SQUEEZING THE MOST OUT OF ATM - Columbia University

SQUEEZING THE MOST OUT OF ATM

Gagan L. Choudhury,1 David M. Lucantoni2 and Ward Whitt3

Abstract—Even though ATM seems to be clearly the wave of thefuture, one performance analysis indicates that the combinationof stringent performance requirements (e.g., 10 − 9 cell blockingprobabilities), moderate-size buffers and highly bursty traffic willrequire that the utilization of the network be quite low. Thatperformance analysis is based on asymptotic decay rates ofsteady-state distributions used to develop a concept of effectivebandwidths for connection admission control. However, we havedeveloped an exact numerical algorithm that shows that theeffective-bandwidth approximation can overestimate the targetsmall blocking probabilities by several orders of magnitude whenthere are many sources that are more bursty than Poisson. Thebad news is that the appealing simple connection-admission-control algorithm using effective bandwidths based solely on tail-probability asymptotic decay rates may actually not be aseffective as many have hoped. The good news is that thestatistical multiplexing gain on ATM networks may actually behigher than some have feared. For one example, thought to berealistic, our analysis indicates that the network actually cansupport twice as many sources as predicted by the effective-bandwidth approximation. That discrepancy occurs because fora large number of bursty sources the asymptotic constant in thetail probability exponential asymptote is extremely small. That inturn can be explained by the observation that the asymptoticconstant decays exponentially in the number of sources when thesources are scaled to keep the total arrival rate fixed. We alsoshow that the effective-bandwidth approximation is not alwaysconservative. Specifically, for sources less bursty than Poisson,the asymptotic constant grows exponentially in the number ofsources (when they are scaled as above) and the effective-bandwidth approximation can greatly underestimate the targetblocking probabilities. Finally, we develop new approximationsthat work much better than the pure effective-bandwidthapproximation.

Keywords—ATM, connection admission control, statisticalmultiplexing, effective bandwidths, performance analysis,queueing models, asymptotic decay rates, numerical transforminversion

_ ____________________1 AT&T Bell Laboratories, Room 1L-238, Holmdel, NJ 07733-3030; email: [email protected] IsoQuantic Technologies, LLC, 10 Oak Tree Lane, Wayside, NJ 07712; email:[email protected] AT&T Bell Laboratories, Room 2C-178, Murray Hill, NJ 07974-0636; email: [email protected]

I. INTRODUCTION AND SUMMARY

Great energy is being devoted to studying the promising newasynchronous transfer mode (ATM) technology for supportingmulti-service high-speed communication networks; e.g., seeRoberts [39]. As indicated in [39], interest in ATM is stimulated bytwo factors: first, by new technology making it possible to transmitand switch at very high bandwidths and, second, by growing demandfor more sophisticated and powerful communication services.

Even though ATM seems to be well on its way to widespread use,a performance analysis based on asymptotic decay rates of steady-state distributions indicates that the combination of stringentperformance requirements (e.g., 10 − 9 cell blocking probabilities),moderate-size buffers and highly bursty traffic associated with thesenew services will necessitate operating the networks at very lowutilizations. (The basic model represents an ATM switch receivingfixed-size ATM cells from several sources and transmitting them overan output channel in a first-in-first-out fashion. The basic question is:How many sources can be admitted for a fixed buffer size with aspecified small cell blocking probability?) A main message of thispaper is that ATM networks may be able to achieve higher utilizationsthan that asymptotic analysis indicates. In other words, there seemsto be more potential for statistical multiplexing gain.

The approximation based on asymptotic decay rates of tailprobabilities is very appealing because it supports a concept ofeffective bandwidths. From an engineering perspective, the notion ofeffective bandwidths is very natural. The idea is to assign eachsource an effective bandwidth requirement, and then consider anysubset of sources feasible (admissible) if the sum of the requiredeffective bandwidths is less than the total available bandwidth. Thus,a suitable notion of effective bandwidths could go a long way towardsolving the connection admission control problem. For instance,once effective bandwidths have been assigned, we can approachengineering and design problems using multi-rate loss networkmodels, e.g., as in Choudhury, Leung and Whitt [12,13] andreferences therein.

A large-deviations asymptotic analysis provides strong supportfor the simple effective-bandwidth procedure, because it shows that itis asymptotically correct as the buffer size gets large and the tailprobabilities get small, and because it provides a basis for assigningactual effective bandwidth values to different sources (voice, data,video, etc.); e.g., see Hui [31], Gibbens and Hunt [26], Guerin,Ahmadi and Naghshineh [28], Kelly [32], Sohraby [41], [42], Chang[10], Whitt [45], Elwalid and Mitra [23], Kesidis, Walrand and Chang[33], Glynn and Whitt [27], Courcoubetis, Kesidis, Ridder, Walrandand Weber [20], and Chang, Heidelberger, Juneja and Shahabuddin[11].

The additive nature of effective bandwidths is clearly appropriateif we let the effective bandwidths be either the source peak rates orthe source average rates. However, it seems intuitively clear thatworking with peak rates is far too conservative, while working withaverage rates is far too optimistic. (We show this later in ourexamples.) Most of the recent work has been aimed at findingappropriate effective bandwidths in between the peak and averagerates.

Unfortunately, however, from the outset, teletraffic engineeringexperience suggests that there may be a flaw in the effectivebandwidth concept because it corresponds to having no trafficsmoothing from multiplexing. In particular, it is known that whenmany separate bursty sources are multiplexed (superposed), the totalis less bursty than the components; i.e., there is a basis for morestatistical multiplexing gain than with Poisson sources. This istheoretically supported by the classical limit theorem stating thatsuperpositions of arrival processes, suitably scaled, converge to aPoisson process as the number of component arrival processesincreases; e.g., see C, inlar [19]. See Heffes and Lucantoni [30],Sriram and Whitt [43] and Fendick, Saksena and Whitt [25] forrelated performance studies. In contrast, with the effective-bandwidth approximation, the burstiness of n superposed independentand identically distributed sources is the same as for a single source(e.g., see p. 76 of [45]). This implies that the effective-bandwidthapproximation will predict greater congestion for any fixed arrivalrate than it should.

Nevertheless, there is a case for the effective bandwidths, becauseprevious teletraffic analysis (such as in [25], [30], [43]) did not focuson extremely small loss probabilities such as 10 − 9 . Such very smallloss probabilities naturally suggest that appropriate asymptoticsshould provide what we want. And the asymptotic analysisassociated with effective bandwidths indicates no traffic smoothing.

The question, then, is what will actually happen in real systems?Will there be significant traffic smoothing or not? Is the asymptoticanalysis supporting effective bandwidths sufficiently accurate or is itnot?

These questions have plagued researchers in recent years becausethe natural models are very difficult to analyze. It is difficult tocalculate very small tail probabilities in models with manyindependent bursty sources, by either exact analytical formulas or bycomputer simulation.

Our main contribution is a new algorithm for a queueing modelthat enables us to compute the desired small tail probabilities exactlywhen there are many bursty sources. From this exact analysis, wefind that in some cases the effective-bandwidth approximation isexcellent, but in other cases (which we think are realistic), it is not.Thus, we conclude that the notion of effective bandwidths baseddirectly on large-deviation asymptotics may not be effective forconnection admission control.

However, from an engineering perspective, the notion of effectivebandwidths remains very appealing. Thus it is natural to look formodifications of the proposed effective bandwidth approximationwhich will work well; e.g., as in Guerin and Gun [29]. It is importantto note that our analysis does not rule out the general notion ofeffective bandwidths. Instead, our analysis only indicates that theremay be serious difficulties with the implementation based directly ona particular kind of asymptotics, without refinements.

A second main contribution of this paper is to developmodifications of the basic effective-bandwidth approximation for tailprobabilities that are more accurate (namely, (1.1), (1.5) and (1.7)below). However, these new approximations do not support thesimple notion of effective bandwidths discussed above. Nevertheless,the approximate tail probabilities can be used for admission control.

The key to our analysis here is the exact numerical solution of the

i = 1Σn

G i/G/1 queueing model, which has a single server, unlimited

waiting room, the first-in first-out service discipline and i.i.d.(independent and identically distributed) service times that areindependent of a superposition arrival process. The arrival process isthe superposition of n independent component general arrivalprocesses. Our algorithm permits these component arrival processesto be heterogeneous Markovian arrival processes (MAPs), as inLucantoni [35,36]; see Section 8 here. (The general theory alsoallows batch arrivals, but in this paper we consider only singlearrivals.) Since superpositions of independent MAPs are againMAPs, it suffices to consider the MAP/G/1 queue. However, thecomputational effort increases when the number of sources increases.Our algorithm is based on Lucantoni [35]. A major new idea is thecombination of the matrix-analytic results in [35] with numericaltransform inversion. In particular, we use the Fourier-series methodin Abate and Whitt [5]. We also use a scheme for accuracyenhancement for computing small probabilities as described in [15].Moreover, we use a number of techniques for speeding upcomputation, in order to treat a large number of sources. Thesetechniques are described in Section 8. For the computation of theasymptotic parameters we use algorithms in [1], [2], [14].

