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Teletraffic Engineering in a Broad-Band Era

HERMAN MICHIEL AND KOEN LAEVENS

This paper gives an overview of the methods and models used forthe performance evaluation of asynchronous transfer mode (ATM)-based broad-band integrated services digital networks (B-ISDN’s).Because these methods rely heavily upon stochastic processestheory, the relevant concepts from this field are succinctly recalled.

The first main topic is traffic modeling. Because of the largeimpact upon the performance of B-ISDN systems, the concept oftraffic burstiness is discussed, and various definitions are given andcompared. The most common stochastic traffic models are thendiscussed:ON–OFF, Markov modulated sources, and Markovianarrival process. The recent field of self-similar traffic is alsoreviewed. Another approach to source characterization is reviewedas well: bounded traffic models, which instead of a precise statisti-cal description give a set of bounds on the cell generation process.

The second main topic is the use of advanced queueing modelsfor the performance evaluation of the ATM statistical multiplexer,the basic B-ISDN modeling concept underlying the design ofswitching elements, output buffers, rate adapters, etc. The threemain solution methods for the statistical multiplexer are reviewed:matrix methods, generating function approach, and fluid flowapproximation.

This paper concludes with an overview of current topics in B-ISDN teletraffic research: connection admission control, bufferdimensioning, multiquality of service, cell discard schemes, rate-based flow control, per-virtual-connection queueing, and weightedfair queueing. The conciseness of this discussion is partially set offby references to recent review papers.

Keywords—ATM, B-ISDN, performance evaluation, queueingtheory, statistical modeling, teletraffic.

I. INTRODUCTION

“Teletraffic” is an old engineering branch, born with thework of Danish engineer A. K. Erlang on queueing systemsin the 1910’s and 1920’s. This amounted to characteriz-ing telephone communications at the call level, i.e., theexpected number of calls per hour and their distribution,

Manuscript received April 7, 1997; revised August 12, 1997. Thiswork was supported in part by the Flemish Institute for the Promotion ofScientific and Technological Research in Industry (in the framework of aproject on analytical modeling techniques for asynchronous transfer modeswitching systems and networks) and in part by the Fund for ScientificResearch (in the framework of a project on applications of stochasticanalysis). K. Laevens’ work was supported by the Belgian Federal Officefor Scientific, Technical, and Cultural Affairs.

H. Michiel is with Alcatel Bell Telephone, Antwerpen B-2018 Belgium(e-mail: [email protected]).

K. Laevens is with the Stochastic Modeling and Analysis of Commu-nication Systems Research Group, University of Ghent, Ghent B-9000Belgium (e-mail: [email protected]).

Publisher Item Identifier S 0018-9219(97)08821-X.

duration (“holding time”), determination of busy hour, etc.The statistical study of telephone calls as a sequence ofrandom events was motivated by the need to providesufficient resources in order to obtain a certain qualityof the telephone service, which, apart from the speechquality, meant essentially a small blocking probability anda small delay. As there was empirical evidence that callarrivals behave more or less as a Poisson process, relativelysimple queueing models could be used that exploit its“memorylessness.” So “teletraffic engineering” became, toa certain extent, a routine business, canonized in Erlangtables and Engset formulas.

With the advent of computers and computer communica-tions in the 1960’s, however, a new kind of problem arosein a field that at that time was not immediately related totelecommunications. Waiting times in time-sharing systems,storing of irregularly arriving digital information beforeforwarding it to the next station, performance of broadcastprotocols (ALOHA)—all these topics initiated a secondyouth for teletraffic engineering. And as telecommunica-tions and data communications became more and moreintertwined in the 1980’s, resulting in integrated broad-band networks carrying multimedia traffic (voice, data,video, etc.), “traffic engineering” nowadays is ubiquitousin communications and networking and is supported byfundamental mathematical research in many academic andcompany research centers. Witness the avalanche of papersin general and specialized journals on this subject matterin the past three decades. In the telephone era, teletrafficengineering was concerned with the statistical behavior onthe call level only; a telephone call required a fixed amountof bandwidth, and consequently, knowledge of the numberand duration of calls was sufficient to determine the neededresources. In contrast, multimedia traffic is characterizedby a high variability in its bandwidth needs. To give afew examples, during the transmission of an encoded videofilm, the needed bandwidth may easily fluctuate by a factorof ten; data communications between computer terminalsusually result in short periods of high activity, followedby long silence periods. So, in the multimedia era, oneis interested not only in the number and duration of callsbut also in the statistical properties of the informationflow during the call, in order to make efficient use of theresources while guaranteeing a high quality of service.

0018–9219/97$10.00 1997 IEEE

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Thus, the study of the statistical aspects of multimediatraffic has become an important field of investigation, andthe findings in this domain have an immediate practicalsignificance for the engineering of switches, multiplexers,flow-control algorithms, etc. In a broad-band integratedservices digital network (B-ISDN), for example, instead ofallocating the highest possible amount of bandwidth thatcould be required during a connection, one can exploitthe knowledge of the statistical properties and accept moreconnections than the sum of the peak bandwidths wouldpermit, so obtaining a “statistical gain.” Guaranteeing thequality then means reducing the probability of loss ofinformation, or the probability of excessive delay, to avery low level. This illustrates nicely the need for astatistical assessment of the problem. As another example,the Internet is increasingly plagued by its success, resultingin long delays. Especially when real-time services areto be offered, along with nondelay-sensitive services, astatistical approach is required in order to determine theneeded resources (link capacity, buffer space, etc.). In theremainder of this paper, the setting will mainly be that ofB-ISDN, for which asynchronous transfer mode (ATM) hasbeen adopted as the standardized transfer mode. With itsfixed size packets, calledcells and virtual circuits, ATM isa specific technology that differs in several respects fromothers [frame relay, transmission control protocol/Internetprotocol (TCP/IP), etc.], but for traffic-engineering pur-poses, this is a matter of detail, as the mathematicalmethods are sufficiently general. In some sense, ATM isthe simpler technology, as the fixed size packets implyconstant processing times. Leaving aside one degree offreedom (variable packet length) may therefore simplify anintroduction to the topic.

Based on statistical empirical data on multimedia traffic,a number of models have been advanced that try to capturetheir salient features. Source models should simulate thebehavior of the traffic generated by a terminal (a computerterminal transferring files, a video-on-demand server, etc.);more general models should represent the multiplexed traf-fic of many (and not necessarily identical) sources. Thesemodels are in the form of stochastic processes. “Good”models should satisfy two criteria: 1) they should renderthe relevant statistical properties of the real-life traffic and2) they should allow one to study the behavior in a network.The latter condition means essentially that one should beable to study the performance of a multiplexer, modeledas a queueing system, when fed by such a traffic model.Thus, queueing theory is another important ingredient inthe traffic engineer’s toolbox.

The set of traffic models advanced so far in the lit-erature is rather large, but we can nevertheless make abroad distinction between two categories, according to thepurpose for which they are mostly used. On the one hand,there is the search for stochastic models that capture asaccurately as possible the relevant statistical properties ofa traffic source of a particular kind. There have been,for example, many proposals for modeling variable-bit-rate(VBR) video traffic. A good model will not only capture the

first few moments of the arrival distribution but will alsorender well the autocorrelation, and possibly higher orderstatistics. Such sophisticated models may be very useful fortraffic-engineering case studies, but it is clear that they areinappropriate in the context of a live switching networkwhere connections are requesting their admission; theseconnections are not going to communicate to connectionadmission control (CAC) their power spectrum and so forth.

So, on the other hand, we have the models that are basedon the limited knowledge that the switching network haswith respect to the traffic characteristics of connections,which are to be found in the traffic contract. These char-acteristics (peak rate, average rate, and some tolerances)are largely insufficient to describe a traffic stream fully.But as this is the only information that CAC obtainsfrom a candidate connection, it is of utmost practicalimportance to study upper bounds on loss and delay oftraffic characterized by this information. We call these“bounded traffic models.” Both stochastic and boundedmodels will be discussed in the paper.

As for the queueing models, there are essentially threeanalytical/numerical solution methods: the matrix method,probability generating function (pgf), and fluid flow ap-proximation.1

1) Matrix Method: The steady-state probabilities for afinite buffer may be obtained as the solution of a systemof linear equations. This is elementary in principle, butdepending on the buffer size and the number of statesin the Markov chain, finding this solution with sufficientnumerical accuracy may be hard or impossible.

2) PGF: The use of the pgf results in reformulating theproblem as a functional equation in the complex plane. Theadvantage is that many results of complex analysis maybe applied, but the extraction of the probabilities from thepgf is not always simple. Many analyses based on the pgfonly calculate the first few moments of the distribution,which is unsatisfactory, as one is usually interested inlow-probability quantiles. Some possible solutions to thisproblem will be discussed in Section XVI-C.

3) Fluid Flow Approximation: A third possibility is toreplace the discrete cell arrival process by a continuous“information fluid.” The queueing problem is then formu-lated as a number of linear differential equations, whichmay be solved by appropriate means.

This paper is organized as follows. We start with arecapitulation of relevant concepts from stochastic pro-cesses theory. We then give an overview of the most usedtraffic models in B-ISDN. First, we discuss the so-calledburstinessof arrival processes; this is the most importantcharacteristic of broad-band traffic, and has a decisiveimpact upon performance. Then we review the importantON–OFF model, which is in some sense the workhorse ofmuch B-ISDN traffic engineering;ON–OFF sources alsoserve as an introduction to the more general class ofMarkovmodulated sources, which are further generalized in the

1By solving a queueing system, we mean finding the equilibriumdistributions (queue length, waiting time).

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“Markovian arrival process.” Last, the recent field of self-similar traffic is reviewed.

Next, techniques used for solving queueing models arediscussed. For each of the three methods mentioned above(matrix, generating function, and fluid flow approximation),a typical example will be worked out in some detail inorder to give an idea of the mathematical machinery.Some successful models published in the literature will behighlighted as well.

Last, diverse problems encountered in broad-band trafficengineering will be discussed, which illustrate some of themethods explained in the paper.

II. WHY STOCHASTIC PROCESSES?

In contrast with the synchronous transfer mode, ATMdoes not reserve particular time slots for a connection. Cellsare transmitted as they present themselves; consequently,every connection consists of a virtually unique successionin time of data cells, and as such is not well suited forscientific scrutiny. Therefore, one considers a connectionas a particular realization of astochastic process, andone studies the properties of this process. We will brieflyrecall a few concepts from stochastic processes theory here.For a short introduction, we refer to [1]; somewhat fullertreatments are found in [2] and [3].

Formally, a stochastic process is a parametrized familyof random variables . The parameter isusually time. is the index set,and if it is countable, wehave adiscrete parameter process,which may be denotedas ; in the opposite case, we have acontinuousparameter process. Thestate spaceis the set of all possiblevalues the random variables may assume; it may bediscrete (countable number of states) or continuous.

