Tài liệu Mạng lưới giao thông và đánh giá hiệu suất P17 ppt

The bandwidth demand for each source as a function of time, fB t†: t 0g, is represented as the sum of two stochastic processes: 1 a scopic longer-time-scale level process fL t†: t 0g a

NETWORK DESIGN AND CONTROL USING ON=OFF AND MULTILEVEL SOURCE TRAFFIC MODELS WITH

HEAVY-TAILED DISTRIBUTIONS

N G DUFFIELD ANDW WHITT

AT&T Labs±Research, Florham Park, NJ 07392

17.1 INTRODUCTION

In order to help design and control the emerging high-speed communicationnetworks, we want source traf®c models (also called offered load models orbandwidth demand models) that can be both realistically ®t to data and successfullyanalyzed Many recent traf®c measurements have shown that networktraf®c is quitecomplex, exhibiting phenomena such as heavy-tailed probability distributions, long-range dependence, and self similarity; for example, see CaÂceres et al [7], Leland

et al [23], Paxson and Floyd [24], and Crovella and Bestavros [10]

In fact, the heavy-tailed distributions may be the cause of all these phenomena,because they tend to cause long-range dependence and (asymptotic) self-similarity.For example, the input and buffer content processes associated with an on=off sourceexhibit long-range dependence when the on and off times have heavy-tailedprobability distributions; for example, see Section 17.9 Heavy-tailed distributionsare known to cause self-similarity in models of (asymptotically) aggregated traf®c;see Willinger et al [27]

In this chapter we propose a way to analyze the performance of a networkwithmultiple on=off sources and more general multilevel sources in which the on-time,off-time, and level-holding-time distributions are allowed to have heavy tails To do

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

421

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

so we must go be beyond the familiar Markovian analysis To achieve the requiredanalyzability with this added model complexity, we propose a simpli®ed kind ofanalysis In particular, we avoid the customary queueing detail (and its focus onbuffer content and over¯ow) and instead concentrate on the instantaneous offeredload We describe the probability that aggregate demand (the input rate from acollection of sources) exceeds capacity (the maximum possible output rate) at anytime Focusing on the probability that aggregate demand exceeds capacity istantamount to considering a bufferless model, which we believe is often justi®ed.

By also considering the probability that aggregate demand exceeds other levels, weprovide a quite ¯exible performance characterization This approach also cangenerate approximations describing loss and delay with ®nite capacity; for example,see Duf®eld and Whitt [14], Section 5 To a large extent, the present chapter is areview of our recent work[14, 15], to which we refer the reader for additionaldiscussion In Duf®eld et al [16] the model is extended to include a nonhomoge-neous Poisson connection arrival process Then each active connection may generatetraf®c according to one of the source traf®c models presented here It is signi®cantthat we are able to obtain useful descriptions of the offered load in the nonstationarycontext

17.2 A GENERAL SOURCE MODEL

Motivation for considering on=off and multilevel models as source models comesfrom traces of frame sizes generated by certain video encoders; for example, seeGrasse et al [19] Shifts between levels in mean frame size appear to arise fromscene changes in the video, with the distribution of scene durations heavy-tailed.Indeed, the expectation that scene durations will have heavy-tailed distributions isone of the motivations behind the renegotiated constant bit rate (RCBR) proposal ofGrossglauser et al [20]

Our approach is interesting for on=off and multilevel source models, but withlittle extra effort we can treat a wider class The general model we consider has twocomponents The bandwidth demand for each source as a function of time,

fB…t†: t  0g, is represented as the sum of two stochastic processes: (1) a scopic (longer-time-scale) level process fL…t†: t  0g and (2) a microscopic (shorter-time-scale) within-level variation process fW…t†: t  0g, that is,

We let the macroscopic level process fL…t†: t  0g be a semi-Markov process (SMP)

as in CËinlar [9, Chap 10]; that is, the level process is constant except for jumps, withthe jump transitions governed by a Markov process, while the level holding times(times between jumps) are allowed to have general distributions depending on theoriginating level and the next level Given a transition from level j to level k, theholding time in level j has cumulative distribution function (cdf) Fjk Conditional onthe sequence of successive levels, the holding times are mutually independent To

obtain models compatible with traf®c measurements cited earlier, we allow theholding-time cdf's Fjk to have heavy tails.