We have also extended the numerical transform inversionalgorithms to multidimensional transforms [15] and applied themultidimensional inversion to calculate transient performancemeasures in the MAP/G/1 queue [37]. Thus, we are in a position tostudy the transient behavior as well as the steady-state behavior.With the ability to calculate time-dependent tail probabilities, we canstudy whether or not transient analysis is needed in the admissioncontrol problem. Here, however, we only discuss steady-statebehavior.

In our examples here, we consider only homogeneous sources,but in Section 7 we also indicate how to treat large systems withheterogeneous sources approximately. We primarily consider sourcesthat are more bursty than a Poisson process, in particular, Markovmodulated Poisson processes (MMPPs) with two-state Markovenvironment processes. (The MMPP is a Poisson process whose rateis itself a continuous-time Markov chain.) For the main example, weconsider on-off sources in which the arrival rate in one environmentstate is zero, i.e., interrupted Poisson processes, which are known tobe renewal processes having hyperexponential interarrival-timedistributions [34]. These bursty source models are often used tomodel ATM traffic. They can represent quite bursty traffic. We havealso obtained similar results for other (non-renewal) MMPP sourcesand multiple numbers of more than one kind (two or three) of MMPPsources.

We also consider sources that are less bursty than a Poissonprocess. In particular, we consider renewal processes with E 2

(Erlang of order 2) interarrival times. Such less bursty sources mightappear in ATM networks as a consequence of traffic shaping at theedge of the network. Contrary to conventional wisdom, we show thatthe effective-bandwidth approximation underestimates the exact tailprobabilities with these less bursty sources, so that the effectivebandwidth approximation need not be conservative.

As a surrogate for the steady-state blocking probability with afinite buffer, we consider the tail probability of the steady-statewaiting time in our infinite-capacity queue. When the service timesare deterministic with mean 1, as is the case with ATM cells (with theappropriate time units), the least integer above the steady-statewaiting time coincides with the steady-state queue length or buffercontent seen by an arrival. (We can also directly compute the queue-length distribution in the MAP/G/1 queue.) We are thusapproximating the steady-state blocking probability in the finite-capacity system by the steady-state probability that the buffer contentexceeds that capacity level in the infinite-capacity system at an arrivalepoch.

Hence, we let W be the steady-state waiting time until beginningservice and we focus on the tail probabilities P(W > x). It turns outthat in considerable generality these waiting-time tail probabilities areasymptotically exponential, i.e.,

P(W > x) ∼ αe − ηx as x → ∞ , (1.1)

where η is a positive constant called the asymptotic decay rate, α is apositive constant called the asymptotic constant, and f (x) ∼ g(x) asx → ∞ means that f (x)/ g(x) → 1 as x → ∞; see Abate, Choudhuryand Whitt [2]. In standard single-source examples the approximationprovided by (1.1) is often remarkably accurate [2]–[4]. Moreover,numerical experience indicates that for bursty sources theapproximation provided by (1.1) with α replaced by 1 often tends tobe conservative, i.e.,

P(W > x) ≤ e − ηx for all x . (1.2)

This is a basis for the simple one-parameter approximation

P(W > x) ∼∼ e − ηx (1.3)

The asymptotic decay rate η in (1.1)–(1.3) is the basis for the conceptof effective bandwidths. The simple one-parameter approximation(1.3) is appealing because the key parameter η in (1.1)–(1.3) isrelatively easy to determine, exactly or approximately. Indeed, it istypically much easier to obtain than the asymptotic constant α.Moreover, the resulting admission-control algorithm using effectivebandwidths based on (1.3) is remarkably simple. Hence, we call (1.3)the effective-bandwidth approximation.

It is important to note that much of the effective-bandwidthliterature has not arrived at (1.3) via (1.1). The large-deviation resultsupporting (1.3) is the weaker limit

x − 1 log P(W > x) → − η as x → ∞ . (1.4)

General sufficient conditions for (1.4) are given in Chang [10] andGlynn and Whitt [27]. Clearly (1.1) implies (1.4), but not conversely.Indeed, the weaker limit (1.4) yields no information about theasymptotic constant α in (1.1).

We have indicated that most of the effective-bandwidth literaturehas arrived at approximation (1.3) via the limit (1.4) or relatedbounds such as (1.2). However, several people have focused on (1.1)too, namely, Abate et al. [2], Baiocchi [7], Elwalid and Mitra [23,24]and Neuts [38]. Given (1.1), (1.3) is a convenient simpleapproximation because α is much harder to obtain than η, and

because (1.3) is consistent with the notion of effective bandwidths.

As discussed in [3], for standard single-source modelsapproximation (1.3) is often a reasonable substitute for approximation(1.1)–(1.4) when we are primarily interested in percentiles of thedistribution, rather than the probabilities themselves. If theasymptotic constant α in (1.1) is not too different from 1, then it playsa relatively small role in higher percentiles.

It should be noted that the asymptotic decay rate η in (1.1) isidentical to the asymptotic decay rate for the blocking probability inthe corresponding finite-buffer model as the buffer size gets large; seeBaiocchi [7]. Hence, our analysis of the infinite-capacity model isdirectly relevant to the finite-capacity model with large buffer sizes.

In order to do better than (1.1), without calculating the exact tailprobabilities, we have also developed a refined three-termapproximation of the form

P(W > x) ∼∼ α 1 e − η1 x + α 2 e − η2 x + α 3 e − η3 x , (1.5)

where α 1 and η 1 are the asymptotic constant and asymptotic decayrate in (1.1), while α 2 , α 3 η 2 and η 3 are chosen to match theprobability of delay P(W > 0 ) and the first three moments EW,E(W 2 ) and E(W 3 ), with η 2 and η 3 required to satisfyη 1 ≤ min {η 2 ,η 3 }; see Choudhury, Lucantoni and Whitt [17].

We thus have four ways to ‘‘calculate’’ the tail probabilityP(W > x) : (1.3), (1.1), (1.5) and the full numerical algorithm. Eachsuccessive method tends to be more accurate, but each successivemethod requires substantial more computational effort. Theeffective-bandwidth approximation (1.3) is by far the easiest tocompute, with there being virtually no limit to the size of the model;e.g., see [45].

Using our exact numerical algorithm for the MAP/G/1 queue, wehave investigated the three approximations in (1.3), (1.1) and (1.5).For many standard single-source models, the refinement in (1.1) isremarkably accurate even for x not too large, e.g., for the 90th

percentile of the distribution. For standard single-source models forwhich (1.1) is excellent at the 90th percentile and beyond, (1.5)typically is excellent for the entire distribution.

However, we have found that the story changes dramaticallywhen the arrival process is the superposition of many componentprocesses. Then the effective bandwidth approximation (1.3) canperform very badly. It is even possible for (1.1) and (1.5) to performbadly, but there is a substantial region where (1.3) performs badlyand (1.1) performs well. This is the basis for our new improvedapproximations beyond (1.3).

We emphasize that, for the model being considered, the limits in(1.1) and (1.4) do indeed hold, and we are indeed interested in verysmall tail probabilities, and relatively large x. However, there is aserious difficulty (evidently pointed out for the first time here): Theasymptotic constant α in (1.1) can be very different from 1 when thenumber of component arrival processes is large.

We also explain why the asymptotic constant α can be verydifferent from 1 with many sources. To do so, we consider the caseof n identical sources with fixed total rate. As n increases, the totalrate is kept fixed by properly scaling the individual streams. Withthis structure, it is known that the asymptotic decay rate η in (1.1) isactually independent of n. We give numerical examples showing thatthe asymptotic constant with n sources (and this scaling), α n , is itselfasymptotically exponential in n, i.e.,

α n ∼ βe − nγ as n → ∞ , (1.6)

where β > 0 and, for sources more bursty than Poisson, γ ≥ 0, while

for sources less bursty than Poisson, γ ≤ 0. Moreover, γ in (1.6)tends to be larger when the burstiness gets further from Poisson andthe traffic intensity decreases, which is a likely operating conditionfor ATM networks.