III. CHARACTERIZATION OF STOCHASTIC PROCESSES

How can we capture the properties of a stochasticprocess? Let us consider a cell arrival process, whichis an example of apoint process.2 Very often, the cellinterarrival time (IAT) distribution is used to “characterize”the arrival process, i.e., the probability that the timebetween two consecutive cell arrivals is 1, 2, 3, celltime units. This is certainly an important quantity, butsome caveats are in order. Unless one studies a fullytheoretically specified model, it is in general not possibleto obtain the full IAT distribution from observed trafficstreams. One should then make do with a small numberof moments of the distribution (mean, variance, and soforth) and thus with an incomplete knowledge. But moreimportant, one should realize that even a perfect knowledgeof the IAT distribution does not at all specify the arrivalprocess in its entirety. What is entirely absent from the IAT

2A point processis not so much a definite kind of random process buta way that a random process is looked at. When events happen at definitetime instants and one is interested precisely in this sequence of events, therandom process is called a point process. But when the event is the birthor death of a population element and one is interested in the evolutionof the population (rather than the dates of birth and death), the (same)process is known as a birth-death process.

distribution is information on theorder in which IAT valuesappear. One possibility is that IAT values are sampledindependently. In that case, one has arenewal process(seeSection VI). But the same IAT distribution is compatiblewith a deterministic process, which periodically sendsthe same sequence of cells. In fact, a (discrete) randomprocess is in general not completelyspecified unless the multivariate probability mass functions(pmf’s) are known forall -tuples and all positive integers

[3].

IV. SECOND-ORDER STATISTICS

Determination of all the multivariate pmf’s is quiteformidable and not feasible in practice. From a theoreticalpoint of view, correlation may take an infinity of formsand degrees. A process may depend on its entire history,or it may show some hysteresis, or it may depend onpart of its recent past, etc. Fortunately, one normally isalready well under way whensecond-order statisticsareavailable, which follow from thebivariate pmf. Second-order quantities express relations between the occurrencesat two time instants. Following are second-order propertiesof a discrete stochastic process .

• Autocorrelation , defined as the expected value of theproduct . This may depend in principle on both

and , but for a wide-sense stationary (wss) process[3], only the “lag” is relevant. Therefore, theautocorrelation of the wss stochastic process isa discrete function of only the lag

(1)

• Autocovariance , analogous to the autocorrelation,but for instead of

(2)

from which it follows that

(3)

• Instead of the autocorrelation, one may use a mathe-matically equivalent expression in the frequency do-main by taking the Fourier transform. The ensuingquantity is called thepower spectrumor power spectraldensity. As we restrict ourselves to discrete-time pro-cesses, the discrete-time Fourier transformis used,and the power spectrum is defined3 as

(4)

3What is given here as adefinition is a theoremin most digital signalprocessing (DSP) treatments; the reason is that the power spectrum is thendefined differently.

MICHIEL AND LAEVENS: TELETRAFFIC ENGINEERING 2009

• Analogously, one may take the discrete-transformof the autocovariance and obtain theautocovariancespectrum 4

(5)

In this context, a few notions from DSP are useful.Some references relevant for our purposes are [3]–[5]for generalities and [6, ch. 2] and [7] for the problemof estimating spectral properties from experimental data.It is clear from (4) and (5) that and areperiodic in with period . Moreover, assuming that theprocess under consideration is stationary,and . Consequently, one has

, which shows that the knowledge of foris sufficient. Another useful property is the

Wiener–Khinchin theorem. A version of it appropriate forour purposes is as follows (see [6, pp. 59–60]). Let ,

be a realization of a randomprocess [ is, for example, the number of arrivals intime slot at a certain buffer]. Then, for the autocovariancespectrum , one may instead of (5) use the equivalentexpression

(6)

In other words, for obtaining the autocovariance spectrum,one may take the discrete-transform of the experimentalseries , i.e.,

Its square divided by the record length should thenbe averaged over different realizations; for sufficiently longrecords, one will obtain the autocovariance spectrum. TheWiener–Khinchin theorem therefore is of utmost practicalimportance. Whereas it is clear that the direct calculationof an estimate of the autocovariance from a recorded datasequence of length would require operations, theWiener–Khinchin theorem shows how an-transform (forwhich efficient fast Fourier transform algorithms exist) issufficient. As shown above, the squared-transform of

(divided by the record length) may be used as anestimator of the autocovariance spectrum. As such, it isknown as theperiodogram

(7)

It is important to know that the accuracy of the periodogramas an estimator of doesnot improve by increasingthe record length ; a large record length allows one toobtain a higher resolution (observation of narrow peaksin the spectrum), but it does not reduce the variance of

4Some authors use the term “power spectrum” for the autocovariancespectrum.

the estimate (the reason being that for a larger recordlength, there are more frequency components). The wayout is to average a number of periodograms. This methodis known as theBartlett procedure. What has been saidon the autocovariance spectrum is of course valid for thepower spectrum of processes with zero mean, because theautocovariance and the autocorrelation coincide in that case.In general, however, (6) may not be used for the powerspectrum when the mean is different from zero. Anotherreason for often preferring the autocovariance spectrum tothe power spectrum is that the latter has asingularity atthe origin. Indeed, -transformation of (3), together withthe identity , results in

(8)

V. MARKOV PROCESSES

We already mentioned that full characterization of a(discrete) stochastic process requires the knowledge of allthe multivariate pmf’s, which is hardly possible in practice.However, forMarkov processes, one has

(9)

In other words, the present state determines the future ofthe process, and full knowledge of its past is not required.Therefore, Markov processes are much more amenableto mathematical analysis and find many applications alsoin teletraffic engineering. A Markov process is called aMarkov chainif its state space is discrete; the state spaceof a Markov chain may therefore always be taken tobe , with possibly infinite. In case theMarkov chain makes only transitions to neighboring states

or , we have abirth and death process.In Section V-A and V-B, we will recall some conceptsconcerning discrete and continuous-time Markov chains.For a fuller mathematical treatment, see [2] and [8]–[11].

A. Discrete Time Markov Chain (DTMC)

A DTMC may be seen as a process making transitionsfrom one state to another (possibly the same) at well-definedinstants . The DTMC is fully determined when theone-step transition probabilitiesProb areknown. Very often, the system ishomogeneous,whichmeans that the transition probabilities are independent oftime. The one-step transition probabilities are then thenumbers , which may be arranged into theone-steptransition matrixor simply thetransition probability matrix

. It follows that

(10)

(For a finite number of states, the sum of course has only afinite number of terms.) Therefore, is called astochasticmatrix (nonnegative elements and row sums equal to one).If the initial state vector is known, we


may calculate the state vector at time instantby applyingonce the one-step transition matrix

(11)

The equilibrium distribution is often designated as(with ). Because the equi-

librium distribution is by definition invariant under theone-step transition matrix, one should have

(12)

B. Continuous-Time Markov Chain (CTMC)

Transitions from stateto state now occur in continuoustime, and this requires some additional machinery [11].Assuming that the chain is temporally homogeneous,we may define

Prob

(13)

As for the discrete-time case, we have

(14)

(15)

and, in addition

(16)

One can now define the matrix of transition proba-bilities with

(17)

and it follows from (16) that

(18)

where is the unit matrix. From , one can deducethe transition density matrix , also called theinfinitesimalgeneratorof the Markov chain. It is defined as

(19)

The interpretation of the elements is as follows. Forsmall , the transition probability from stateto statein an interval is approximately equal to . If wefurther define

(20)

the infinitesimal generator of a finite chain may be writtenas

(21)

The infinitesimal generator is sufficient to reconstruct thetransition matrix by a (matrix) Taylor series

(22)

which may be written concisely as

(23)

Note also that the condition for the equilibrium probabilitiesin the continuous-time case is

(24)

rather than (12); is the zero matrix. It is important to notethat thesojourn time5 in a state of a CTMC is exponentiallydistributed, with mean sojourn time equal to

mean sojourn time in state

(25)

where is the diagonal element of the infinitesimal gen-erator (21).

VI. RENEWAL PROCESSES

One can look at a point process by considering theintervals between successive time points .If the are independent random variables with the samedistribution [independently identically distributed (i.i.d.)],the process is a renewal process,and the “time points”are called renewal epochs. Thus, a renewal process isderived from a generalization of the Poisson process byhaving interarrival times that are i.i.d. but not necessarilyexponential.

VII. SEMI-MARKOV PROCESSES(SMP’s)

An SMP is a marriage between a Markov process and arenewal process. When we discussed the DTMC, transitionstook place at instants , which are further undefined. Andfor the CTMC, the sojourn time is necessarily exponentiallydistributed. More freedom is obtained by specifying thetime between transitions. For an SMP, the probabilitydensity function (pdf) of the time spent in statebefore jumping to state is a function , whichdepends in general on both the initial state and thefinal state . Thus, an SMP is a continuous-time process

with discrete state space. It is not a Markov process,because knowledge of the present stateis not sufficientto predict the future (because of the renewal processesthat guide the time between transitions). If transitions takeplace at the instants , however, we may consider theprocess , which is a DTMC, the Markov chainembeddedin the process . In the special case where thedistribution depends only on the present state,the process is called a special (S)SMP. Another way to lookat an SMP is to consider the sequence of couples .This is sometimes called aMarkov renewal process(MRP)(see [12, p. 220]).

5The sojourn time is the random variable describing the time spent ina state before entering another state.


VIII. A RRIVAL PROCESSES ANDBURSTINESS

In the following sections, we will highlight the maintraffic models used in practical traffic engineering and inresearch studies on the performance of B-ISDN. For an ex-tended reference list on traffic characterization techniques,we refer to [13]. We also mention an excellent overview ofbroad-band traffic models, which appeared when we werecompleting the present paper [14].

We will first review the characterization of the burstinessof a traffic stream. Although the term is somewhat vague,burstiness is widely recognized as the single most importanttraffic feature when dealing with the performance of B-ISDN systems.

In the B-ISDN context, the termburstiness is usedto indicate that cells tend to arrive in clusters ratherthan in a more-or-less smooth way. In this sense, evena Bernoulli process (or Poisson process in continuoustime) is more “bursty” than a periodic cell stream; thecommon understanding, however, is that Bernoulli sourcesand periodic sources are not bursty. The question then ishow to quantify the burstiness. First note that it is notthe irregularity of the traffic stream as such that intereststhe traffic engineer, but rather its implications for thebuffer occupation when such streams are multiplexed. Inthis sense, “burstiness” should be an indicator for “greedybuffer utilization.” Many proposals have been formulatedfor a quantitative definition of burstiness (for a review, see[15]–[17]); we discuss a number of them below.

A. Peak Rate/Mean Rate

The ratio of the peak rate to the mean rate is a verypopular definition of the burstiness of a source, and it isclear that a high value ( 1) of this ratio indicates a highdegree of variability in the arrival pattern.

This definition, however, leaves out many importantaspects of a traffic stream. It fails, for example, to catchthe lengthof the ON period of anON–OFF source.6 Anotherproblem is how one defines the peak rate. If no time intervalis specified in which to evaluate the peak rate, meaninglessresults may be obtained, as it suffices to receive two cells“head to tail” for having a peak rate that equals the link rate.Experience also shows that multiplexedON–OFF sourcesbehave almost identically whether they emit a periodicstream or a Bernoulli stream in theON period, although thepeak rate may be quite different in both cases. In summary,the peak-to-mean ratio is not an adequate parameter for adescription of the burstiness of a source.

B. Coefficient of Variation of the Traffic Load

The coefficient of variation of a random variable isdefined as the ratio of its standard deviation to its mean.Alternatively, thesquaredcoefficient of variation (scv) isused. Let be the number of cells transmitted per cell time,

6Take twoON–OFF sources A and B, which emit periodic traffic in theirON-period. The only difference between them is that B hasON periods andOFF periods that are ten times as long as for A. So mean rate and peakrate are the same, but it is well known that the mean burst length is animportant parameter for buffer performance.

such that and .Then the scv of the arrival process equals . It is akind of normalized variance; as such, it has little to dowith burstiness. Much more information is contained instatistical measures, which depend on the time scale (orfrequency) of the variation; such is the case for the “index ofdispersion” or the spectral functions, to be discussed below.

