packetloss rate combination would be low

How severely packet loss impacts the throughput experienced by an applicationwill depend on the characteristics of the application at hand To better re¯ect suchcosts, we will use a throughput measure gk,

that (polynomially) penalizes raw throughput n by packet loss rate 0  c  1, wherethe severity can be set by the parameter k  0 Thus raw throughput n is a specialinstance of gk with k ˆ 0 We will measure instantaneous throughput gk at thereceiver and feedback to the sender for use in the control law (18.4) Figure 18.6illustrates the relationship between gk and l for an M=M=1=n queueing system,which shows that for c > 0 the load-throughput curve gkˆ gk…l† is unimodal Notethat c is a monotone decreasing function of l while n is monotone increasing In thecase of M=M=1=n and most other network systems, raw bandwidth is upperbounded by the service rate or link speedÐthat is, n  mÐand thus most load-throughput functions of interest (not just Eq (18.5)) will be unimodal due to theabove montonicity properties

18.4 SAC AND PREDICTIVE CONGESTION CONTROL 459

18.4.2 Selective Aggressiveness Control (SAC)

Assuming that future network contention is predictable with a suf®cient degree ofaccuracy, there remains the question of what to do with this information forperformance enhancement purposes The choice of actions, to a large measure, isconstrained by the networking context and what degree of freedom it allows In thetraditional end-to-end congestion control setting, the network is a shared resourcetreated as a black box, and the only control variable available to a ¯ow is its traf®crate l

In this chapter, the target mechanism to be improved using predictability is theperformance loss stemming from conservative bandwidth usage during the linearincrease phase of linear increase=exponential decrease congestion control algo-rithms [26] Feedback congestion control protocols, including TCP, implementvariants of this basic control law due to well-established stability reasons In Kim[26], however, it was shown in the context of TCP Reno that the asymmetrystemming from linear increase after exponential back-off ends up signi®cantlyunderutilizing bandwidth such that, in some situations, a simple nonfeedback controlwas shown to be more effective.2

Given that linear increase=exponential decrease is widely used in congestioncontrol protocols including TCP, we seek to target the linear increase part of suchprotocols such that, when deemed bene®cial, and only then, a more aggressive band-width consumption is facilitated This selective application of aggressiveness, whencoupled with predictive capability, will hopefully lead to a more effective use of

bandwidth, resulting in improved performance Without selective, controlled cation of aggressiveness, however, the gain from aggressiveness may be canceled out(or even dominated) by its costÐaggressiveness, under high network contentionconditions, can lead to deteriorated performance, even congestion collapseÐtherebymaking predictability and its appropriate exploitation a nontrivial problem.Our protocolÐselective aggressiveness control (SAC)Ðis composed of twoparts, prediction and application of aggression, and they are described next.

appli-18.4.2.1 Per-Connection On-Line Estimation of Future Contention In the to-end feedback congestion control context, the two principal problems that aconnection faces when estimating future network contention are:

end-1 The need to estimate ``global'' network contention using ``local'' tion information

per-connec-2 The need to perform on-line prediction

First, with respect to requirement (1), since the network is a black box as far as anend-to-end connection or ¯ow is concerned, we cannot rely on internal networksupport such as congestion noti®cation via router support to reveal network stateinformation Instead, we need to gleamÐin our case, predictÐfuture network stateusing information obtained from a ¯ow's input=output interaction with the network.For this to work, two assumptions need to hold in practice First, due to the couplinginduced by sharing of common resources, a connection's individual throughput whenengaging in feedback congestion control (such as Eq (18.4)) is correlated with theaggregate ¯ow accessing the same resources Second, aggregate traf®c level, whenpartitioned according to the quantization scheme Lk…Vk† of Section 18.3.1, iscorrelated to the contention level at the router that the aggregate traf®c enters.Second, with respect to requirement (2), it turns out that on-line estimation of theconditional probability density PrfL2jL1ˆ lg is easily and ef®ciently accomplishedusing O…1† cost update operations On the sender side, SAC maintains a two-dimensional array or table