Combining (1.1) and (1.6), we obtain the refined asymptoticapproximation (with the scaling for (1.6))

P(W > x) ∼∼ βe − nγ e − ηx . (1.7)

Our numerical results supporting (1.6) and (1.7) naturally suggesta limit theorem, on which we were working but others havedelivered. The idea is still to consider large-deviation asymptotics,but now letting both n and x go to infinity. Since this paper wasoriginally submitted, large deviation limits supporting (1.6) and (1.7)have been reported by several authors, the first known to us being A.Dembo and O. Zeitouni (talk at National Meeting of the OperationsResearch Society of America, Boston, April 1994). Large-deviationpapers supporting (1.6) and (1.7) have been written by Botvich andDuffield [9], Courcoubetis and Weber [21], Simonian [40] and Tse,Gallager and Tsitsiklis [44]. To state their result, let W n be thesteady-state waiting time with n sources. Paralleling (1.4), theirresult is

n − 1 logP(W n > nx) = I(x) as n → ∞ , (1.8)

for an appropriate function I(x), yielding the approximation

P(W n > x) ∼∼ e − nI(x / n) . (1.9)

Clearly (1.8) provides strong theoretical support for (1.6) and (1.7),but (1.8) is substantially weaker than (1.1) and does not yield theasymptotic constant β in (1.6) and (1.7). Nevertheless, the form ofthe limit function I(x) provides useful insights, as can be seen fromthe references.

We propose (1.7) as a basis for developing useful approximationsfor the tail probabilities when there are large numbers of each of afew types of sources. In particular, we suggest using (1.7) to estimateβ and γ by extrapolating using (1.6) and exact calculations for smalln; see Section 7 for further discussion.

Approximation (1.7) can be regarded as an approximation to thetrue asymptote (1.1). Our numerical experience indicates that forsources more bursty than Poisson the true asymptote (1.1) is a goodapproximation at 10 − 9 if α is not too small, e.g., α ≥ 10 − 4 . For awide range of problems, this is the region of interest. Our experiencealso indicates that the true asymptote (1.1) is a poor approximation at10 − 9 if α is too small, e.g., α ≤ 10 − 8 . For estimated values of α inbetween, e.g., 10 − 8 < α < 10 − 4 , the true asymptote is onlymoderately good as an approximation. In this region, we proposeusing the true asymptote with α replaced by 10 − 4 as a roughconservative heuristic approximation. For sources less bursty thanPoisson, (1.1) tends to be a good (bad) approximation at a targetblocking probability 10 − 9 if α < 104 (α > 108 ).

Here is how the rest of the paper is organized. In Section 2 weconsider a specific example with bursty sources to demonstrate thatthe effective-bandwidth approximation (1.3) can perform poorly. InSection 3 we consider the case of n identical sources scaled so thatthe total rate is independent of n, and give examples supporting (1.6).Next in Section 4 we present modifications of the example inSection 2 with different numbers of sources in which the twoasymptotic approximations in (1.1) and (1.3) are, first, both good and,second, both bad. These cases seem identifiable by looking at theobserved value of the asymptotic constant α. In Section 5 weinvestigate the impact of changing other variables. We show that theasymptotic approximations tend to get worse as the number ofsources increases, the buffer size decreases, the channel utilization

decreases, the target blocking probability increases, and the sourcegets further from Poisson, either more bursty or less bursty.

In Section 6 we consider sources less variable than Poisson. Forthese sources, we show that the effective-bandwidth approximationneed not be conservative; i.e., (1.2) does not hold. In Section 7 weshow how to do approximations in large systems with heterogeneoussources. In Section 8 we briefly describe the computationalalgorithm. Finally, in Section 9 we state our conclusions.

II. AN EXAMPLE WHERE THE EFFECTIVE-BANDWIDTHAPPROXIMATION FAILS

In this section we consider an example with homogeneous on-offsources. (The on-off property is not essential for our model or for theconclusions we draw from it. A source can have multiple states, eachcharacterized by a different arrival rate. We use such sources in laterexamples.) Our primary purpose is to show that the effective-bandwidth approximation (1.3) can be a bad approximation for arealistic example and can lead to seriously underestimating thenumber of sources the queue can accommodate.

We let the service times be i.i.d. with mean 1. Thus the unit oftime is the time required to transmit one ATM cell. We let theservice time have an Erlang distribution of order 16 (E 16 ), which hassquared coefficient of variation (SCV, variance divided by the squareof the mean) cs

2 = 1/16. This distribution is similar to thedeterministic (D) distribution of ATM cells, but better behaved fornumerical calculations. (It is possible to do calculations for Ddistributions, but it requires more computational effort to achievecomparable precision.) In fact, we have computed using the E 1024

distribution and have found for the examples in this paper withMMPP sources that there is little difference between E 16 and E 1024 ,supporting our hypothesis that D is well approximated by E 16 .

The on-off source has exponentially distributed on and offperiods. During the on periods, cells arrive according to a Poissonprocess; during the off periods there are no arrivals. Each on-offsource is characterized by three parameters, the mean on period ω,the mean off period ζ and the (peak) rate during the on period, p.Here we let the on and off periods have means ω = 436. 36 andζ = 4363. 63, respectively. We let the peak rate be p = 0. 1375.The overall average rate is thus λ = pω/(ω + ζ) = 0. 0125. Theratio of peak to mean rates is thus p /λ = 11. 0. The mean number ofcells during an on period is pω = 60. 0.

We feel that the level of source burstiness considered above canhappen quite naturally in bursty data sources. Furthermore, theexample is chosen mainly to illustrate that bad behavior can occur.Later we discuss how the basic behavior of superposed sourceschanges with change in burstiness, buffer size and other parameters.

The above on-off source (an interrupted Poisson process) can alsobe characterized as a renewal process with a hyperexponential (H 2 ,mixture of two exponentials) interarrival time distribution [34]. Themean interarrival time for one source is thus 1/λ = 80 and the SCVis ca

2 = 100. 2. Scaled to have mean 1, the first four moments of aninterarrival time are: m 1 = 1, m 2 = 101. 2, m 3 = 1. 68 × 104 andm 4 = 3. 73 × 106 . This is another way to see that the individualsources are quite bursty. (For a simple exponential random variablewith mean 1, the n th moment is n!)

Since the arrival process is a renewal process, The SCV coincideswith the normalized asymptotic variance or limiting index ofdispersion for counts for the counting process N(t), i.e.,

I c (∞) ≡t→ ∞lim

EN(t)Var N(t)_ _______ = ca

2 = 100. 2 , (2.1)

see [25,43]. The value I c (∞) = 100. 2 is large, so that each source isindeed quite bursty.

We let the buffer size be 600 cells, which falls in the rangeconsidered by switch manufacturers. The buffer size 600 is ten timesthe mean number of cells in an on period; i.e., the buffer contains 10bursts. The quality of the approximations depends strongly on thebuffer size. (We discuss this further in Section 5.) Currently buffersize is limited by the availability of fast memory. We regard 600 as arepresentative moderate buffer size.

As is customary in ATM, we let the target blocking probability be10 − 9 . Hence, we choose the number of sources so that

P(W > 600 ) ∼∼ 10 − 9 . (2.2)

Our exact numerical results indicate that (2.2) is attained with n = 24sources, yielding a utilization of 30%.

Figure 1 displays the exact tail probabilities P(W > x) and theapproximations (1.1), (1.3) and (1.5). In addition, Figure 1 displaysthe Poisson approximation, which is the exact tail probability in theM/E 16/1 queue, computed by numerical transform inversion. The tailprobability P(W > x) is our approximate probability of bufferoverflow with x being the buffer size.

________________________Insert Figure 1 about here.

________________________

The (exact) asymptotic parameters in (1.1) are η = 0. 01809 andα = 1. 453 × 10 − 5 . The three-term approximation is also fullydetermined, based on the exact probability of delayP(W > 0 ) = 0. 4242 and first three moments EW = 0. 6924,E(W 2 ) = 8. 573 and E(W 3 ) = 579. 3.

Of course, there is the question of numerical accuracy of the exactresults reported in Figure 1. We verified them using a built-inaccuracy check in our numerical procedures, as explained in [15, 37]and Section 8 here. In the related case of 64 sources with exponentialservice times, numerical results by Elwalid and Mitra [24] confirmedour results.

Figure 1 indicates that (1.1) and (1.5) are still in error by a factorof 3.9 at x = 600. This error is relatively small, though, compared tothe error in (1.3), which is by a factor of 105 . The Poissonapproximation obviously is even worse than (1.3).

It is interesting to consider Figure 1 in relation to previousdiscussions of ‘‘cell-level congestion’’ and ‘‘burst-level congestion,’’e.g., in [39]. To a large extent, Figure 1 indicates two nearly linearregions for the exact curve. As discussed on p. 19 of [39], there is aninitial period of steep nearly linear decline corresponding to ‘‘cell-level congestion’’ and a later period of more gradual nearly lineardecline corresponding to ‘‘burst-level congestion.’’ This burst levelcongestion corresponds to the true asymptote (1.1). However, thetransition between these two regimes is smooth and fairly long. As in[39], much of the ATM literature considers separate models torepresent cell-level and burst-level congestion. In contrast, werepresent both within a single model. Indeed, we consider preciselythe multiple-MMPP-source model described as difficult on p. 185 of[39]. In this context, a major contribution here is to point out that theintercept with the y-axis of the asymptote, which is the asymptoticconstant α in (1.1), may well be very small.