C. Index of Dispersion

There are two equivalent ways to analyze a point process:one can give statistics on the interarrival times or on thenumber of events in a fixed interval of time. This explainsthe use of two related concepts: theindex of dispersion forintervals(IDI) and theindex of dispersion for counts(IDC).For a full discussion, see [2], [7], and [18].

1) IDI: Let be the successive intervals between ar-rivals. We do not assume that the are necessarily i.i.d.;nevertheless, under rather loose conditions (so-calledweakstationarityof the process7), the sum of successive ’shas a definite mean and a definite variance

, which is independent of the consideredtime instant . Moreover, for , we see that a definitecommon mean and a definite common variance

exists for the . Unless the are uncorrelated,however, the variance of the sum is not simplybut rather

(26)

where is the covariance of and, . So the

variance function (26) depends on the correlation betweensuccessive interarrival times, and a normalized version ofit is called the IDI, given by

(27)

Note that the index of dispersion is a (discrete) function of, in contrast with, for example, the coefficient of variation,

which is just a number. The scv of intervals equals.The normalization in (27) is chosen such that for a Poissonprocess, the index of dispersion for intervals is identicallyone.

2) IDC: An alternative way to express correlation in thearrival stream is to count the number of arrivals in aninterval of length . The index of dispersion for counts isdefined as the variance of the number of arrivals in aninterval of length divided by the mean number of arrivals

(28)

Note that is a function of the continuous parameter;for a Poisson process, it is identically one. It can be proved

7See [7, ch. 4] for the concept of stationarity.


[18] that the limits of the IDI and the IDC are equal andgiven by

(29)

where is the interarrival time autocovariance [see(2)]. In [18], the IDI for processes with hyperexponentialinterarrival time, the IDC for the batch Poisson process,and the Markov modulated Poisson process (MMPP) arecalculated; an IDC-based fitting procedure for arbitrarytraffic to a two-state MMPP model is proposed. In [19],the IDC for experimental local-area network traffic isdetermined and fitted to a superposition of homogeneousON–OFF sources by matching the IDC at two time instants.

D. Spectral Properties

A brief introduction on the concept of the autocovariancespectrum (“power spectrum”) was given in Section IV.The spectral characterization of broad-band traffic is arecent field of investigation. A series of papers by S.-Q.Li [20]–[23] tries to transfer as much as possible fromthe DSP tool box to queueing performance analysis. In[21], after having investigated the influence of not onlythe input traffic second-order but also third- and fourth-order statistics (i.e., correlations , , and

, respectively) through the power spectrum,bispectrum, and trispectrum, respectively, the conclusion isreached that the input power spectrum is most essential toqueueing analysis and that input power in the low-frequencyband has a dominant impact on queueing performance,whereas high-frequency power to a large extent can beneglected. Grunenfelder in [24] comes to the same conclu-sion; he applies spectral methods in [25]–[28] to practicalmeasurement problems.

We conclude this section on spectral methods by notingthat its essence is not the fact that one switches from thetime domain to frequency domain. This is a mathematicaloperation that may ease the analysis. What is important isthe use of higher order statistics (in practice, second order)as the autocorrelation function. In this sense, the IDI andIDC discussed in Section VIII-C are spectral measures, justas the power spectrum. We show an interesting relation be-tween the autocovariance spectrum at zero frequencyand the limit for of the IDC . In [24], is ad-vocated as a traffic descriptor of prime importance. Its rela-tion to the earlier proposed IDC is as follows. From (5), wehave ( being the random number of arrivals per time unit)

Therefore

(30)

Comparing with (29), it is seen that and arepractically equivalent as traffic descriptors.

E. Other Approaches

Other methods have been applied to traffic characteriza-tion that we cannot discuss in detail but that are interestingas well. In [29], theentropy rateof a stochastic process

is defined as the single numberwhere

(31)

The evolution of the traffic stream’s entropy is studied asit traverses successive queues. In [30], it is proposed to usethe entropy ratedirectly (without model fitting) to obtainthe quality of service (QoS) parameters of interest. It is notclear how “entropy” relates to the other traffic descriptorssuch as power spectrum at low frequency or the index ofdispersion. From its definition, however, it is clear thatcorrelation is rendered by in some way.

Another approach [31], [32] consists of catching statis-tical parameters of buffer underload and overload periods.This is not a description of the traffic as an isolated process,but it makes the link with a buffering system from theoutset. In an underload period, the traffic intensity is lowenough to cause no congestion; during overload, congestionis to be expected. Parameters as the mean and variance ofthe underload and overload periods are then used as trafficdescriptors. A recent field of research is due to the discoveryof “fractal” (self-similar) properties in broad-band traffic;this topic will be discussed in Section XIII.

IX. THE SIGNATURE OF BURSTINESS

Apart from the quantitative characterization of a source’sburstiness, it is instructive to see how burstiness manifestsitself in a queueing system. Fig. 1(a) shows schematicallythe queue contents tail distribution in a multiplexer wheretraffic consists of “smooth” sources, for example, Poissontraffic. Fig. 1(b) is obtained when the sources generate glob-ally the same traffic, but in a “bursty” manner. The queueingbehavior is typical, the “signature of burstiness.” The taildistribution consists of acell component,which coincidesinitially with the tail distribution in the Poisson case. Thedistribution then gradually changes to a second branch withsmaller slope, theburst component. The intersection of thetangents to both branches is often designated as “the knee.”A qualitative explanation of this behavior is as follows. Thecell component is similar as in the case of smooth traffic;its origin is the accidental coincidence of arrivals of two ormore sources. In contrast, when several sources emit burstsof cells simultaneously, a new kind of behavior shows up.The load on the system is larger than one for a considerable


(a) (b)

Fig. 1. (a) Queue contents tail distribution in a multiplexer where traffic consists of “smooth”sources, for example, Poisson traffic. (b) When the sources generate globally the same traffic, butin a “bursty” manner, a typical queueing behavior is observed. The initial part of the curve (“cellcomponent”) is unaltered but is followed by a flatter part, the “burst component.

amount of time, and the queue length grows. The moresources are in a burst period simultaneously, the larger thequeue length, but as it is unlikely thatall sources sendbursts at the same time, the very large queue lengths havea small probability. As a rule of thumb, the knee positionis where just enough sources are in a burst period as to“fill the link.”

There are some general trends in this behavior. Whenthe number of sources or their peak rate increases, the kneemoves upward, reducing the cell component, and the burstcomponent becomes flatter. Increasing the burst length orits variance while keeping the load fixed (by having longerperiods of low or no source activity as well) leaves the kneemore or less unchanged but makes the burst componentflatter.

The practical implications of this state of affairs areapparent. While a buffer of size [Fig. 1(a)] would besufficient to guarantee a cell loss of at most 10in the caseof smooth traffic, a much larger buffer would be requiredin the bursty case. It is also clear that bursty behavior, butwith a knee at very low probability (for instance, at 10),may be considered from a practical point of view as smoothtraffic. Methods that allow one to determine an approximateposition of the knee are therefore quite useful.

X. ON–OFF SOURCES

A. Types ofON–OFF Sources

The ON–OFF source is the prototype of a bursty sourceand has been used extensively in B-ISDN traffic modeling.The essence of anON–OFF source is that informationis sent in a succession of active (ON) periods separatedby silent (OFF) periods. ON–OFF sources may further bedistinguished according to the way information is sentduring theON period. The most used cases are the strictlyperiodic (“deterministic”) cell emission on the one handand the Bernoulli (or Poisson in the continuous-time case)process on the other hand. They may also be distinguished

according to the statistical properties of theON and OFF

periods. Several cases are possible here, too. The sourcemay switch from theON to the OFF state according to aCTMC with two states. As the start of anON period isthen an instant in continuous time, this is best combinedwith a Poisson emission process; we have then a specialcase of the MMPP, that is, a two-state MMPP with zerorate in one state, also called an interrupted Poisson process(IPP).8 Because the sojourn time in a state of a continuousMarkov chain is exponentially distributed, burst and silenceperiods are necessarily exponential. A discrete analog ofthis source type is obtained by having a DTMC with twostates whose transitions (possibly to the same state) occurat periodic instants . The number of periodsthe chain remains in the same state is then geometricallydistributed, which is the discrete version of the exponentialburst and silence length above. And because of this slottedoperation, the cell emission process in theON state isbest defined as a Bernoulli process, the discrete versionof the Poisson process. We thus obtain a special case ofthe Markov modulated Bernoulli process with two states[switched Bernoulli process (SBP)] and zero parameterin the OFF state [interrupted Bernoulli process (IBP)]. Ahybrid combination of a CTMC and a deterministic cellemission process has also been considered [33] and is calleda Markov modulated deterministic process (MMDP). Thus,the ON–OFF source with exponentially distributedON andOFF periods may also be considered as a special case ofan MMDP (2). As discussed above, a Markov modulatorimplies an exponentially distributedON andOFF period. Toallow more general distributions, one may use an SMP (seeSection VII), with a generally distributed sojourn time ineach of the two states. The SMP as a traffic source (notnecessarily restricted to two states) was introduced by Ding[34]; assuming that the emission process after a transition

depends on only (instead of the more general case,

8An MMPP with two states is sometimes called a switched Poissonprocess (SPP); an IPP is then a special SPP with zero rate in one state.


Fig. 2. Transition diagram of an ON–OFF source.

where it could depend on bothand ), this process is calledan SSMP. AnON–OFF source with generally distributedON

and OFF periods therefore is a special case of an SSMP.In the remainder of this section, we will restrict ourselves

to the ON–OFF source with exponential sojourn times.Although this model could be considered as a specialcase of the more general models to be discussed below(without the restriction of only two states, of which one isinactive), theON–OFF source is so often used that a separatepresentation is desirable.

B. Characterization of a Discrete-TimeON–OFF Sourcewith Geometrically DistributedON and OFF Periods

Let the transition matrix be

(32)

The meaning of the entries is made clear in the diagramof Fig. 2. The equilibrium distribution obeys (12)and is given by

Prob[state isON]

Prob[state isOFF] (33)

This is valid for , and a glance at Fig. 2 shows thatin the opposite case, we are left with two “once in neverout” (absorbing) states without interesting applications. Thevalue ( ) gives the probability that the chain observedat an arbitrary instant is in state 0 (1). It is interestingto calculate this probability in an alternative way. Let theaverage length of theON (OFF) period be ( ). Tocalculate, for example, , we need , the probabilitythat anON state has length. Thus, the chain, being in theON state, should stay in theON state for anothertime steps and then switch to theOFF state. From Fig. 2,we see that this probability is given by

(34)

The averageON period is then

(35)

and similarly, for theOFF period

(36)

As the time axis is covered by a succession ofON andOFF

periods, the probability of finding the chain in theON stateequals

Prob[chain isON]

If the average interarrival time in theON period is (cellperiods), the average loadin the ON period is

(37)

and the overall average loadproduced by such a sourceis then

(38)

Calculation of the correlation properties of theON–OFF

source is straightforward, and one can give a closed-formexpression of the autocorrelation function of the arrivalprocess. This is because higher powers of the 22transition matrix are readily obtained (see [10, p. 9])

(39)where

and and are defined in (33). The autocorrelation of thearrival process of theON–OFF source thus can be written as

(40)with as in (37).