We assume that the within-level variation process fW…t†: t  0g is a zero-meanpiecewise-stationary process During each holding-time interval in a level, thewithin-level variation process is an independent segment of a zero-mean stationaryprocess, with the distribution of each segment being allowed to depend on the level

We allow the distribution of the stationary process segment to depend on the level,because it is natural for the variation about any level to vary from level to level

We will require only a limited characterization of the within-level variationprocess; it turns out that the ®ne structure of the within-level variation process plays

no role in our analysis Indeed, that is one of our main conclusions In severalexamples of processes that we envisage modeling by these methods, there will only

be the level process First, the level process may be some smoothed functional of araw bandwidth process This is the case with algorithms for smoothing stored video

by converting into piecewise constant rate segments in some optimal manner subject

to buffering and delay constraints; see Salehi et al [25] With such smoothing, theinput rate will directly be a level process as we have de®ned it Alternatively, thelevel process may stem from rate reservation over the period between level-shifts,rather than the bandwidth actually used This would be the case for RCBRpreviously mentioned In this situation we act as if the reservation level is theactual demand, and thus again have a level process

A key to being able to analyze the system with such complex sources represented

by our traf®c model is exploiting asymptotics associated with multiplexing a largenumber of sources The ever-increasing networkbandwidth implies that more andmore sources will be able to be multiplexed This gain is generally possible, even inthe presence of heavy-tailed distributions and more general long-range dependence;for example, see Duf®eld [12, 13] for demonstration of the multiplexing gainsavailable for long-range dependent traf®c in shared buffers As the scale increases,describing the detailed behavior of all sources become prohibitively dif®cult, butfortunately it becomes easier to describe the aggregate, because the large numbersproduce statistical regularity As the size increases, the aggregate demand can bewell described by laws of large numbers, central limit theorems, and large deviationprinciples

We have in mind two problems: ®rst, we want to do capacity planning and,second, we want to do real-time connection-admission control and congestioncontrol In both cases, we want to determine whether any candidate capacity isadequate to meet the aggregate demand associated with a set of sources In bothcases, we represent the aggregate demand simply as the sum of the bandwidthrequirements of all sources In forming this sum, we regard the bandwidth processes

of the different sources as probabilistically independent

The performance analysis for capacity planning is coarser, involving a longertime scale, so that it may be appropriate to do a steady-state analysis However, when

we consider connection-admission control and congestion control, we suggestfocusing on a shorter time scale We are still concerned with the relatively longtime scale of connections, or scene times in video, instead of the shorter time scales

of cells or bursts, but admission control and congestion control are suf®ciently term that we propose focusing on the transient behavior of the aggregate demandprocess In fact, even for capacity planning the transient analysis plays an importantrole The transient analysis determines how long it takes to recover from rarecongestion events One application we have in mind is that of networks carryingrate-adaptive traf®c In this case the bandwidth process could represent the idealdemand of a source, even though it is able to function when allocated somewhat lessbandwidth So from the point of view of quality, excursion of aggregate bandwidthdemand above available supply may be acceptable in the short-term, but one wouldwant to dimension the linkso that such excursions are suf®ciently short-lived In this

short-or other contexts, if the recovery time from overload is relatively long, then we mayelect to provide extra capacity (or reduce demand) so that overload becomes lesslikely However, we do not focus speci®cally on actual design and control here; seeDuf®eld and Whitt [14] for some speci®c examples Our main contribution here is toshow how the transient analysis for design and control can be done

The remainder of this chapter is devoted to showing how to do transient analysiswith the source traf®c model We suggest focusing on the future time-dependentmean conditional on the present state The present state of each level processconsists of the level and age (elapsed holding time in that level) Because of theanticipated large number of sources, the actual bandwidth process should be closelyapproximated by its mean, by the law of large numbers (LLN) As in Duf®eld andWhitt [14], the conditional mean can be thought of as a deterministic ¯uidapproximation; for example, see Chen and Mandelbaum [8] Since the within-level variation process has mean zero, the within-level variation process has no effect

on this conditional mean Hence, the conditional mean of the aggregate bandwidthprocess is just the sum of the conditional means of the component level processes.Unlike the more elementary M=G=1 model considered in Duf®eld and Whitt [14],however, the conditional mean here is not available in closed form

In order to rapidly compute the time-dependent conditional mean aggregatedemand, we exploit numerical inversion of Laplace transforms It follows quitedirectly from the classical theory of semi-Markov process that explicit expressionscan be given for the Laplace transform of the conditional mean More recently, it hasbeen shown that numerical inversion can be an effective algorithm; see Abate et al.[1]

For related discussions of transient analysis, design and control, see Chapters 13,

16, and 18 in this volume

17.3 OUTLINE OF THE CHAPTER

The rest of this chapter is organized as follows In Section 17.4, we show that theLaplace transform of the mean of the transient conditional aggregate demand can beexpressed concisely This is the main enabling result for the remainder of the chapter.The conditional mean itself can be very ef®ciently computed by numericallyinverting its Laplace transform To carry out the inversion, we use the Fourier-

series method in Abate and Whitt [2] (the algorithm Euler exploiting Eulersummation), although alternative methods could be used The inversion algorithm

is remarkably fast; computation for each time point corresponds simply to a sum of