Now suppose, instead, that we use approximation (1.3). Let η n

be η in (1.1) as a function of n (without any rescaling of individual

sources). As indicated above, η 24 = 0. 01809, but η increases as n(and ρ) decreases. Figure 2 displays η n as a function of n (withoutany rescaling of individual sources). It turns out that

e − 600ηn ≤ 10 − 9 for all n ≤ 12 , (2.3)

but not for n ≥ 13; η 12 = 0. 03749 and η 13 = 0. 03354. Hence,using the effective-bandwidth approximation (1.3), we conclude thatthe queue can accommodate only 12 sources with criterion (2.2).Hence, (1.3) indeed significantly underestimates the capacity. Theactual capacity is two times that predicted by (1.3).


________________________

Table 1 compares seven different procedures for determining thenumber of sources that can be supported:

(i) The exact tail probabilities

(ii) the effective-bandwidth approximation (1.3)

(iii) the full asymptotic approximation (1.1)

(iv) the three-term approximation (1.5)

(v) the Poisson approximation

(vi) average-rate engineering

(vii) peak-rate engineering.

________________________Insert Table 1 about here.

________________________

From Figure 1 and Table 1, we see that even though the trueasymptote is not too accurate for the tail probability (being off by afactor of 4), it produces a good estimate for the number of sources(being off by only one).

Since the average rate is λ = 0. 0125, if we could size by averagerate, then the queue could accommodate 80 sources. Since the peakrate is p = 0. 1375, if we sized by peak rate, then the queue canaccommodate 7 sources. The numbers 7 and 80 help put the realizedimprovement from 12 to 24 going from approximation (1.3) to theexact numerical result in perspective.

There is a simple explanation for the poor performance ofapproximation (1.3). For n = 24, the asymptotic constant in (1.1) isα 24 = 1. 453 × 10 − 5 . As can be seen from Figure 1, in this case theone-term asymptotic approximation αe − ηx based on (1.1) is a goodapproximation, but replacing α by 1 introduces a large error.

In customary models with a single source, the asymptotic constantα is usually not too different from 1. Hence, it is interesting to seehow α n depends on n (again, without rescaling the individualsources). This is shown in Figure 3. There we see that α n declinesfrom α 1 = 0. 340 to a minimum value of α 11 = 6 × 10 − 8 and thenincreases toward 1 again as the traffic intensity approaches 1. Fromheavy-traffic theory, we anticipate that α n → 1 as ρ n → 1.Assuming that α n stays well above the target value 10 − 9 for all n, asit does in Figure 3, we can anticipate that the asymptote (1.1) will bea reasonably good approximation, but we clearly cannot simplyreplace the asymptotic constant α by 1.


________________________

Instead of fixing the buffer size at 600 and asking how manysources the queue can accommodate, we could instead fix the number

of sources at 24 and ask what size buffer is required. The exactnumerical results indicate a buffer size of 600. The approximations(1.1) and (1.5) indicate that a buffer size of 530 is required,approximation (1.3) indicates 1146 and the Poisson approximationindicates only about 20. These results are also displayed in Table 1.This is another way to look at the weakness of the effective-bandwidth and Poisson approximations. This view also shows thatapproximations (1.1) and (1.5) are reasonably good.

As shown in [4], the asymptotics (1.1) for the steady-state waitingtime are closely related to corresponding asymptotics for othersteady-state quantities, such as the workload, sojourn time and queuelength (at arrivals and at arbitrary times). For this example, the queuelength decay rate parameter σ in [4] is σ = 0. 9821. Although weconsider only the waiting time here, our experience indicates that adetailed analysis of one of the other steady-state distributions tellsessentially the same story. For the example considered in thissection, if we look at the queue length at an arbitrary time (instead ofat the arrival instant) then both the exact value and approximation(1.1) drop roughly by a factor of 3, but the effective-bandwidthapproximation (1.3) remains unchanged (another weakness of (1.3)).However this further difference by a factor of 3 is small compared tothe already-established difference by a factor of 105 .

III. ASYMPTOTICS FOR THE ASYMPTOTIC CONSTANT INSCALED SUPERPOSITION PROCESSES

As shown in [10], [18], [23], [24], [27], [45], the asymptotic

decay rate η n in thei = 1Σn

G i/G/1 model with the superposition of n

i.i.d. arrival processes is independent of n when we fix the totalarrival rate by scaling the component arrival processes. In particular,let {N(t) : t ≥ 0 } be the arrival counting process with only onesource. The proper scaling is achieved with n sources by letting eachcomponent arrival process be distributed as {N(t / n) : t ≥ 0 }. If thearrival rate of N(t) is λ, then the arrival rate of N(t / n) is λ / n, so thatthe total arrival rate with n sources is λ for all n. (This scaling is alsodiscussed previously in [43] and 25].)

Of course, when we add sources in a real network, we do not doany rescaling, so we did not do any rescaling in Section 2. However,since η n is independent of n with this rescaling, the rescaling helps usunderstand what is happening with the various approximations. Asdiscussed in [43], a key theoretical reference point is the fact that,with the scaling, the superposition process approaches a Poissonprocess as n → ∞. Since η n does not change with n, we see that thetwo limits x → ∞ and n → ∞ do not interchange. This is a source ofour difficulties.

Our numerical experience indicates that with this rescaling asn → ∞ the asymptotic constant α n in (1.1) approaches 0 for sourcesmore bursty than Poisson and approaches infinity for sources lessbursty than Poisson. More precisely, α n appears to decay or growexponentially as in (1.6) as n → ∞. Numerical evidence supporting(1.6) is given in Figures 4 and 5 for sources more bursty thanPoisson. There we display α n as a function of n in log scale fordifferent examples. Figure 4 displays α n as a function of n for theexample in Section 2, while Figure 5 displays α n as a function of n infour other examples. For Figure 4, the reference case is the case withn = 24 sources and ρ = 0. 30. All other cases in Figure 4 arerescaled to have this same ρ. In each example we rescale the arrivalprocesses as n changes, so that η n is independent of n. The linearityin Figures 4 and 5 for n not too small provides strong support for(1.6). (Figure 12 in Section 6 provides similar support for sourcesless bursty than Poisson.)

________________________Insert Figures 4 and 5 about here.

________________________

The four examples depicted in Figure 5 represent all combinationsof two different traffic intensities, ρ = 0. 5 and ρ = 0. 8, and twoMMPP arrival processes. The service-time distributions always areE 16 with mean 1. As before, these represent nearly deterministicservice times. The two arrival processes are two-state MMPPs withasymptotic variance constants in (2.1) of 2.8 and 28. These representmoderately bursty and substantially more bursty sources, respectively(but less bursty than the example in Section 2). These examples havepositive arrival rates in both environment states, so that thecomponent MMPPs are not renewal processes.

In particular, the MMPPs are characterized by four parameters,one of which can be taken to be the arrival rate. The ratio of thearrival rates in the two environment states is fixed at 4. The expectednumbers of arrivals during visits to the two environment states areequal, 5 in the less bursty example with I c (∞) = 2. 8 and 75 in themore bursty example with I c (∞) = 28.

From Figure 5, we see that the key decay rate parameter γ in (1.6)is decreasing in the traffic intensity ρ but increasing in the burstiness.This appears to be a general tendency. This means that thephenomenon in Section 2 is most likely to occur with low ρ and highburstiness, which is what we anticipate for ATM. The phenomenonmight have been missed by others, because they focused on higher ρand less bursty sources. For instance, the examples in Section 5 ofElwalid and Mitra [23] all have high ρ (above 0.75) and lowerburstiness, so that α > 10 − 2 .

IV. THREE REGIMES FOR THE ASYMPTOTIC APPROXIMATIONS

The example in Section 2 was one for which the asymptoticapproximation αe − ηx in (1.1) is pretty good, but the simpleapproximation e − ηx in (1.3) is bad (because α ∼∼ 10 − 5). In thissection we present modifications of this example in which, first, bothapproximations (1.1) and (1.3) are good and, second, bothapproximations (1.1) and (1.3) are bad.

To obtain these alternative cases, we simply modify the numberof sources in the example with n = 24 in Section 2, scaling thearrival processes as in Section 3 to keep the total arrival rate fixed atλ = 0. 3 and η fixed at η = 0. 01809. In particular, the twoalternative regimes are obtained by letting n = 2 and n = 60. Theresults are displayed in Figures 6 and 7.