XI. M ARKOV MODULATED SOURCES

Markov modulated source models cover a very large areaand have been extensively used in broad-band performanceevaluation. The special case where the Markov chain hasonly two states, of which one is “silent,” has already beendiscussed in Section X. In this section, we will reviewa generalization of theON–OFF source, where there are

, states. When in state, the source emitscells with a definite rate, say, . After a certain time (theresidence timein state ), the process switches to anotherstate and cells are generated with rate. The imbeddedprocess consisting of the changes of state is assumed tobe a Markov chain, in discrete or continuous time. (Forclarity, the state of this Markov chain is mostly designatedas itsphase, to be distinguished from thestateof the overallarrival process.) The arrival process may therefore be calleda Markov modulated rate process (MMRP). For practicalreasons, we exclude from this section the Markovian arrivalprocess (MAP) and related models, which will be discussedseparately. A Markov modulated source model has a cellgeneration process that changes according to a Markov


chain. This implies the following “degrees of freedom” todefine Markov modulated sources.

• The number of phases of the Markov chain.

• The modulator structure (transition probabilities). Inparticular, one may consider chains where only tran-sitions to nearest neighbors are possible (birth/deathprocess). This may simplify the math and be physicallymore appropriate, for example, when a rather simplemodel is to represent the aggregate load of a number ofsources; a queueing system may evolve from an “un-derload state” to “congestion” and back to “underload”(see [31]).

• The processes in each phase. This will in the firstplace depend on the kind of time used. In continuoustime, cells may be generated according to, for example,a Poisson process, and we then have an MMPP.In discrete time, the natural choice—considering theslotted nature of ATM—is for time to beslotted,

. . . Cells might, for example, begenerated according to a Bernoulli process, MMBP, oras a deterministic (periodic) process, and one has anMMDP. Batch arrivals are also possible, in discrete aswell as in continuous time, for example Markov modu-lated batch Poisson process. In principle, the processesin the different phases might be quite different (forexample, Poisson in phase 1, deterministic in phase 2,etc.), but this is not usually considered; one normallyhas a single kind of process that differs only in theparameter values from phase to phase, for example,Poisson processes with parameters, , etc.

• The time evolution of the Markov chain. There areseveral possibilities. If the modulator is a CTMC,one has exponentially distributed sojourn times, andtherefore the phases last for an exponential time. Fora DTMC, we have the following cases.

— The chain may make transitions at “slotted”instants . . . If transitions to thesame state are allowed, the sojourn time willthen be geometrically distributed.

— The sojourn time may be specified by somegeneral distribution. The phase process is thenan SMP (see Section VII). This gives rise toa semi-Markovian arrival process. When thesojourn time depends only on the present phaseand not on the next, one has an SSMP.

Table 1 gives a summary of the types of Markov modu-lated processes usually considered in the literature. Severalsolution methods exist for finding the equilibrium distribu-tions of a statistical multiplexer fed by Markov modulatedsources; some will be discussed in Section XVI.

XII. M ARKOVIAN ARRIVAL PROCESSES

It is already clear at this point that ever new traffic modelsmay be defined for ever changing purposes, requiringad hoc approaches for solving the associated multiplexerproblem. One would wish to have a more universal model

that encompasses many special cases, such that solutionmethods may also be standardized. Neuts [35], [36] hasinitiated research in this direction, which has led to therather broad class of arrival processes known as MAP.Explaining the rather complex mathematics is beyond thescope of this paper, but a bit of history is useful to situatea number of models that are related to Neuts’ MAP.

When analyzing a queue with server vacations, Lucantoni[37] devised a process ofsingle arrivals, which containedboth the phase type renewal process (PH process, notfurther discussed here) and the MMPP. It was christened the“Markovian arrival process.” In contrast with Neuts’ ver-satile MAP, this model was conceptually and notationallyrather simple and served the same purpose as Neuts’, i.e.,unifying and generalizing a number of arrival processes.The discrete-time version of MAP is known as DMAP.The MAP was later generalized to includebatch arrivals,and this was called BMAP. Later on, it was shown that theBMAP is mathematically equivalent with Neuts’ versatileMarkovian arrival process. Last, a discrete-time version ofthe BMAP was given by Blondia [38], [39] and calledthe D-BMAP. This is a quite general and useful modelfor sources with slotted time (as ATM) and includes manyspecial cases such as the Bernoulli source, Markov modu-lated Bernoulli sources, and Markov modulated Bernoullisources with (correlated) batch arrivals. This generalizedsource model, combined with efficient computer programsfor solving systems of linear equations, shows the way fora more unified approach to the performance evaluation ofB-ISDN systems.

XIII. SELF-SIMILAR AND LONG-RANGE-DEPENDENT

PROCESSES

Interest in self-similar arrival processes was stimulatedby measurements on Ethernet traffic at Bellcore [40], [41].The measurements indicated that “traffic looks the same onall time scales”; hence the term self-similar (or “fractal”)traffic. Since then, this feature has been established in manyother experiments, for example, in TCP [42] or MotionPictures Experts Group (MPEG) video traffic [43], [45].An important characteristic of self-similar traffic is its long-range dependence, i.e., the existence of correlation over abroad range of time scales. An excellent introduction to thetopic can be found in [40]. Some of the important conceptsare summarized below.

A. Some Concepts

The following definition of self-similarity for stochasticprocesses [40], [41] is widely adopted. Assume tobe a wide-sense stationary process with mean

and autocorrelation function .9

Consider next the processes ( ) that areconstructed out of as ,i.e., by averaging over nonoverlapping blocks of size.

9Note that this autocorrelation function differs from the one defined inSection IV. The specific form used here leads to an elegant definition ofself-similarity.


Table 1 Classification of Markov Modulated Source Models

The processes are also wide-sense stationary, withmean and autocorrelation function . The process

is called (asymptotically) second-order self-similar if(for ).10 Self-similarity manifests

itself in a number of equivalent ways [40], [41], [48]:

• slowly decaying variances: if, whereby ;

• a slowly decaying autocorrelation function:if ;

• a power spectral density11 that behaves like thatof noise around the origin: if

.

Self-similarity also implies long-range dependence, i.e.,. A process for which

is said to be short-range dependent. Such a process differsfrom a long-range-dependent process in the sense that [40]

• the variances decay as ;

• decays exponentially fast: for large ;

10“All-order” self-similarity is present when, for allm, the processmX

(m)k

is statistically identical to the processmHXk, i.e., when allmultivariate distributions of both processes are the same [46], [47]. Note,however, that this definition requiresEfXkg to be zero.

11In fact, the autocovariance spectrum as defined in Section IV.

• the power spectral density remains finite (and approx-imately constant) around the origin;

• the process behaves like (second-order) pure noise forlarge ( ( ) if ).

An important parameter of a long-range-dependentprocess is the so-called self-similarity or Hurst parameter

. It is named after H. Hurst, who observedthe following fact. Given a set of experimental data( ), with sample meanand sample variance ,define the rescaled adjusted range (R/S) statistic as

whereby . (Thequantities measure the deviation of the processfrom its “expected value.” then measures the “record”values of this deviation.) For many “naturally” occurringprocesses, one has whenwith “typically” around 0.7. If is a short-range-dependent process, i.e., with a correlation structure overonly small time scales, one would have . The


larger experimental values can be explained by assumingthat is self-similar [40].

Some “tools” to asses the self-similar nature of a giventrace are:

• visual inspection—plots of samples of the processesfor a wide range of values of all look

similar for fractal traffic, while those for short-range-dependent traffic “flatten out” when gets large (see[40], [41] for some examples);

• the plot or pox plot—plottingversus for various subsets of the available dataallows one to determine, by linear regression, the Hurstparameter ;

• the variance-time plot—plotting ver-sus allows one to estimate ;

• the spectral power density, which can be estimatedby means of the periodogram, as explained inSection IV—plotting versus allowsan estimate for . (In [40], it is argued thatsince for the periodogram statistics are well knownand studied, the latter approach also allows one todetermine confidence intervals for the estimates.)

Studies like [40] or [43] show that the various estimationmethods outlined above might yield substantially differentvalues of the parameter or estimates with relatively largeconfidence intervals. This, and the fact that these methodsrequire the processing of a large amount of data, questionsthe viability of the Hurst parameter as a practical trafficdescriptor, a conclusion also reached in [44].

B. Models for Long-Range-Dependent Processes

Since the introduction of the notion of “self-similarity”into the field of teletraffic engineering, a number of sourcemodels exhibiting long-range dependence, and their queue-ing behavior, have been analyzed by various authors. Someof them are discussed in the following sections. Note thatthe diversity of these approaches is less inspired by thesearch for a better fit to real traffic than by reasons ofmathematical preference or tractability, and it is an openquestion which model is best suited for describing forexample MPEG video.

1) Fractional Brownian Motion (FBM): In [47], FBM isdefined as a stochastic process with the followingproperties: has stationary increments, and

for all , for all , hascontinuous paths, and is Gaussian. (In [40], FBM isdefined as a mean-zero Gaussian process with correlationfunction with

. In [50], it is defined as the th derivativeof standard Brownian motion.) This process is exactly self-similar. The derivative of is called fractional Gaussiannoise. In [47], a fluid arrival process , the number ofarrivals in , is then constructed as ,with a positive drift and an FBM. Negative incrementsof are thus possible, but this has no serious implications.

2) Fractional Autoregressive Integrated Moving Average(ARIMA) Processes:An ARMA ( ) processis defined by the recursive relation

with and .The “inputs” are i.i.d. random variables, i.e., “whitenoise.” The recursive relation can also be denoted as

, with and polynomialsin the backward-shift operator . AnARIMA ( ) process is defined by ,where is an ARMA process. The differenceoperator of order , denoted , is defined recursivelyas and .(This is equivalent with the definition .)Fractional ARIMA ( ) processes [50] are extensionsof the ARIMA ( ) processes by allowing nonintegervalues of , using the identity

For , such a process is asymptotically second-order self-similar with Hurst parameter [40],[50].

3) Chaotic Maps: This approach [49] makes use of thepossible fractal-like behavior of nonlinear systems. The ideais to associate with different regions of the state spaceof such systems different (distributions for the) numbersof arrivals. The trajectory of the system can be chaoticand may, as such, introduce long-range dependence in thearrival stream. An example given in [49] is the so-calledintermittency map

ifif .

An ON–OFF arrival stream exhibiting long-range depen-dence is then obtained by assuming ifand if .

4) ON–OFF Sources with Heavy Tails:ON–OFF sources(continuous time, discrete time, or fluid flow) exhibit long-range dependence if the distributions of the duration of theON (or OFF) periods have so-called heavy tails that resultin infinite variances. (A distribution is said to have a heavytail if Prob with . A popular exampleis the Pareto distribution [42], [48].) Proper aggregation ofa large, i.e., infinite number of such sources may lead toan FBM process. Similar results are obtained by means ofrenewal reward processes [40] or processes derived fromthe buffer occupancy of the M/G/ queue. Recent studiesof the latter model can be found in, e.g., [51]–[54].

5) Quasi-Self-Similar MMRP’s:An approach to synthe-size (quasi) self-similar traffic is to modulate the arrivalstream by a Markov chain, as explained in Section XI,in such a way that correlation is introduced over a suf-ficiently wide range of time scales, by properly choosingthe transition probabilities (see, e.g., [46], [55], and [56]).