50 terms We provide numerical examples in Examples 17.6.2 and 17.8.1 Example17.8.1 is of special interest, because the level-holding-time distribution there isPareto

In Section 17.5 we show that in some cases we can avoid the inversion entirelyand treat much larger models We can avoid the inversion if we can assume that thelevel holding times are relatively long compared to the times of interest for control.Then we can apply a single-transition approximation, which amounts to assumingthat the Markov chain is absorbing after one transition Then the conditional mean isdirectly expressible in terms of the level-holding-time distributions Alternatively, wecan perform a two-transition approximation, which only involves one-dimensionalconvolution integrals

In Section 17.6 we describe the value of having more detailed state information,speci®cally the current ages of levels With heavy-tailed distributions, a large elapsedholding time means that a large remaining holding time is very likely; for examplessee Duf®eld and Whitt [14, Section 8] for background, and Harchol-Balter andDowney [21] for an application in another setting

In Section 17.7 we turn to applications to capacity planning The idea is toapproximate the probability of an excursion in demand using Chernoff bounds andother large deviation approximations, then chart its recovery to a target acceptablelevel using the results on transience Interestingly, the time to recover fromexcursions suf®ciently close to the target level depends on the level durationsessentially only through their mean Correspondingly, the conditional mean demandrelaxes linearly from its excursion, at least approximately so, for suf®ciently smalltimes If the chance for a larger excursion is negligible (as determined by the largedeviation approximation mentioned) then this simple description may suf®ce Anexample is given in Section 17.8

In Section 17.9 we show how long-range dependence in the level process arisesthrough heavy-tailed level-holding-time distributions Finally, we draw conclusions

in Section 17.10

17.4 TRANSIENT ANALYSIS

17.4.1 Approximation by the Conditional Mean Bandwidth

Throughout this chapter, the state information on which we condition will be eitherthe current level of each source or the current level and age (current time) in thatlevel of each source No state from the within-level variation process is assumed.Conditional on that state information, we can compute the probability that eachsource will be in each possible level at any time in the future, from which we cancalculate the conditional mean and variance of the aggregate required bandwidth byadding

The Lindberg±Feller central limit theorem (CLT) for non-identically-distributedsummands can be applied to generate a normal approximation characterized by theconditional mean and conditional variance; see Feller [18, p 262] For the normalapproximation to be appropriate, we should checkthat the aggregate is notdominated by only a few sources.

Let B…t† denote the (random) aggregate required bandwidth at time t, and let I…0†denote the (known deterministic) state information at time 0 Let …B…t†jI…0††represent a random variable with the conditional distribution of B…t† given theinformation I…0† By the CLT, the normalized random variable

…B…t†jI…0†† E…B…t†jI…0††

Var…B…t†jI…0††

is approximately normally distributed with mean 0 and variance 1 when the number

of sources is suitably large

Since the conditional mean alone tends to be very descriptive, we use theapproximation

O…pn†, while the conditional mean is O…n†

Given that our approximation is the conditional mean, and given that our stateinformation does not include the state of the within-level variation process, thewithin-level variation process plays no role because it has zero mean Let i index thesource Since the required bandwidths need not have integer values, we index thelevel by the integer j; 1  j  Ji, and indicate the associated required bandwidths inthe level by bi Hence, instead of Eq (17.1), the required bandwidth for source i can

information is I…0† ˆ … j; x† ˆ … j1; ; jn; x1; ; xn† and the conditional aggregatemean is

E…B…t†jI…0††  M…tjj; x† ˆPn

iˆ1

PJikiˆ1P…i†jiki…tjxi†bi

From Eq (17.5), we see that we need to compute the conditional distribution ofthe level, that is, the probabilities P…i†jk…tjx†, for each source i However, we can ®ndrelatively simple expressions for the Laplace transform of P…i†jk…tjx† with respect totime because the level process of each source has been assumed to be a semi-Markovprocess

We now consider a single source and assume that its required bandwidth process

is a semi-Markov process (SMP) (We now have no within-level variation process.)

We now omit the superscript i Let L…t† and B…t† be the level and required bandwidth,respectively, at time t as in Eq (17.4) The process fL…t†: t  0g is assumed to be anSMP, while the process fB…t†: t  0g is a function of an SMP, that is B…t† ˆ bL…t†,where bjis the required bandwidth in level j If bj6ˆ bkfor j 6ˆ k, then fB…t†: t  0gitself is an SMP, but if bjˆ bkfor some j 6ˆ k, then in general fB…t†: t  0g is not anSMP

17.4.2 Laplace Transform Analysis

Let A…t† be the age of the level holding time at time t We are interested in calculating

as a function of j; k; x, and t The state information at time 0 is the pair … j; x† Let P

be the transition matrix of the discrete-time Markov chain governing level transitionsand let Fjk…t† be the holding-time cdf given that there is a transition from level j tolevel k For simplicity, we assume that Fc

jk…t† ˆ 1 Fjk…t† > 0 for all j; k, and t, sothat all positive x can be level holding times Let P…tjx† be the matrix with elements