Since η is independent of n, approximation (1.3) is the same forthe three cases n = 2, n = 24 and n = 60. However, α 2 = 0. 505,α 24 = 1. 45 × 10 − 5 and α 60 = 2. 22 × 10 − 12 . (For n = 1,α 1 = 0. 992. ) From Figures 1, 6 and 7, it is evident that we have thethree regimes as claimed. In Figure 6 with n = 2, approximations(1.1), (1.3) and (1.5) are all very close, while in Figure 7 it is evidentthat approximations (1.1), (1.3) and (1.5) are very far apart.


________________________

Since α 60∼∼ 10 − 12 < 10 − 9 when n = 60, it should come as little

surprise that the asymptotics (1.1) have not kicked in by the time thetail probabilities reach 10 − 9 in this case. As a rough rule of thumb, itappears that approximation (1.1) tends to be good only when α n isgreater than the desired tail probability P(W > x). To a large extent,it appears that we are able to understand when the variousapproximations will be sufficiently accurate by computing only theasymptotic constant α. (This computation is possible, even for very

large n, by computing α n for small values of n and then extrapolatingusing (1.6).)

From Figure 7, we also see that the three-term approximation(1.5) is not accurate at 10 − 9 . The performance of approximation(1.5) in Figure 7 with n = 60 is much worse than in Figure 1 withn = 24. It is reassuring that its poor performance is signalled by thefact that it is not possible to find parameters exactly matching thethird moment in this case; see [17]. Moreover, the three-termapproximation performs pretty well even in Figure 7 for tailprobabilities above 10 − 4 , for which it was originally designed. Thepoor performance at 10 − 9 in Figure 7 suggests that alternative fittingprocedures should be considered for extremely small probabilitiessuch as 10 − 9 . (This is being investigated.)

V. CHANGING OTHER PARAMETERS

In Section 4 we saw what happens as we changed the number ofsources, scaling them in the manner of Section 3 so that the totalarrival rate remains unchanged. We saw that the quality of theasymptotic approximations decline as n increases.

More generally, we conclude that the asymptotic approximationstend to get worse as the number of sources increases, the buffer sizedecreases, the channel utilization decreases, the target blockingprobability increases, and the source gets further from Poisson.However, we would also like to point out that there is significantdifference in the accuracy and the region of applicability of theeffective bandwidth asymptotic (1.3) and the other asymptoticapproximations (1.1) and (1.5). Roughly speaking, (1.3) is not badwhen 10 − 2 < α < 102 , whereas (1.1) and (1.5) are not bad when10 − 8 < α < 108 . We expect that in practical engineering situationswith multiple sources (1.1) and (1.5) will typically be a reasonablygood approximation, while (1.3) may be quite bad.

In this section we consider the effects of buffer size andburstiness. Figures 8 and 9 compare the approximations with exactvalues when the buffer size is 6000 and 60, respectively, instead of600 as in Section 2. Here we keep the number of sources fixed atn = 24. We scale the sources in the manner of Section 3 until theblocking probability at the indicated capacity is 10 − 9 .


________________________

So far, we have focused on the steady-state waiting time at arrivalepoch. Similar results hold for the steady-state workload at anarbitrary time (the virtual waiting time). To demonstrate this, theremaining numerical results in this paper, including those in Figures 8and 9, are for the steady-state workload. (The differences betweenthe waiting time and workload are negligible compared to the mainphenomena being discussed.)

When the buffer size is increased to 6000, the buffer holds 100bursts instead of 10. Figure 8 shows that all the approximationsperform very well with this larger buffer, just as in Figure 6. Nowthe utilization is 0.835. The effective-bandwidth approximationworks very well; it predicts that the system can support 23 sources,which is within 1 of the exact value.

In contrast, Figure 9 shows that the approximations get evenworse when we decrease the buffer size from 600 to 60, whichcorresponds to just one burst. Figure 9 parallels Figure 7. Now theutilization is only 0.18. The effective-bandwidth approximationwould now only admit 10 sources.

Suppose now we keep the buffer size at the high value of 6000and the number of sources at n = 24, but increase the burstiness.Suppose that we increase the burstiness by multiplying the meannumber of bursts in an on period by 5. We keep the arrival rate fixedby multiplying the ratio of the off period to the on period by 5. Thismakes the mean number of arrivals in an on period 300, which meansthat the buffer now holds 6000/300 = 20 bursts.

Figure 10 compares the approximations with exact values in thiscase. Figure 10 shows that there is once again a big differencebetween the effective-bandwidth approximation and the exact result,just as in Figure 1. For this example, the effective-bandwidthapproximation would admit only 11 sources. Hence, the advantage ofthe larger buffer of size 6000 is offset by the larger burstiness. Theactual performance of the approximations obviously depends on thecombination of variables that actually prevails.


________________________

VI. SOURCES LESS BURSTY THAN POISSON

In this section we consider sources that are less bursty thanPoisson. In particular, we assume that each source is renewal withinterarrival times that are E 2 . As before, we assume that the overallarrival process is the superposition of independent versions of thesingle source process. It seems unlikely that the sources in an ATMnetwork will actually be less bursty than Poisson, but this is apossibility, due to traffic shaping at the network edge.

In the earlier bursty model we approximated the deterministiccell-length distribution by an E 16 distribution and commented thatE 16 and D give about the same result. However, this is no longer truewith less bursty arrival processes. Therefore, in order to remainpretty close to D, in this section we assume an E 1024 service timedistribution.

Figure 11 displays the approximations and exact tail probabilitiesfor this case assuming 24 sources and a buffer size of 8. In order tomeet the 10 − 9 buffer overflow probability, the channel utilization hasto be 29%. In this case the buffer overflow probability is greatlyunderestimated by the effective-bandwidth approximation (1.3). Forthis case of less bursty sources, the effective-bandwidthapproximation (1.3) would admit 39 sources in order to meet thebuffer overflow requirement with a buffer of size 8, instead of theproper number of 24.


________________________

Figure 11 also shows the probability of buffer overflow forPoisson arrivals. For large tail probabilities, the exact result is closeto the Poisson prediction and, for small tail probabilities, the exactresult approaches the true asymptote. It is also interesting to observethat the true asymptote is about ten orders of magnitude higher thanthe effective bandwidth approximation. This is in sharp contrast withthe earlier highly bursty examples where the true asymptote wasalways smaller than the effective bandwidth approximation. Also,note that there is a qualitative change in the shape of the exact curve.It is concave in Figure 11 as opposed to being convex for the morebursty sources.

The behavior of the tail probabilities in Figure 11 may beunderstood by plotting, in log scale, the asymptotic constant α n as afunction of n, the number of sources. This we do in Figure 12 for theE 2 sources at channel utilizations 0.3 and 0.7 respectively. This is

similar to what we did in Figure 5, and indeed Figures 5 and 12 looksimilar in the sense that in both cases the logarithm of α n isasymptotically linear with n, i.e., α n changes exponentially with n.However, the striking difference is that with more bursty sources(Figure 5) α n decays exponentially with n and approaches zero, whilein the less bursty E 2 case, α n grows exponentially with n andapproaches infinity. The growth rate increases as the channelutilization decreases. We have also plotted how the α n changes withn for Poisson sources with channel utilizations of 0.3 and 0.7respectively. Of course, α n does not change with n in the Poissoncase. The Poisson case places in perspective the spectacular growthrate of α n with n in the non-Poisson case.


________________________

As before, the effective-bandwidth approximation (1.3) performsbetter with larger buffer sizes. This is shown in Figure 13 where thebuffer size is 100 instead of 8. As before, the number of sources isn = 24. The sources have been scaled in the manner of Section 3 sothat P(W > 100 ) = 10 − 9 . This drives the channel utilization up to0.949. Figure 13 shows that all the approximations are close to theexact values in this case.


________________________

VII. APPROXIMATIONS FOR LARGE SYSTEMS WITHHETEROGENEOUS SOURCES

In this section we propose a method for obtaining usefulapproximations for tail probabilities in large systems.Approximations are needed, because our exact algorithm cannothandle large numbers of sources, since the number of phases of thesuperposed Markovian Arrival Process (MAP) grows rapidly with thenumber of sources. Specifically, it can be shown that if there are Ltypes of sources, k i sources of type i, and each source of type i has m i

phases, then the total number, P, of phases of the superposed MAP isgiven by

P =i = 1Π

L m i − 1k i + m i − 1

. (7.1)

Note that for Poisson sources m i = 1 and m i − 1k i + m i − 1

= 1, so that

we can add any number of them without increasing P. To run theexact model in reasonable time, we need P to be at most about 100.In all the numerical examples in this paper we assumed L = 1 andm i = 2 so that we could treat up to about K 1 = 99 sources. (Weactually considered up to 60 sources). However, with L = 2 and 3(with m i still fixed at 2) we can treat only up to K i = 9 and K i = 3sources of each type.