The advantage of this approach is that the “classical”methods to analyze the queueing behavior (see Section XV)remain valid, whereas the use of models such as theFBM or the M/G/ processes may involve rather complexmathematics.

C. Implications

Experimental evidence for long-range dependence invarious kinds of traffic is already overwhelming and con-tinues to grow. Clearly, the phenomenon can no longer beignored in teletraffic engineering. So far, simulations andanalytical studies have shown that it may have a consid-erable impact on network performance not predicted bythe more traditional short-range-dependent models. How-ever, its implications on, for example, CAC or networkdimensioning are still not well understood. In particular,a very practical question is how long-range dependence isaffected by queueing and traffic shaping (as, for example,in a “leaky bucket,” the device used to do the policingin an ATM node). Long-range dependence seems to berather immune against traffic shaping [57], which wouldexclude simple countermeasures. On the other hand, hastyconclusions such as self-similarity’s being the grave ofATM are not warranted, either. In [58], for example, a“cross-over effect” is discussed, whereby multiplexing gainincreaseswith increasing Hurst parameter. It is clear thatfurther research will be needed to find how to live withlong-range-dependent traffic.

XIV. B OUNDED TRAFFIC MODELS

The ever growing amount of studies, of increasing com-plexity and sophistication, devoted to the definition oftraffic models for B-ISDN systems should not lead oneto the conclusion that such complicated models will beused in real-life B-ISDN networks in order to characterizeconnections. As already mentioned in the introduction,“exhaustive” models, which aim at a high degree of realism,are mainly intended for “off-line” performance studies. Onthe other hand, there is the problem of announcing tothe network the request for a new connection setup; thisrequires a succinct description, with a limited number of“simple” parameters, of the expected traffic pattern of theconnection. Simplicity is required not only for reasons ofeconomy (to avoid overhead, to establish connections ina very short time) but also because the precise statisticalproperties of the connection are not known in advance(think of a telephone call).

This leads us to the topic of what we will call “boundedtraffic models.” Stochastic traffic requires a multitude ofparameters for a more or less thorough description; a“bounded” traffic modelof the same traffic streamspecifiesonly a few parameters, and instead of trying to givean accurate statistical description of the connection’s cellstream, it defines some bounds on the number of cells sentin a given time interval.

Of course, such an approach cannot be optimal, as for re-source utilization; it becomes clear, however, that a feasible

and relatively efficient approach toward CAC necessitatessome tradeoffs. A bounded traffic model may be such atradeoff. Note that bounded traffic models have been mostlyintroduced in conjunction with a class ofservice disciplines(for a definition of this term, see Section XV). This allowsone to establish bounds on buffer occupation and delayfor connections described by a bounded traffic model andmultiplexed in a buffer, which is served according to theparticular service discipline. This is a possible approachtoward guaranteeing QoS on a per-connection basis (seeSection XVIII-D). A number of bounded traffic modelswill be presented now; some models have absolute bounds(“deterministic bounds”), others have bounds expressed instatistical terms (“statistical bounds”).

A. The Deterministic Calculus of Cruz

Let be the instantaneous rate of a connection at time; the amount of traffic generated in the interval is

then

traffic generated in

(41)

Cruz [59], [60] considers traffic streams that have a ratefunction with the following property

(42)

where are constants and . This property isalso written as .

The meaning of the parameters may be understood asfollows. For , one may neglect the finitequantity in (42), and one sees thatis the long-term averagerate. On the other hand, one sees that in an infinitesimalinterval of time, an amount of traffic may be generated;

therefore may be related to the burst size. The practicaluse of this model is apparent if one realizes that traffic“policed” by a leaky bucket is precisely characterized byan average rate and a maximum burst size [62], [63].

Cruz also considers traffic streams with a more generalconstraint on the rate function

(43)

also written as , where is any function definedon the nonnegative reals.

The relative simplicity of this approach allows one toreach general theorems and to study systematically the kindof transformation ( ) undergoes when the traffic traversesnetwork elements (multiplexers, shapers, etc.). Considerthe following example to appreciate the elegance of thisapproach. Suppose a certain traffic stream enters a networkelement with rate function (the same networkelement may be traversed by other traffic streams as well).If it is known that the maximum delay that any bit of theconnection suffers in the network element is bounded by


, then one can write for the rate functionof the connection when leaving the network element

If, for example, , then, implying that the burst-size upper bound

has increased by the amount . There is a price to be paidfor this elegance, however; the bounds may be loose, suchthat part of the resources (buffers, bandwidth) are wasted.

B. The Statistical Exponentially BoundedBurstiness (EBB) Model

Better resource utilization will be obtained by usingstatistical bounds (but at the price of higher complexity).Such an approach is presented in [64] and [65]. The EBBprocess is defined by the condition that the traffic generatedin the time interval , satisfies the followingrelation:

Prob

(44)

where and are positive constants. The work in[64] and [65] is devoted to EBB sources multiplexed in abuffer with a “weighted fair queueing” (WFQ) discipline.Statistical bounds are given for the backlog, delay, andend-to-end delay in a small network.

C. The Model

An extension of the calculus is provided in [66],where the purpose is to describe VBR video traffic. The

model maintains pairs , where the amountof traffic in an interval is bounded by

(45)

A procedure is given for obtaining a set starting froman empirical envelope; the latter is defined as

Note that this procedure may only be applied to trafficstreams that are knowna priori, as is the case for anMPEG-compressed video trace.

D. The “Burstiness Curve”

Instead of a discrete set of values, as in theprevious section, one can consider the entire curve;this is the approach in [67] and [68]. Some of the conceptsused in this work are summarized below.

A message of duration is specified by its ratefunction , . If the message is forwardedto an (infinite) buffer read out by a server at a fixed rate,then Reich’s backlog theorem (see, for example, [59]) tellsthat the backlog at time will be

(46)

The maximum buffer occupancy for a fixed service rate,denoted , is therefore

(47)

is called theburstiness curveof the message . Forevery value of the service rate, it tells how much bufferspace should be available for preventing loss. It is also clearthat , the backlog when the server is “dead,” is thetotal size of the message.

As argued in Section VIII, “burstiness” is a rather vagueconcept. Low and Varaiya define burstiness in a way that isimmediately related to the queueing properties of the traffic.Messages can be (partially) ordered according to theirburstiness as follows. Let and be two burstinesscurves. Only equal-length messages are compared; thus,we assume that . Message 1 is less burstythan message 2, denoted as or if

(48)

This formalism allows some general theorems. To give anexample, consider a number of messageswith burstiness curves and construct the “mul-tiplex” ( ) with burstinesscurve . Then , which may beinterpreted as “multiplexing saves resources.”

E. The Stochastic Bounding-Interval-Dependent(S-BIND) Model

H. Zhang and E. Knightly [69] elaborate upon a schemeproposed by Kurose [70]. This scheme models a source witha family of random variables thatstochasticallybounds thenumber of bits sent over various interval lengths.

More specifically, a sourceis characterized by a familyof two-tuples

where is a random variable that is stochastically largerthan the number of bits generated over any interval of length

12: with , the traffic generated in , onehas

Prob Prob

(49)

F. The Deterministic (D)-BIND Model

The D-BIND model ([71], see also [66]) is analogousto the previous S-BIND model but aims at providingdeterministicperformance guarantees. Instead of (49), onenow has

(50)

For practical applications, only a restricted number ofwould be used for characterizing a source and interpolation

12A random variableX is stochastically larger than a random variableY if Prob[X > s] � Prob[Y > s] 8 s. This may be written asX � Y .


for the intermediate time intervals; the model is then similarto the model.

Examples are given in [71] that show how the D-BINDmodel may be applied to MPEG compressed video traces.

XV. THE ROLE OF QUEUEING MODELS

IN BROAD-BAND NETWORKS

In a packet switching system, there are several pointswhere packets have to undergo some processing beforecontinuing on their way from source to destination (analysisof the routing information, multiplexing, rate adaptation,etc.). In all these cases, the instantaneous offer of workto the processor may be larger than the instantaneousprocessing capacity, although on a longer time scale, theoffered work is less than the processing capacity. Thesolution is then temporarily to store the packets into abuffer and process them one by one. This is what is called a“statistical multiplexer,” and it is at this point that queueingtheory enters the field of B-ISDN performance evaluation.

An important question is in which order the queuedpackets will be served; this is called theservice discipline.The most obvious and common discipline is first in, firstout (FIFO), in which packets are served in the order inwhich they arrived. The main reason for FIFO’s long-standing popularity is the simplicity of its implementation.However, guaranteeing the QoS while keeping the resourceutilization high may necessitate other service disciplines.Even nonwork-conserving service disciplines are beingconsidered now. (A service discipline is said to be workconserving if it never happens that there are packets in thebuffer and the server is idle. In other words, the serverimmediately takes a new packet for service, if one isavailable, after finishing the previous one.) A reason fordoing so might be, for example, that the outgoing trafficstream becomes less bursty.

In the queueing literature, a queueing system is conciselydescribed by theKendall notation. This is in the form

, where describes the arrival process, theservice time distribution, the number of servers, and

the system capacity (maximum number of packets); ifis omitted, the capacity is infinite. The symbols are

more or less standard, (for or ) indicating anexponential interarrival or service time distribution and

a constant interarrival or service time. Thus,is an infinite buffer system with one server, the packetsarriving according to a Poisson process, and the servicetime being exponentially distributed. For constant lengthpackets as ATM cells, is often appropriate. Foran introduction to queueing theory, Kleinrock’s book [73]is still one of the best known.

One of the findings that has become increasingly im-portant in broad-band traffic engineering is the role ofthe burstiness—or, in more statistical terms, thecorrela-tion—in the arrival process to a queueing system. Whereasthe older (pre-B-ISDN) teletraffic queueing literature wasalmost exclusively devoted to Poisson or Bernoulli arrivals(because it described rather well the call arrival process in

Fig. 3. Buffer overflow probability in the case of “smooth” trafficand “bursty” traffic. The smooth traffic is generated by 150 sources,which emit cells according to a Bernoulli process (discrete versionof the Poisson process). The bursty traffic is from 150ON–OFF

sources with an exponentially distributedON period (average length100 cell periods) and exponentially distributedOFF period (averagelength 400 cell periods). The aggregate load is 0.75 erlang in bothcases.

a telephone exchange and because of the equally importantreason that these processes lead to a mathematically sim-ple queueing model), more complicated arrival processes,which feature correlation, are essential for B-ISDN perfor-mance evaluation. The impact of correlation may be seenfrom Fig. 3, where the buffer overflow probability (“taildistribution”) as a function of buffer size is compared fora queue with Bernoulli arrivals and one fed byON–OFF

sources, both with a load of 0.75 erlang (meaning thatthe server is busy 75% of the time on average). It isseen that cell loss will be less than 10 for a buffersize of about 40 in the case of Bernoulli sources; forON–OFF sources, however, a much larger buffer is requiredto keep cell loss at the same low level. (Note that the “cellcomponent” discussed in Section IX and shown in Fig. 1 isnot displayed in theON–OFF case here. This is because thedistribution was obtained using a fluid flow approximation,discussed further, which is unable to calculate this part ofthe curve.)