CondProb‰
Š‰
Š

of size 8  9, one row for each l 2 ‰1; 8Š The last column of CondProb,CondProb[l][9], is used to keep track of hl, the number of blocks observedthus far whose traf®c level maps to l, that is, L1…V1† ˆ l (see Section 18.3.1).For each l02 ‰1; 8Š, CondProb[l][l'] maintains the count hl0 SincePrfL2ˆ l0jL1ˆ lg ˆ hl0=hl, having the table CondProb means having the condi-tional probability densities

All that is needed to maintain CondProb is a clock or alarm of period 2, which,starting at time t ˆ 0, goes off at times

t ˆ T1; T1‡ T2; T1‡ T2‡ T1; T1‡ T2‡ T1‡ T2;

18.4 SAC AND PREDICTIVE CONGESTION CONTROL 461

If a feedback packet containing an instantaneous throughput g measured at thereceiver arrives during the period

‰i…T1‡ T2†; i…T1‡ T2† ‡ T1Š; i  0;

it is added to V1 When the alarm goes off at t ˆ i…T1‡ T2† ‡ T1, V1 is used tocompute the updated Vt

min; Vt max and the quantization step m which can be easilydone incrementally by using O…1† operations Now l ˆ L1…V1† is computed using theupdated Vt

O…1†

It should be noted that the conditional densities computed from CondProb attime t are approximations to the conditional probability densities computed off-linefor the period ‰0; tŠ since in the on-line algorithm running sums are used to computeand update Vt

The actuation part of the interface between SAC and Eq (18.4) is de®ned asfollows Let ltdenote the newly updated rate value at time tÐby Eq (18.4)Ðand let

lt0 be the most recently …t0< t† updated rate value previous to t

SAC (Actuation Interface)

1 If lt > lt0 then update lt lt‡ Et

2 Else do nothing

Here, Et 0 is an aggressiveness factor that is determined by SAC based on thecurrent state of CondProb Note that SAC kicks into action only during the linearincrease phase of Eq (18.4), that is, when lt> lt0 The magnitude of Et determinesthe degree of aggressiveness, and it is determined as a function of the predictednetwork state as captured by CondProb and its conditional probability densities

At time t, the algorithm used to determine E is as follows Let Stbe the aggregatethroughput reported by the receiver via feedback over time interval ‰t T1; tŠ

462 CONGESTION CONTROL FOR SELF-SIMILAR NETWORK TRAFFIC

E…l0† ˆ 1=l0:Other schedules of interest include the threshold schedule with threshold y 2 ‰1; 8Šand aggressiveness factor y*, where E ˆ y* if l0 y, and 0 otherwise.

Table 18.1 shows the CondProb table for two runs corresponding to a ˆ 1:05(top) and a ˆ 1:95 (bottom) traf®c conditions The column containing hl has beenomitted and the entries show actual relative frequencies rather than hl0 counts forillustrative purposes Clearly, the conditional probability densities are skeweddiagonally for a ˆ 1:05 traf®c, whereas they are roughly invariant for a ˆ 1:95traf®c The expected future contention level l0ˆ E…L2jL1 ˆ l† and aggressivenessschedule (inverse) are shown as separate columns For a ˆ 1:05 traf®c, the expectedfuture contention level E‰L2j
Š varies over a wide range, which is a directconsequence of the predictabilityÐthat is, skewednessÐpresent in the correlationstructure For a ˆ 1:95 traf®c, however, E‰L2j
Š is fairly ``¯at,'' indicating thatconditioning on the present does not aid signi®cantly in predicting the future

18.5 SIMULATION RESULTS

18.5.1 Congestion Control Evaluation Setup

We use the LBNL Network Simulator, ns (version 2), as the basis of our simulationenvironment ns is an event-driven simulator derived from Keshav's REAL networksimulator supporting several ¯avors of TCP and router packet scheduling algo-rithms We have modi®ed ns in order to model a bottleneck network environmentwhere several concurrent connections are multiplexed over a shared bottleneck link

A UDP-based unreliable transport protocol was added to the existing protocol suite,and our congestion control and predictive control were implemented on top of it

An important feature of the setup is the mechanism whereby self-similar traf®cconditions are induced in the network One possibility is to have a host inject self-

18.5 SIMULATION RESULTS 463