Pjk…tjx† and let ^P…sjx† be the Laplace transform (LT) of P…tjx†, that is, the matrix withelements that are the Laplace transforms of Pjk…tjx† with respect to time:

For any cdf G, let Gcbe the complementary cdf, that is, Gc…x† ˆ 1 G…x† Also, let

Theorem 17.4.1 The transient probabilities for a single SMP source have thematrix of Laplace transforms

0 dQjl…u†Plj…t uj0†;

so that

P…tj0† ˆ D…t† ‡ Q…t†  P…tj0†

To compute the LT ^P…sj0†, we only need the LSTs ^fjk…s† and ^gj…s† associated withthe basic holding-time cdf's Fjk and Gj Abate and Whitt [3±5] give special attention

to heavy-tail probability densities whose Laplace transforms can be computed and,thus, inverted However, to compute ^P…sjx†, we also need to compute ^D…sjx† and

^h…sjx†, which require computing the LSTs of the conditional cdf's Hjk…tjx† and

Gj…tjx† in Eq (17.9) In general, even if we know the LST of a cdf, we do notnecesssarily know the LST of an associated conditional cdf However, in specialcases, the LSTs of conditional cdf's are easy to obtain because the cdf's inherit theirstructure upon conditioning For example, this is true for phase-type, hyperexpo-nential and Pareto distributions; Duf®eld and Whitt [15, Section 4] Moreover, othercdf's can be approximated by hyperexponential or phase-type cdf's; see Asmussen et

al [6] and Feldman and Whitt [17]

If the number of levels is not too large, then it will not be dif®cult to compute therequired matrix inverse …I q…s†† 1 for all required s Note that, because of theprobability structure, the inverse is well de®ned for all complex s with Re…s† > 0

To illustrate with an important simple example, we next give the explicit formulafor an on=off source Suppose that there are two states with transition probabilities

P12ˆ P21ˆ 1 and holding time cdf's G1 and G2 From Eq (17.9) or by directcalculation,

vj…t; x† so that wj…t; x† can be omitted, but it could be included.

Finally, we consider the aggregate bandwidth associated with n sources Again let

a superscript i index the sources The conditional aggregate mean and variance are

It is signi®cant that we can calculate the conditional aggregate mean at any time t

by performing a single numerical inversion, for example, by using the Euleralgorithm in Abate and Whitt [2] We summarize this elementary but importantconsequence as a theorem

Theorem 17.4.2 The Laplace transform of the n-source conditional meanaggregate required bandwidth as a function of time is

^M…sjj; x† …1

0 e stM…tjj; x† dt ˆPn

iˆ1

PJi kiˆ1^P…i†

Jiki…sjxi†bki; …17:25†

where the single-source transform ^P…i†jiki…sjxi† is given in Theorem 17.4.1

Unlike for the aggregate mean, for the aggregate variance we evidently need toperform n separate inversions to calculate vi

i…tjxi† for each i and then add to calculate

V…tjj; x† in Eq (17.24) (We assume that the within-level variances wi

i…tjxi†, ifincluded, are speci®ed directly.) Hence, we suggest calculating only the conditionalmean in real time to perform control, and occasionally calculating the conditionalvariance to evaluate the accuracy of the conditional mean

17.5 APPROXIMATIONS USING FEW TRANSITIONS

The most complicated part of the conditional aggregate mean transform ^M…sjj; x† in

Eq (17.25) is the matrix inverse …I ^q…s†† 1 in the transform of the single-sourcetransition probability in Eq (17.14) Since the matrix inverse calculation can be acomputational burden when the number of levels is large, it is natural to seekapproximations that avoid this matrix inverse We describe such approximations inthis section

nˆ0q…s†n For P…tjx†, it captures the possibility of any number of transitions up

to time t However, if the holding times in the levels are relatively long in the timescales relevant for control, then the mean for times t of interest will only besigni®cantly affected by a very few transitions Indeed, often only a single transitionneed be considered

The single-transition approximation is obtained by making the Markov chainabsorbing after one transition Hence, the single-transition approximation is simply

Pjk…tjx†  Hjk…tjx†; j 6ˆ k; and Pjj…tjx†  Gc

j…tjx† ‡ Hjj…tjx† …17:26†for Hjk…tjx† in Eq (17.9) and Gj…tjx† in Eq (17.9) From Eq (17.26) we see that noinversion is needed

Alternatively, we can develop a two-transition approximation (Extensions tohigher numbers are straightforward.) Modifying the proof of Theorem 17.4.1 in astraightforward manner, we obtain