XVI. A NALYTICAL SOLUTION METHODS

The determination of the equilibrium distributions ofoccupancy and waiting time in a queueing system fed bygiven types of traffic is an important element in assessing itsperformance. (There are many reasons, however, why thisis not sufficient. The equilibrium condition of the systemis assumed more for mathematical convenience than basedon real-life experience. And it is not clear how stringentconditions as a maximum average loss of one cell in 10should be interpreted if a connection sends much less than10 cells before being released. We will not enter into thisdiscussion here, but limit ourselves to an introduction ofthe basic methods.)

An unsophisticated method is Monte Carlo simulationon a computer. Simulation allows one to model the sys-tem to any degree of accuracy, including special servicedisciplines, arbitrary traffic models (for example, the useof experimental traces obtained from life systems), etc. As


the QoS is expressed in terms of very low probabilities,however, simulation is not often up to this task. Simu-lation may be used when the overall behavior is knownin advance, such that extrapolation of a tail distributionis warranted. And there exist methods for speeding upsimulations in order to determine frequencies ofrare events(see, for example, [74]–[77]), but they are not easily appliedto queueing systems (see, however, [78]).

Therefore, it is highly desirable to have analyti-cal/numerical methods for solving queueing systems. Inthe following sections, we will present the three mainmethods for doing so, i.e., the matrix method, pgf, andfluid flow approximation.

A. Matrix Method

We illustrate the main features by considering a statisticalmultiplexer in discrete time, fed by a Markov modulatedsource. The steady-state buffer occupation probabilitiesmay be obtained as the solution of a system of linear equa-tions. This is elementary in principle, but depending on thebuffer size and the number of states in the Markov chain,finding this solution with sufficient numerical accuracy maybe difficult or impossible.

We assume that time is slotted; the source may emitzero, one, or more cells per time slot with a probabilitythat depends on the state of the modulator. (A source thatemits more than one cell per time slot does not representa physical source such as a terminal but may model theaggregate traffic of a number of terminals.) Service time isconstant and equal to one time slot. The system has a buffercapacity of cells (excluding the server itself). Let us findthe queue contents after arrivals (which are assumed to beinstantaneous, at the beginning of a slot, just after servicecompletion).

First, consider the elementary case with only one modu-lator phase. The nature of the arrival process in this state isimmaterial, but the probability that cells are generated ina time slot is assumed to be known and independent fromslot to slot

Prob cells generated in a time-slot (51)

such that . The occupancy of the queue at theinstants defined above is a random variable whose evolutionis determined by the following transition probability matrix

, as shown in (52) at the bottom of the page. The rowsand columns are indexed by the number of cells in the

queue, and the entry of should represent theprobability of a transition of cells in the queue at slot

to cells in the queue at slot . For example, thetransition (upper left entry) means from zero in thequeue to zero in the queue a slot later. It is clear that thisrequires that there were no arrivals and, indeed, this entryis . Continuing in the row “0,” transitions require

arrivals, and this as long as . As the buffer is finite( positions), all cases where more thancells arrive in abuffer with cells result in a transition , asseen in column . Note that this implies a row sum equalto one, as expected for a stochastic matrix. The row “1” isfor transitions . As one cell may always be served, thesituation is the same as for transitions , and row “1”is identical with row “0.” Row “2” designates transitions

; as it is impossible to end with zero cells in thebuffer, the next slot, the first entry is zero; requireszero arrivals, and therefore has the entry, and so on.

Because of its staircase structure, (52) is known as anupper Hessenbergmatrix in numerical analysis. Because itis also the matrix structure appearing in the solution of the

queue, it is also called amatrix of type.Let the modulator now have phases; phase transitions

occur at the beginning of a slot. Insofar as the queueingsystem is concerned, this is irrelevant. A transitionstill requires the arrival of cells. The arrivalof cells, however, now has a probability that dependson the phase of the modulator; moreover, the phase of themodulator itself may have changed in the last time slot. Thatis to say, one can no longer describe the queue occupancy asan (embedded) Markov chain. However, we may considerthe bivariate process with the queue contentsas the first component and the phase of the modulatoras the second one. It is easily seen that knowledge of

is sufficient to determine the process (ina probabilistic sense) for , and we therefore haveagain a Markov chain. If the state vector for a single phasemodulator is written as , for phases itshould be enhanced by an index indicating the state of themodulator

Thus, is the probability that there are cells in thebuffer while the modulator is in state. For the transitionprobability matrix, this enhancement of the state vector is

in queue

in queue

in queue

in queue

in queue

(52)


mirrored in the replacement of the scalarby anmatrix . The index still indicates that the number ofarrivals is , but the inner machinery of takes intoaccount the phase change of the modulator

Probmodulator makes the transition

and, per time-slot cells are generated

in phase (53)

As the modulatordynamicsare totally independent fromthe cell generation process, may be further detailed. Ifthe modulator has transition matrix, and the probabilityof generating cells when the modulator is in state is

, we have

(54)

In summary, the Markov modulation only entails the matrixin (52) to be replaced by (55), shown at the bottom of

the page.This matrix structure is known asblock Hessenberg, i.e.,

there are no blocks below the ones parallel to the diagonal.This structure may be exploited in order to solve the systemof linear equations efficiently; this is the purpose of theMarkov block Hessenberg algorithm described in [79] and[80]. An extensive study of different algorithms for solvingMarkov chains is given in [81]. Remarks on numericalprocedures are also found in the books by Neuts [35], [36].

An important problem for the matrix method is thestate space explosion,which occurs when several Markovmodulated sources are superposed. Thus, superposing 30MMPP(2) sources as the input for a statistical multiplexerwith 100 buffer places leads to a matrix of dimension10 . By exploiting the structure of the Markov chain,however, the complexity may be reduced enormously.Thus, in [82], the problem is reduced to the inversion of 2

2 matrices. A few interesting papers that apply matrixmethods for solving the statistical multiplexer problem,some including systems with priorities, are [79], [83], and[84].

B. PGF Approach

Use of the pgf results in reformulating the problemas a functional equation in the complex plane, such thatmany results of complex analysis may be applied. Becausethe pgf is a transform method, one should be able to

“transform back” for obtaining the probability distribu-tion. This requires appropriate methods; in the past, manyanalyses based on the pgf were satisfied with only thefirst few moments of the distribution (which follow easilyfrom the pgf), but for broad-band traffic engineering, oneis interested in low probability quantiles, and the wholedistribution should be extracted from the pgf. Some possiblesolutions to this problem will also be discussed, but first weexplain the essence of the method in a simple case wheretransforming back is elementary. A mathematical treatmentof the pgf method is found in [9], and applications to ATMsystems are found in [85].

The pgf of a discrete distribution is afunction defined as the-transform of the probabilitymass function

(56)

where may be complex. Knowledge of is in prin-ciple equivalent with knowledge of because it is easilyseen that

(57)

Similarly, the mean of a random variable may bededuced from the pgf of

(58)

For higher derivatives, however, the expressions becomeunwieldy and this is not a viable way to deduce theprobability distribution.

We illustrate the use of the pgf by considering a singleserver queueing system operating in discrete time, withconstant service time equal to one time slot and a generalindependent (GI) arrival process (i.e., interarrival times arei.i.d. random variables with a general distribution).

If represents the system contents at time, the timeevolution will be governed by

(59)

where is the number of arrivals in time slot and. From (59), we deduce that in equilib-

rium (i.e., when and have the same distribution)

Prob Prob (60)

in queue

in queue

in queue

in queue

in queue

(55)


Let us temporarily represent the random variableby , and let Prob Prob and

Prob . As the two random variables andare assumed independent (because of the GI assumption),the distribution of their sum is the convolutionof their respective distributions, and the-transform ofis the product of the -transforms

(61)

If and are the pgf’s of andrespectively, (61) becomes

(62)

To complete the derivation, we have only to expressin terms of . A little algebra shows thattranslates into

and we have the following relation between the pgf ofthe system contents and the pgf of the arrival process:

Prob (63)

(The factor Prob , the probability for an emptysystem, equals one minus the load; according to (58),the latter is , and is fully expressed in terms ofthe arrival process.)

To illustrate further the use of the pgf, let us specializeand consider a geometrical distribution of the number

of arrivals per time slot

Prob (64)

with mean (load)

(65)

The pgf of is

(66)

Equation (63) then becomes, after some simplification

(67)

Comparison with the defining relationshows that

Prob (68)

In this example, inversion of the pgf was straightforward,but this usually is not the case. The present example,however, shows a possible approximation. If, for example,

the pgf is a rational expression , it may beexpanded in partial fractions

(69)

where are zeros of . Each term in the expansion isthe -transform of a geometrical distribution, as was usedin the example. The probability may then be writtenas a sum of terms . For sufficiently large (tailof the distribution), it is sufficient to restrict the sum toa single term, i.e., the one involving the smallest zero.For some more advanced applications of this technique,see, for example, [85, p. 147 and the following pages].Another powerful approach for inverting the-transformis the Fourier series method, for which we refer to theexcellent paper [86].

Some interesting papers using the pgf approach are[87]–[90].

C. Fluid Flow Approximation

Just as it is quite irrelevant that an inundation comesabout by individual raindrops, the discrete nature of thearrival process may be neglected when interest is focusedon a buffer in congestion. This gives rise to the fluid flowapproximation, which models the queueing system as a“leaky bucket,” receiving a continuous information fluidof varying intensity and being emptied at a certain (usuallyconstant) rate. Note that it was via a fluid flow approach thatthe first really useful analysis of a statistical multiplexer, fedby bursty sources, was performed. This was the seminalpaper by Anicket al. of 1982, treating an FIFO queuewith (identical) exponentially distributedON–OFF sources[91]. Following [92], we derive the differential equationdescribing the buffer contents assuming the filling processto be modulated by a Markov process and the service time(leak rate) to be constant.

Let be a CTMC that takes the val-ues , and let the infinitesimal generator (seeSection V-B) be the matrix with elements . When theMarkov chain is in state, fluid arrives with a rate ; aslong as the buffer is not empty, it is drained at a constantrate , and therefore the net rate of change of the buffercontents is

(70)

This net rate may be positive as well as negative; when, is an underload state; when , is an

overload state.Let denote the buffer contents at time. is

a continuous random variable satisfying ,where is the buffer size with possibly . We will,for example, be interested in the quantity

Prob (71)

the equilibrium overflow probability beyond the level.


As discussed in Section XVI-A, for determining thequeue contents, we have to deal with the bivariate stochasticprocess with a joint pdf-pmf

Prob

(72)

(In words: is the probability that, at time, thebuffer is filled to at most the level and the modulator isin phase .) When the system has reached equilibrium, wemay define

Prob (73)

For conciseness, one may define an -dimensionalrow vector

(74)

which allows one to express the equilibrium overflowprobability (71) as

Prob (75)

where , for transpose and for scalarproduct.

The time evolution of is governed by the fol-lowing equation:

(76)

In words, the probability that at time the buffer isfilled to at most and the modulator is in stateconsistsof two terms. The first one applies to the case where themodulator was in one of the other states, , attime . To end up in state at , there should bea phase transition , which has a probability of [towithin ] to make a transition . Moreover,somewhere in the time interval , the buffer contents willchange. Say the net rate of change is. To end up with

at , one should have at . Thisaccounts for the first term. The second term is similar, butwithout phase transition of the modulator. The termvanishes faster than for .