Now we specify our approximation procedure in terms of twoexamples. First, suppose that there are three classes, each with 100homogeneous sources. We can calculate the asymptotic decay rate ηexactly by considering the three-class system with one source in eachclass, with each source scaled appropriately, as in Section 3, so thatthe arrival rate of each single source equals the total arrival rate forall 100 sources, for each class, in the original system.

In order to approximately determine the asymptotic constant α inthe original system, we calculate the exact asymptotic constants in thesystems with three classes and k sources in each class, again scaled to

be consistent with the original system according to Section 3, forseveral feasible k, e.g., k = 1 , 2 and 3. (Note that here L = 3 andm i = 2, so that k i = 3 for all i is feasible.) Then, assuming (1.6),we obtain an approximate asymptotic constant for the original systemby fitting β and γ in (1.6) to the data. If the resulting estimatedasymptotic constant α is substantially greater than the target tailprobability, then we can apply approximation (1.1) with confidence.If the estimated asymptotic constant α is less than the target tailprobability, then we note that (1.1) is probably not appropriate. If theestimated asymptotic constant is greater than the target tailprobability, but not much greater, then we might use the heuristicsuggested in Section 1; i.e., we might use (1.1) with a higher value ofα as a rough conservative estimate.

The approach we have just described is satisfactory if all classeshave many sources, as when there are 100 sources from each of thethree classes. However, in actual applications, e.g., with videosources, there may be only a few sources from some classes. Toillustrate, suppose that we have four classes, with 1 source in the firstclass, 2 sources in the second class and 100 sources each in the lasttwo classes. Let all sources have two states. The method we havejust applied does not work for this example, but a modification does.

We have found that the asymptotic relation (1.6) still holds if wedivide the sources into two groups, and hold one group of sourcesfixed, while we multiply the number of sources in the second groupby n. As before, we scale the sources in the second group, so that thetotal arrival rate for each class remains fixed, independent of n.

As before, (1.1) holds for each n and the asymptotic decay rate ηis independent of n. Moreover, numerical experience indicates thatthe asymptotic relation (1.6) still holds, but now β in (1.6) is afunction of the fixed sources. Now to estimate α in the originalsystem we at first estimate β and γ in (1.6) using n = 2 and 3 andthen estimate α from those using n = 100. Note that here L = 4,m i = 2 for each i, k 1 = 1, k 2 = 2, k 3 = k 4 = n and hence n = 2and 3 are feasible for the exact model.

More generally, the asymptotic result just described suggests anapproximation for m classes with multiplicities n 1 ,n 2 , . . . , n m of theform

α n1 , . . . , nm∼∼ βe − (γ 1 n1 + . . . + γm nm ) . (7.2)

For multiple classes, approximation (7.2) is convenient because wecan determine the asymptotic decay rates γ i by changing one class ata time. However, more work is needed on this.

VIII. THE ALGORITHM

In this section we describe our algorithm for numericallycomputing the exact tail probabilities. To compute the tailprobabilities, we draw heavily on Lucantoni [35]; see [36] for areview. We model each source as a Markovian Arrival Process(MAP), which is a two-dimensional Markov process {N(t) , J(t) } onthe state space { (i , j) :i ≥ 0 , 1 ≤ j ≤ m} with an infinitesimalgenerator having the structure

Q =

...

0

0

D 0

...

0

D 0

D 1

...

D 0

D 1

0

...

D 1

0

0

. . .

. . .

. . .

(8.1)

where 0 , D 0 and D 1 in (8.1) are m×m matrices; D 0 has negativediagonal elements and nonnegative off-diagonal elements; andD ≡ D 0 + D 1 is an irreducible infinitesimal generator. We alsoassume that D ≠ D 0 , which ensures that arrivals will occur. The

variable N(t) counts the number of arrivals in the interval [ 0 ,t],while J(t) gives the arrival ‘‘phase’’ at time t.

In this paper we consider the superposition of many sources. Ifthe individual sources are characterized by matrices D 0i and D 1i fori = 1 , 2 , . . . , K, where K is the number of sources, then it can beshown [35] that the superposed system is also a MAP characterizedby matrices

D 0 = D 01 ⊕ D 02 ⊕ . . . ⊕ D 0K (8.2)

D 1 = D 11 ⊕ D 12 ⊕ . . . ⊕ D 1K (8.3)

where ⊕ is the Kronecker sum. We do all computations on thesuperposed MAP characterized by D 0 and D 1 .

As explained in [35], the Laplace-Stieltjes transform (LST) of thesteady-state waiting-time cumulative distribution function (CDF) isgiven by

W(s) = (πd) − 1 s( 1 − ρ) g[sI + D( h(s) ) ] − 1 e , (8.4)

where e is a vector of all 1’s, h(s) is service time LST, ρ is the trafficintensity (the mean arrival rate times the mean service time),d = D 1 e, π is the stationary probability vector of the Markov processwith generator D (i.e., π satisfies πD = 0 and πe = 1), g satisfies theequations gG = g and ge = 1, and G satisfies the equation

G =0∫∞

e (D0 + D1 G) xdH(x) , (8.5)

where H(x) is the service-time CDF and

D( h(s) ) = D 0 + D 1 h(s) . (8.6)

Our computation is based on, first, fast and accurate computationof W(s) using (8.4) and, second, fast and accurate transforminversion to get the waiting time CDF from W(s). To do theinversion, we use the Fourier-series method in [5] along with theround-off error control procedure in [15]. The procedure in [15] alsoprovides a self-contained accuracy check by doing the calculationtwice with different round-off control parameters. This amounts tousing different contours for complex inversion integration. With thisprocedure, we achieve high accuracy even at the tail probability of10 − 9 .

Next we describe two techniques for greatly speeding up thecomputation of the LST W(s). First, note that all individual sourcesare two-phase sources. The Kronecker sum operations in (8.2) and(8.3) makes the superposed source a 2K-phase source. This wouldprevent us from using large K, but we can take advantage of the factthat the sources are homogeneous to greatly reduce thedimensionality of the superposed source. We can define the phase ofthe superposed source as one plus the number of component sourcesin phase 1. Therefore, the total number of phases is K + 1 instead of2K . The D 0 and D 1 matrices of the new superposed sources aregiven in terms of the component D 01 and D 11 matrices of the(identical) component sources as follows:

(D 1 ) i ,i = (i − 1 ) (D 11 ) 1 , 1 + (K − i + 1 ) (D 11 ) 2 , 2

for i = 1 , 2 , . . . , K + 1

(D 1 ) i ,i + 1 = (K − i + 1 ) (D 11 ) 2 , 1 for i = 1 , 2 , . . . , K

(D 1 ) i ,i − 1 = (i − 1 ) (D 11 ) 1 , 2 for i = 2 , 3 , . . . , K + 1 (8.7)

with (D 1 ) i , j = 0 for all other pairs (i , j),

(D 0 ) i ,i + 1 = (K − i + 1 ) (D 01 ) 2 , 1 for i = 1 , 2 , . . . , K

(D 0 ) i ,i − 1 = (i − 1 ) (D 01 ) 1 , 2 for i = 2 , 3 , . . . , K + 1 (8.8)

with (D 0 ) i , j = 0 for all other pairs (i , j) with i ≠ j, and

(D 0 ) i ,i = − [ (D 0 ) i ,i − 1 + (D 0 ) i ,i + 1

+ (D 1 ) i ,i + (D 1 ) i ,i − 1 + (D 1 ) i ,i + 1 ] . (8.9)

For the service-time distribution, we approximate thedeterministic distribution by an Erlangian distribution of orderk (E k ). A significant computational burden is the computation of Gfrom the matrix integral equation (8.5). The uniformizationprocedure recommended in [35] for general service-time distributionscan be significantly improved for Erlang distributions by noting thatfor the E k distribution with mean 1 (8.5) reduces to

G = [I − k − 1 (D 0 + D 1 G) ] − k . (8.10)

Equation (8.10) can easily be solved by successive substitution, i.e.,

G n + 1 = [I − k − 1 (D 0 + D 1 G n ) ] − k , (8.11)

where G 0 is chosen to be a stochastic matrix. Note that since wechoose k = 2m (specifically 24 for bursty sources and 210 for smoothsources), each iteration in (8.11) involves only a single matrixinversion and m matrix multiplications.

In order to compute the asymptotic parameters α and η in (1.1),we use two independent algorithms (that cross-check each other): thealgorithm in [2] and the moment-based procedure in [14] and [1].

IX. CONCLUSIONS

Our first main conclusion here is that the effective-bandwidthapproximation (1.3) can break down when there is a large number ofindependent sources. Approximation (1.3) tends to get worse as thenumber of sources increases, the channel utilization decreases, thebuffer size decreases and the source gets further from Poisson, eithermore bursty or less bursty.

If the sources are more bursty than Poisson, as is anticipated forATM networks, then the effective-bandwidth approximation (1.3) isconservative. When the approximation is bad, there may besubstantially more statistical multiplexing gain than approximation(1.3) predicts. On the other hand, if the sources are less bursty thanPoisson, then the effective-bandwidth approximation is no longerconservative, and may also be bad. In general, contrary to manystatements, the effective-bandwidth approximation need not beconservative.

Typically, the exact tail probabilities lie between the effective-bandwidth approximation (1.3) and the true asymptote (1.1). If thesetwo curves are very close to each other (differ by less than a factor of10), then (1.3) is usually reasonably accurate. If the two curves arefar from each other, but not too far (differ by less than 108), then(1.3) is bad, but (1.1) and (1.5) are reasonably accurate. Finally, ifthe two curves are extremely far apart (differ by more than 108), theneven (1.1) and (1.5) are not accurate. Our second main conclusion isthat, even though (1.3) may not be a good approximation, the trueasymptote (1.1) often is a good approximation. Moreover, fromcalculations of the asymptotic parameters α and η, we have a way toestimate whether or not the approximations will be good.

A reason for the degradation of the effective-bandwidthapproximation as the number n of sources increases is that theasymptotic constant α n in (1.1) is itself asymptotically exponential inn. For sources more bursty than Poisson, α n decrease to 0exponentially fast, as shown in Figures 4 and 5. For sources lessbursty than Poisson, α n increases to infinity exponentially in n, asshown in Figure 12. The Poisson case is the reference case, becauseα n there does not change with n. The asymptotically exponentialform for α n in (1.6) allows us to compute it approximately for

arbitrary n by extrapolating based on computed values for small n.For heterogeneous sources with different multiplicities, we exploit(7.1). Having a way to approximate the asymptotic constant α isimportant, not only because we can use it in approximations (1.1) and(1.5), but also because the value of α indicates whether or not theapproximations will be good.

Finally, in addition to gaining a better understanding of theeffective-bandwidth approximation (1.3), we have provided bases formore refined analysis tools via our exact MAP/G/1 numericalalgorithm, the refined approximation (1.5) [17] and the exponentialrelation for α n in (1.6). We have indicated how (1.6) and (7.1) canbe combined with the true asymptote (1.1) to get an approximationthat is almost as simple as the effective-bandwidth approximation(1.3), but is applicable in a substantially wider region.Acknowledgments. This work was done when David Lucantoniwas at AT&T Bell Laboratories. We thank our colleagues BharatDoshi, Ted Eckberg, Dave Houck and Pat Wirth for helpfulcomments.

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[18] G. L. Choudhury and W. Whitt. Heavy-traffic approximationsfor the asymptotic decay rate in BMAP/G/1 queues.Stochastic Models 10 (1994) 453-498.

[19] E. C, inlar. Superposition of point processes, pp. 549-606 inStochastic Point Processes: Statistical Analysis, Theory andApplications, ed. P. A. W. Lewis, Wiley, New York, 1972.

[20] C. Courcoubetis, G. Kesidis, A. Ridder, J. Walrand and R. R.Weber. Admission control and routing in ATM networksusing inferences from measured buffer occupancy. IEEETrans. Commun., to appear.

[21] C. Courcoubetis and R. R. Weber. Buffer overflowasymptotics for a switch handling many traffic sources,University of Cambridge, 1994.

[22] A. Dembo and O. Zeitouni. presentation at the nationalmeeting of the Operations Research Society of America,Boston, April 1994.

[23] A. I. Elwalid and D. Mitra. Effective bandwidth of generalMarkovian traffic sources and admission control of high speednetworks, IEEE/ACM Trans. Networking 1 (1993) 329-343.

[24] A. I. Elwalid and D. Mitra. Markovian arrival and servicecommunication systems: spectral expansions, separability andKronecker-product forms, 1993.

[25] K. W. Fendick, V. R. Saksena and W. Whitt. Investigatingdependence in packet queues with the index of dispersion forwork. IEEE Trans. Commun., 39 (1991) 1231-1244.

[26] R. J. Gibbens and P. J. Hunt. Effective bandwidths for themulti-type UAS channel. Queueing Systems 9 (1991) 17-28.

[27] P. W. Glynn and W. Whitt. Logarithmic asymptotics forsteady-state tail probabilities in a single-server queue. Studiesin Applied Probability, Papers in Honour of Lajos Taka cs, J.Galambos and J. Gani (eds.), Applied Probability Trust,Sheffield, England, 1994, 131-156.

[28] R. Guerin, H. Ahmadi and M. Naghshineh. Equivalentcapacity and its application to bandwidth application in high-speed networks. IEEE J. Sel. Areas Commun. 9 (1991) 968-981.

[29] R. Guerin and L. G. .

un. A unified approach to bandwidthallocation and access control in fast packet-switchednetworks. INFOCOM ’92, Florence, Italy.

[30] H. Heffes and D. M. Lucantoni. A Markov modulatedcharacterization of packetized voice and data traffic andrelated statistical multiplexer performance. IEEE J. Sel.Areas Commun. SAC4 (1986) 856-868.

[31] J. Y. Hui. Resource allocation for broadband networks. IEEEJ. Select. Areas Commun. SAC-6 (1988) 1598-1608.

[32] F. P. Kelly. Effective bandwidths at multi-class queues.Queueing Systems 9 (1991) 5-16.

[33] G. Kesidis, J. Walrand and C.-S. Chang. Effectivebandwidths for multiclass Markov fluids and other ATMsources, IEEE/ACM Trans. Networking 1 (1993) 424-428.

[34] A. Kuczura. The interrupted Poisson process as an overflowprocess. Bell System Tech. J., 52 (1973) 437-448.

[35] D. M. Lucantoni. New results on the single server queue witha batch Markovian arrival process. Stochastic Models 7(1991) 1-46.

[36] D. M. Lucantoni. The BMAP/G/1 queue: a tutorial. Modelsand Techniques for Performance Evaluation of Computer andCommunication Systems, L. Donatiello and R. Nelson, eds.,Springer-Verlag, 1993, 330-358.

[37] D. M. Lucantoni, G. L. Choudhury and W. Whitt. Thetransient BMAP/G/1 queue. Stochastic Models 10 (1994)145-182.

[38] M. F. Neuts. The caudal characteristic curve of queues. Adv.Appl. Prob. 18 (1986) 221-254.

[39] J. W. Roberts. Performance Evaluation and Design ofMultiservice Networks, COST 224 Final Report, Commissionof the European Communities, Luxembourg, 1992.

[40] A. Simonian and J. Guilbert. Large deviations approximationfor fluid queues fed by a large number of on-off sources,preprint, 1994.

[41] K. Sohraby. On the asymptotic behavior of heterogeneousstatistical multiplexer with applications. IEEE INFOCOM ‘92,Florence, Italy.

[42] K. Sohraby. On the theory of general on-off sources withapplications in high speed networks. IEEE INFOCOM ‘93,San Francisco, CA.

[43] K. Sriram and W. Whitt. Characterizing superposition arrivalprocesses in packet multiplexers for voice and data. IEEE J.Sel. Areas. Commun. SAC-4 (1986) 833-846.

[44] D. N. C. Tse, R. G. Gallager and J. N. Tsitsiklis. Statisticalmultiplexing of multiple time-scale Markov streams.Laboratory for Information and Decision Sciences, M.I.T.1994.

[45] W. Whitt. Tail probabilities with statistical multiplexing andeffective bandwidths for multi-class queues.Telecommunication Systems, 2 (1993) 71-107.

Gagan L. Choudhury received theB.Tech. degree in Radio Physics andElectronics from the University ofCalcutta, India in 1979 and the MS andPh.D. degrees in Electrical Engineering

from the State University of New York(SUNY) at Stony Brook in 1981 and1982, respectively. Currently he is atechnical manager at the AT&T BellLaboratories, Holmdel, New Jersey,USA. His group is responsible for theperformance assessment of variousAT&T products and services based onmodeling and simulation. His main

research interest is in the development of multi-dimensionalnumerical transform inversion algorithms and their application to theperformance analysis of telecommunication and computer systems.

David M. Lucantoni is Vice-Presidentand Chief Technical Officer forIsoQuantic Technologies, LLC. IQTech develops network-level softwaresolutions for the wireless and wirelinetelecommunications industry. Dr.Lucantoni received a B.S inMathematics, an M.S. in Statistics and aPh.D. in Operations Research from theUniversity of Delaware in 1976, 1978,and 1981, respectively. He joinedAT&T Bell Laboratories in 1981 andover the next 13 years worked on theperformance analysis of varioustelecommunication systems. He haspublished over forty professional papersranging from multiplexer design tobroadband congestion controlalgorithms to state-of-the-art solutiontechniques to complex stochasticmodels. He was the co-recipient of theIEEE Steven O. Rice Prize paperaward in the area of CommunicationTheory in 1986 and was promoted toDistinguished Member of the TechnicalStaff in 1987.