We subtract from both sides in (76) and divideby ; letting and using the properties of theinfinitesimal generator, we obtain

Last, for the equilibrium solution, , and weobtain the set of equations

(77)

Introducing the rate matrix

(78)

Equation (77) may be written compactly as

(79)

This is a linear first-order differential equation, with so-lution a linear combination of exponentials. This solutionmay be formally written as

(80)

( a constant row vector). It follows that the exponentialsare of the form , where , are theeigenvalues of . An essential part of the work whensolving fluid flow models is therefore devoted to findingeigenvalues and eigenvectors.

The already mentioned paper [91] succeeds in reachingclosed-form expressions for the eigenvalues and eigen-vectors, thanks to the restriction to homogeneous (i.e.,identical) exponentialON–OFF sources multiplexed in anFIFO buffer. More complex cases may be treated byfluid flow methods, although closed-form expressions arethen no longer possible. An advanced application is [93],where sources have a Markov modulator with an arbitrarynumber of phases and there are an arbitrary number of losspriorities.

XVII. F ITTING TO MODELS

Analytic source models may be used in essentially twoways. One can use an analytic model to represent asaccurately as possible anindividual source (for example,one video connection). For the performance evaluation ordimensioning of a statistical multiplexer, however, onehas to consider a superposition of such sources (or ofa mix of diverse sources). In most cases, the aggregatetraffic resulting from the superposition is of a differentnature than the individual sources. The superposition ofexponentialON–OFF sources is not anON–OFF source, etc.Therefore, a second application of analytic source models isto represent an aggregate traffic stream; the latter may arisein analytical work (for example, studying a multiplexerfed by a number of independent VBR sources) or maybe an experimental traffic stream measured in some real-life system. For this second application, one needs ananalytical model for the aggregate traffic, as well as a


Fig. 4. The three steps in the SMAQ approach.

procedure for finding the parameters of the analytical modelsuch that it “fits” the original traffic stream. It will havebecome apparent throughout this paper that capturing atleast approximately some second-order statistics (autocor-relation, index of dispersion, power spectrum, etc.) of atraffic stream is of prime importance for assessing themultiplexer performance. Therefore, one wants a procedureto fit first- and second-order statistics of a given trafficstream or source superposition to some analytical model.There have been many proposals in this direction [24],[31], [34], [94]–[97], but the most encompassing one toour knowledge is that proposed by S.-Q. Li and coworkers[98], called the statistical match queueing tool (SMAQ).The general approach is shown in Fig. 4. The ambitionof SMAQ is to model anarbitrary traffic stream. Thetraffic is assumed to be sufficiently characterized by thedistribution function and the power spectrum ofthe input rate process. The measurement of these quantitiesis the first step. Then comes the modeling of the traffic.It is proposed to use a special kind of MMPP() source,called circulant modulated Poisson processes (CMPP’s).Last, the CMPP traffic model may be used as the inputprocess of a multiplexer; this queueing model is solvedanalytically/numerically and results in the performancemeasures of interest as cell loss, delay, etc.

Without going into the mathematics, one can summarizethe approach as follows. Given the couple ,the input rate process, and the power spectrum of the in-coming traffic, one should find a couple , which are,respectively, the infinitesimal generator and the rate vectorof an MMPP( ) process with the same . Thisis a case of an “inverse eigenvalue problem” and has notbeen solved. To circumvent this difficulty, Li introducesa special MMPP with an infinitesimal generator of thefollowing form:

(81)

It is seen that knowledge of the arrayis sufficient to generate the whole

matrix by simply “circulating” the rows. This specialMMPP ( ) is therefore called circulant modulated Poissonprocess. This special MMPP has several interestingproperties. The equilibrium probabilities are all equal to

, and the eigenvalues and the power spectrum are

simple closed-form expressions. In this way, the inverseeigenvalue problem may be solved. Experiments on realtraffic described in [98] (a video sequence fromStar Warsand real Ethernet traffic) were modeled with CMMP’sof reasonable dimension (100 and 160, respectively) andgave good results for the cell loss. In a recent publication[99], the SMAQ tool is used to evaluate the performanceof an ATM switching element design, including multicastcapabilities.

XVIII. S OME TOPICS IN CURRENT B-ISDNTRAFFIC ENGINEERING

In this paper, we have concentrated on mathematicaltechniques rather than on the kind of problems in the trafficengineering of broad-band systems. In what follows, wewant to give an impression of the diversity of questions tobe solved by the traffic engineer. A very useful and recentoverview of work in this area is [100].

A. CAC

A B-ISDN based on ATM is intended as a reliableand universal vehicle for digital information, whatever thenature of the requested service. It is up to the network toguarantee the requested QoS once a connection has beenaccepted. To this end, a number of traffic and congestioncontrol functions are defined. One is CAC, a preventivefunction that should decide on the setup of a new con-nection, guaranteeing the requested QoS of the newandthe already established connections. Moreover, the networkbandwidth resources should be exploited as efficiently aspossible. This is a difficult problem because the environ-ment is highly “stochastic,” due to the random setup andrelease of connections and the stochastic behavior on thecell level of the connections themselves. The CAC decisionshould be taken swiftly, precluding lengthy calculations.

As a “zeroth-order solution,” one could take as a ruleto accept new connections until the sum of their declaredpeak rates reaches the link capacity. Even suchpeak-rateallocation is not without problems, however, because celldelay variation (CDV) may have introduced cell clumpingas the connection’s traffic traversed access multiplexers,switching nodes, etc. One can take into account the actionof the policing device:after passing through the policer, aconnection’s traffic is guaranteed to be of the “bounded”kind, as explained in Section XIV-A. This constraint maybe taken into account by CAC, in a worst case approach(guaranteeing no loss at all), or in a statistical way (reducingthe loss to an acceptable level).


(a) (b)

Fig. 5. (a) A switching node with output buffering. On a numberof input lines, there are connections destined for output A. Thus,an output line may be modeled as a statistical multiplexer (b).Acceptance of a new connection destined for output A dependsmainly on the state of this statistical multiplexer.

For bursty sources (VBR), peak-rate allocation is tooinefficient, and one wants to take advantage of astatisti-cal gain while guaranteeing the QoS. The possibility fora “statistical gain” stems from the low probability thatindependentbursty sources will emit at their peak ratessimultaneously. As shown in Fig. 5, the CAC problem es-sentially boils down to the study of a statistical multiplexer,which is one reason for the prominent role of queueingtheory in B-ISDN traffic engineering.

Whereas the general statistical multiplexer problem isa difficult one, the development of a CAC strategy iseven more difficult due to the stringent timing constraints.Indeed, one needs a fast algorithm that decides “on thefly” whether a new connection may be accepted or not.Much effort has been spent on defining an “effectivebandwidth” or “equivalent capacity” for bursty connections.In principle, the admissibility of a new connection dependsnot only on the properties of this connection but also on thewhole constellation of the already established connectionsplus the new connection. Effective bandwidth (EB) is anapproximation, anadditivequantity, which depends only onthe characteristics of the connection itself; a new connectionwill be accepted if the sum of the EB’s of the alreadyestablished connections and the EB of the new connection isless than a certain fraction of the link bandwidth. Advancedstatistical methods are used for determining such EB, oftenbased on alarge deviations approximation, a statisticaltheory on rare events. For an excellent introduction tothe topic of large deviations, we refer to [101]. A recentoverview of CAC schemes has been given in [102].

B. Buffer Dimensioning

Buffer dimensioning is another issue that uses the statisti-cal multiplexer model. Given a certain switch architecture,the buffering capacity (in concentrators, switching ele-ments, output buffers, etc.) has to be determined suchthat a certain switch throughput can be obtained whileguaranteeing that cell loss will be below a given level. Untilrecently, only FIFO servers were considered, and as theycannot discriminate between the different connections, thecell loss performance was determined over the whole cellpopulation. We will briefly discuss how buffer dimension-ing is done in this case; the new trend of using “intelligent”service disciplines that “see” the different connections willbe discussed in Section XVIII-D.

Fig. 6. Switching element with four “routing groups” correspond-ing to four logical queues. The queues share the same bufferingspace, resulting in a statistical gain of buffer memory.

Dimensioning always relies on some assumptions as forthe kind of traffic multiplexed in the buffer, and thereforealso depends on switch architecture and offered services.To give an example, if traffic entering a switching fabricis being randomized (as in a multipath switch, wherethe cells of one connection follow different routes frominput to output port), the traffic in the fabric’s switchingelements may be expected to be “smooth”; simple queueingmodels such as may then be used to dimension theswitching element buffer. Whatever the traffic model, whatis needed for buffer dimensioning is the buffer occupancyof a queueing system receiving that type of traffic. Quiteoften, the mathematical model has aninfinite buffer, andinstead of the cell loss proper, the probability for the bufferoccupation to exceed a given level (“overflow probability”)is determined. This is easily accomplished by plotting thebuffer occupancy tail distribution, as in Fig. 3; the buffersize needed in order for the cell loss not to exceed a givenvalue (say, 10 ) is then found as the 10-quantile of thistail distribution.

The use ofshared buffersmay complicate the dimension-ing. Just as statistical multiplexing allows one to optimizethe use of bandwidth, a shared buffer allows one to optimizethe use of buffer capacity. In a switching element, forexample (see Fig. 6), the different “routing groups,” whichthemselves are statistical multiplexers from a logical pointof view (multiplexing the connections destined for the sameoutput port of the switching element), may share the samephysical buffer space, and it is the latter that should bedimensioned properly. There are various possibilities, alsorelated to hardware implementations. The simplest case iswhen the logical queues are identical and independentfrom a traffic point of view. The shared buffer occupation isthen the sum of the occupations of the logical queues. Theseoccupations are i.i.d. random variables, and the sharedbuffer occupation is obtained from the-fold convolutionof the occupancy pmf of a logical queue.

The possible correlation between these queues may com-plicate the analysis. Negative as well as positive correla-


tions may be identified; an early assessment of this problemwas already given in 1988 [103]. What one wants to find isa good compromise between a worst case solution (sharedbuffer times as large as one logical queue) and a naıveapproach that neglects correlation. Some useful referencesare [104]–[107].

C. New Trends in Switch Design

A number of new architectural and hardware features areprogressively introduced in broad-band switches to improveefficiency and quality of service. They require a carefulanalysis from the teletraffic point of view.

If a buffer multiplexes packets with highly different QoSrequirements (for example, speech on the one hand, whichis very delay sensitive but rather loss tolerant, and dataon the other hand, which has the opposite characteristics),the use of a buffer common to both and FIFO serviceevidently results in very poor performance. This is theso-called multi-QoS problem. Different strategies may beused to improve efficiency. A simple method is astaticpriority approach. Each packet belongs to one ofpriorityclasses and is stored in one ofFIFO queues. The commonserver obeys a strict priority rule, first serving packets ofthe highest priority class, switching to the second classonly when no packets are queueing in the highest class,etc. If only two classes are present, say one “real-time”and one “data,” this might work [108], [109], but as moreservice classes have been standardized, the static priorityapproach is not really satisfactory. Moreover, use of theFIFO service discipline for all connections of the samepriority class does not guarantee a good QoS for eachindividual connection; this is why per-virtual-connection(VC) queueing is considered increasingly important (seeSection XVIII-D).

Another feature, mainly intended for the case wherethere are twoloss priority classes, is the use of a push-out mechanism (push-out buffer, POB) or a partial buffersharing (PBS) mechanism [110]–[113]. In the POB scheme,a high-priority packet arriving in a full buffer pushes outa low-priority packet; in the PBS scheme, there are one ormore thresholds, and the part above a threshold is reservedfor the higher priority class(es). A shortcoming of theseapproaches is that they are mainly restricted to loss control,whereas for a truly integrated services network, loss, delay,and delay variation should be controlled simultaneously.Moreover, unless they are for very small buffers, theintroduction of POB or PBS does not lead to a substantialthroughput increase, and interest in these mechanisms hassomewhat waned.

In contrast,cell-discard schemes for packet networksarereceiving considerable attention [114], [115]. This is partof a more general trend, whereby the ATM QoS notiongradually pervades the data communications world; witnessthe evolution of the Internet from a “best effort service”toward a “guaranteed service.” All this is an illustration ofthe commonplace notion that “telecommunications and datacommunications are converging” [116].

The search for efficient cell-discard mechanisms in packetnetworks is logical: in transport-layer protocols such as TCPover ATM, the entire packet is discarded if one or moreof its constituting ATM cells is lost. Discarding cells ofthe corrupted packets will therefore increase the efficiencyof the ATM network. There are several strategies.Earlypacket discardanticipates the loss of part of the packet bydiscarding the entire packet if the queue length exceeds acertain threshold.Partial packet discardand tail droppingdiscard the packet partially. As is often the case in teletrafficengineering, a performance study does not just consistof finding a theoretical “optimum” but must also involvepractical considerations as to implementation complexity,required buffer size, etc.

Packet discard is one of the techniques to counteract con-gestion in the B-ISDN network (for a review on congestioncontrol, see [117]). Another approach is to userate-basedflow control, as is done for the available bit rate (ABR)service category [118]. This data service exploits the time-varying available bandwidth (the residue that remains afterall guaranteed services have taken the needed bandwidth),but to prevent packet loss, a closed control loop approachis used. Congestion information is communicated by thenetwork to the ABR terminals, which may consequentlyadapt their sending rate. Several schemes exist, such asbinary congestion indication (congested or not congested)and explicit rate (explicitly giving the rate at which theterminal may send). Engineering ABR control loops israther complex because of the variable delays of controlinformation (due to the difference in length of the differentpaths in a network and to the random component in thedelay on a given path). Consequently, control theoreticmethods also have recently found their way to trafficengineering [119].

D. Quality Per Individual Connection

QoS is at the heart of ATM. From a user point of view,it seems evident that this statement should be interpretedin terms of the QoS of each individual connection. Never-theless, until recently, ATM switching designs were basedon a less “individualistic” interpretation of QoS. The cellloss figure, for instance, was usually based on the overallperformance by averaging over all connections belongingto a certain QoS class. There were several reasons for this.Taking care of the QoS of individual connections requiresa queue service discipline, which, unlike FIFO, “sees”the different connections. The hardware implementation ofsuch a mechanism, operating at broad-band speed, is atechnological challenge that only now comes within reach.A second reason why QoS figures are usually derived fora certain group of connections is the theoretical difficultyto attach a meaning to, for example, a cell loss probabilityof 10 for a connection that lasts only 10cells. Thisprobably explains why the telecommunications sector ofthe International Telecommunications Union and the ATMForum standardization bodies [121], [122] have not yetstandardized the notion of QoS per connection.


The continuing progress in very large scale integrationtechniques. however, brings non-FIFO service disciplines,which do “see” the individual connections and which takecare of the individual quality, within reach of hardwareimplementation. This is known as per-VC queueing [120].There are two issues in per-VC queueing. One is the fairrepartition of bandwidth; the other is the fair repartition ofbuffer space. As for the bandwidth, it is desirable to gobeyond the bandwidth allocation that is implicit in FIFOqueueing. FIFO allocates bandwidth in proportion to thenumber of packets stored in the buffer, whether they belongto a high or low bit-rate connection. In contrast, “fair”scheduling schemes should allocate bandwidth to eachconnection according to a predetermined rule, for example,the declared bit rate of the connections or a certain fractionof the residual available bit rate for “best effort” services.

When discussing fair scheduling schemes, one should,at least conceptually, consider the different connectionsas queueing in separate buffers. The scheduling schemethen defines the order in which these different queuesare served. This is not unlike the schemes discussed inthe previous section, where separate priority classes areassigned to separate buffers and the different buffers areserved according to a certain rule (as, for example, staticpriority); the difference, however, is that for per-VC queue-ing, there are as many queues are there are connections(which may be several thousands per outlet port), implyinga real challenge for a hardware implementation. A simplemethod is round robin, which spends one time slot to eachqueue in a cyclic manner; this would only be adequatefor connections requiring the same bandwidth. Somewhatmore sophisticated is weighted round robin (WRR), wherethe number of consecutive slots devoted to a queue isdependent on a weight factor. A drawback of WRR isthat it may increase the burstiness, several packets of theconnection being served consecutively. This is why somuch attention has been paid toweighted fair queueing[123]. The idea behind WFQ is an ideal system, consistingof a number of “fluid” connections (which, in contrast withpackets, are infinitely divisible) and a server, which maydivide its service capacity into as many parallel serversas needed. This allows a mathematically precise definitionof fair repartition of the bandwidth, each fluid serverinstantaneously offering service to its fluid connection.The ideal system is then used for defining algorithmsapplicable to real packet systems. Many such derivationsexist, mainly differing in the degree of complexity for ahardware implementation: virtual time [123], self-clockedfair queueing [124], [125], and virtual spacing [126].

Even with a WFQ service discipline, the CDV of theoutgoing connections is not fully controlled, as it dependson the behavior of the other connections. To further reducejitter, one has to resort to nonwork-conserving servicedisciplines, whichshapethe traffic, spacing packets apart bya minimum distance. For a review of nonwork-conservingservice disciplines, see [72].

So far, we have discussed possibilities for the schedulingdiscipline in per-VC queueing; the goal is to share the

bandwidth in a fair way. The other issue is the way thatthe buffer space is assigned. The several logical queuesconsidered above may be implemented in different ways.One extreme is complete sharing; the other extreme is tohave strictly individual queues per connection. The latter,although ideal from a QoS point of view, could lead toexcessive memory requirements; a form of sharing of thebuffer space seems preferable. The same considerationsapply as mentioned in Section XVIII-B concerning thedimensioning of shared buffers. A compromise betweencomplete sharing and complete separation consists in usinga threshold mechanism, whereby a minimum buffer spaceis reserved per connection and the rest of the buffer spaceis shared between all connections.

When attention shifts from performance measures av-eraged over a number of connections to the quality ofan individual connection, one should also question theapplicability of the stationary analysis underlying most oftraffic-engineering work, whereby performance measuresare obtained for a hypothetical system that has reached“statistical equilibrium.” Relatively few studies (see, e.g.,[127]–[129]) are devoted to the “transients” that occur incongestion periods. As the impact may be considerable[129], and as nonstationarity is more realistic in a realnetwork, this topic deserves in fact more attention than isactually the case.

A last item we want to mention concerning QoS guar-antees in a B-ISDN is the issue of end-to-end perfor-mance. The models discussed in this paper all concentrateon a single node of a switching network, and we wereconcerned with the cell loss, delay, and delay jitter insuch a node. What counts is theoverall cell loss, how-ever, the end-to-end delay, the jitter at the destinationnode. Now, just as in the ordinary telephone network,the equipment composing a B-ISDN will normally consistof nodes constructed by different vendors and managedby different operators. Therefore, the interfaces shouldbe standardized and the QoS requirements expressed ona per-node base; this justifies the focus of much of theteletraffic studies on a single node. However, the translationof end-to-end objectives into per-node requirements (atopic par excellencefor standardization bodies) is notstraightforward. The segments of a connection may dif-fer considerably, especially if one or more consist of“satellite links.” Moreover, QoS parameters are in generalnot additive, due to correlations between successive hops.Standardization bodies [121], [122] provisionally take aworst case additive approach, but further study shouldmake clear if this may be improved upon, permittinghigher network utilization. Additivity is especially toostringent for the cell delay variation; the insertion of a singletraffic shaper at the destination may drastically change thepicture. In general, the use of special service disciplinesin combination with bounded traffic models is a possibleway for end-to-end performance guarantees, as mentionedin Section XIV. From a traffic-engineering point of view,the main difficulty of an end-to-end analysis lies in thecharacterization of the output process of a queueing node;


for example, multiplexing may not preserve the Markovproperty of incoming traffic streams [130]. More effort togo beyond the worst case additive approach is also desirablenow that the phenomenon of long-range dependence andits possible negative impact upon network performanceurges a more careful approach. An attempt to integrateself-similarity into an end-to-end analysis is found in [131].

XIX. CONCLUSIONS

Teletraffic engineering is as old as the telephone service,but its role has become increasingly important with theadvent of B-ISDN. This is due to the integration of serviceswith widely varying characteristics and to the stringentrequirements, as for QoS. These evolutions gave birth tonew concepts, methods, and algorithms (Markov modulatedand long-range-dependent source models, statistical gain,EB, WFQ, leaky bucket policing, and so forth), whichshould be evaluated as for their performance.

As has become apparent throughout this paper, a con-siderable part of B-ISDN traffic engineering is devotedto the study of astatistical multiplexer, i.e., a queueingsystem fed by a number of random arrival processes. Thisinvolves the identification and mathematical modeling ofthe arrival processes appropriate for B-ISDN, the choice ofa service discipline (FIFO, round robin, WFQ, etc.) and thedetermination of the queueing performance (loss, delay) ofsuch a system. We have introduced the basic mathematicaltechniques involved, some of which (spectral properties,self-similar processes) entered the field only recently.

We also gave a rather thorough overview of “boundedtraffic models,” which, instead of defining a particularstochastic process, delineate a whole class of models byspecifying bounds on the cell-generation process. While theparticular traffic models with precise statistical propertiesare important for performance case studies, the bounded-traffic-model approach is important for declaring B-ISDNconnections in practice, at the connection setup phase.

The decisive test on the validity and accuracy of all themethods and models introduced in teletraffic engineering inthis “broad-band era” will appear in the near future, whenbroad-band networks will be deployed on a large scale,offering the rich choice of services for which they wereintended.

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Herman Michiel was born in 1951. He receivedthe licentiate degree in mathematics and thePh.D. degree from the University of Leuven,Belgium, in 1974 and 1983, respectively.

He was a mathematics and physics Teacherfor some years and then began research insolid-state physics at the University of Leuven.From 1984 to 1989, he conducted amorphoussemiconductor research at the InteruniversityMicroelectronics Center (IMEC, Leuven). Hethen joined the Teletraffic Group of the Research

Center, Alcatel Bell Telephone (Antwerpen), where he was involved inthe modeling and performance evaluation of ATM switching systems.His current research topics include traffic modeling, “intelligent” servicedisciplines, and queueing architectures.

Koen Laevenswas born in Roeselare, Belgium,in 1967. He received the M.S. degree in elec-trical engineering from the University of Ghent,Belgium, in 1991, where he currently is work-ing toward the Ph.D. degree on performanceanalysis of asynchronous transfer mode (ATM)switches.

In 1991, he joined the Stochastic Model-ing and Analysis of Communication SystemsResearch Group at the University of Ghent.Besides the theoretical aspects of (discrete-time)

queueing theory, his research interests also include the performance evalu-ation of communication systems—in particular of ATM networks—trafficcharacterization and modeling, and the study of long-range dependenceand self-similarity in ATM traffic. He has been involved in a number ofprojects, including the European COST 242 and COST 257 actions onbroad-band multiservice networks.


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