Following a position at Motorola’s Satellite CommunicationsDivision as a performance analyst for the IRIDIUM(tm) Low EarthOrbit satellite system, Dr. Lucantoni co-founded IsoQuanticTechnologies, LLC, in 1994.

Ward Whitt received the A.B. degreein mathematics from DartmouthCollege in 1964 and the Ph.D. degree inoperations research from CornellUniversity in 1969. He taught in theDepartment of Operations Research atStanford University in 1968-1969, andin the Department of AdministrativeSciences at Yale University from 1969to 1977. Since 1977, he has beenemployed by AT&T Bell Laboratories.He currently works in the NetworkServices Research Laboratory, MurrayHill, NJ.

His research interests include queueing theory, stochastic processes,stochastic models in telecommunications and numerical inversion oftransforms.

Dr. Whitt is a member of the Operations Research Society ofAmerica and the Institute of Mathematical Statistics.

Figure 1. A comparison of approximations and exact values of theprobability of buffer overflow (approximated by the tail probabilityP(W > x)) as a function of the buffer size x for the example inSection 2.

Figure 2. The decay rate η n in (1.1) as a function of the number n ofsources (without scaling the sources) for the example in Section 2.

Figure 3. The asymptotic constant α n in (1.1) as a function of thenumber n of sources (without scaling the sources) for the example inSection 2.

Figure 4. The asymptotics constant α n as a function of the number nof sources (with scaling) for the example in Section 2.

Figure 5. The asymptotic constant α n as a function of the number nof sources (with scaling) for four examples in Section 3.

Figure 6. A comparison of approximations and exact values of theprobability of buffer overflow (approximated by the tail probabilityP(W > x) as a function of the buffer size x for the example withn = 2 in Section 4.

Figure 7. A comparison of approximations and exact values of theprobability of buffer overflow (approximated by the tail probabilityP(W > x) as a function of the buffer size x for the example withn = 60 in Section 4.

Figure 8. A comparison of approximations and exact values of theprobability of buffer overflow (approximated by the workload-tailprobability) as a function of the buffer size x for the example withn = 24 in Section 2 and higher buffer capacity 6000 in Section 5.

Figure 9. A comparison of approximations and exact values of theprobability of buffer overflow (approximated by the workload-tailprobability) n = 24 in Section 2 and lower buffer capacity 60 inSection 5.

Figure 10. A comparison of approximations and exact values of theprobability of buffer overflow (approximated by the workload-tailprobability) n = 24 in Section 2, higher buffer capacity 6000 andhigher burstiness in Section 5.

Figure 11. A comparison of approximations with exact values of theprobability of buffer overflow (approximated by the workload tailprobability) as a function of the buffer size x for the example inSection 6 with E 2-renewal-process sources, n = 24 and buffer size 8.

Figure 12. The asymptotic constant α n as a function of the numbern of sources (with scaling for the E 2 examples in Section 6.

Figure 13. A comparison of approximations with exact values of theprobability of buffer overflow (approximated by the workload tailprobability) as a function of the buffer size x for the example inSection 6 with E 2-renewal-process sources, n = 24 and buffer size100.

_ ________________________________________________________________Method of Number of Sources Allowed Buffer Size Required

Computation for Buffer Size x = 600 to Support n = 24 Sources_ ________________________________________________________________exact 24 600_ ________________________________________________________________effective-bandwidthapproximation (1.3) 12 1146_ ________________________________________________________________(1.1) and (1.5) 25 530_ ________________________________________________________________Poisson 78 20_ ________________________________________________________________Average-Rateengineering 80 not applicable_ ________________________________________________________________peak-rateengineering 7 not applicable_ ________________________________________________________________

Table 1.A comparison of different methods for determining (i) the number ofsources for the fixed buffer size x = 600 and (ii) the buffer sizerequired to support n = 24 sources for the example in Section 2.

Even though ATM seems to be clearly the wave of the future, one performanceanalysis indicates that the combination of stringent performance requirements(e.g., 10 − 9 cell blocking probabilities), moderate-size buffers and highly burstytraffic will require that the utilization of the network be quite low. Thatperformance analysis is based on asymptotic decay rates of steady-statedistributions used to develop a concept of effective bandwidths for connectionadmission control. However, we have developed an exact numerical algorithmthat shows that the effective-bandwidth approximation can overestimate thetarget small blocking probabilities by several orders of magnitude when thereare many sources that are more bursty than Poisson. The bad news is that theappealing simple connection-admission-control algorithm using effectivebandwidths based solely on tail-probability asymptotic decay rates mayactually not be as effective as many have hoped. The good news is that thestatistical multiplexing gain on ATM networks may actually be higher thansome have feared. For one example, thought to be realistic, our analysisindicates that the network actually can support twice as many sources aspredicted by the effective-bandwidth approximation. That discrepancy occursbecause for a large number of bursty sources the asymptotic constant in the tailprobability exponential asymptote is extremely small. That in turn can beexplained by the observation that the asymptotic constant decays exponentiallyin the number of sources when the sources are scaled to keep the total arrivalrate fixed. We also show that the effective-bandwidth approximation is notalways conservative. Specifically, for sources less bursty than Poisson, theasymptotic constant grows exponentially in the number of sources (when theyare scaled as above) and the effective-bandwidth approximation can greatlyunderestimate the target blocking probabilities. Finally, we develop newapproximations that work much better than the pure effective-bandwidthapproximation.

I. Introduction and Summary

Great energy is being devoted to studying the promising new asynchronoustransfer mode (ATM) technology for supporting multi-service high-speedcommunication networks; e.g., see Roberts [39]. As indicated in [39], interestin ATM is stimulated by two factors: first, by new technology making itpossible to transmit and switch at very high bandwidths and, second, bygrowing demand for more sophisticated and powerful communication services.

Even though ATM seems to be well on its way to widespread use, aperformance analysis based on asymptotic decay rates of steady-statedistributions indicates that the combination of stringent performancerequirements (e.g., 10 − 9 cell blocking probabilities), moderate-size buffers andhighly bursty traffic associated with these new services will necessitateoperating the networks at very low utilizations. (The basic model represents anATM switch receiving fixed-size ATM cells from several sources andtransmitting them over an output channel in a first-in-first-out fashion. Thebasic question is: How many sources can be admitted for a fixed buffer sizewith a specified small cell blocking probability?) A main message of thispaper is that ATM networks may be able to achieve higher utilizations thanthat asymptotic analysis indicates. In other words, there seems to be morepotential for statistical multiplexing gain.

The approximation based on asymptotic decay rates of tail probabilities isvery appealing because it supports a concept of effective bandwidths. From anengineering perspective, the notion of effective bandwidths is very natural.The idea is to assign each source an effective bandwidth requirement, and thenconsider any subset of sources feasible (admissible) if the sum of the requiredeffective bandwidths is less than the total available bandwidth. Thus, asuitable notion of effective bandwidths could go a long way toward solving theconnection admission control problem. For instance, once effectivebandwidths have been assigned, we can approach engineering and designproblems using multi-rate loss network models, e.g., as in Choudhury, Leungand Whitt [12,13] and references therein.

A large-deviations asymptotic analysis provides strong support for thesimple effective-bandwidth procedure, because it shows that it isasymptotically correct as the buffer size gets large and the tail probabilities getsmall, and because it provides a basis for assigning actual effective bandwidthvalues to different sources (voice, data, video, etc.); e.g., see Hui [31], Gibbensand Hunt [26], Guerin, Ahmadi and Naghshineh [28], Kelly [32], Sohraby[41], [42], Chang [10], Whitt [45], Elwalid and Mitra [23], Kesidis, Walrandand Chang [33], Glynn and Whitt [27], Courcoubetis, Kesidis, Ridder,Walrand and Weber [20], and Chang, Heidelberger, Juneja and Shahabuddin[11].

The additive nature of effective bandwidths is clearly appropriate if we letthe effective bandwidths be either the source peak rates or the source averagerates. However, it seems intuitively clear that working with peak rates is fartoo conservative, while working with average rates is far too optimistic. (Weshow this later in our examples.) Most of the recent work has been aimed atfinding appropriate effective bandwidths in between the peak and averagerates.

Unfortunately, however, from the outset, teletraffic engineering experiencesuggests that there may be a flaw in the effective bandwidth concept because itcorresponds to having no traffic smoothing from multiplexing. In particular, itis known that when many separate bursty sources are multiplexed(superposed), the total is less bursty than the components; i.e., there is a basisfor more statistical multiplexing gain than with Poisson sources. This istheoretically supported by the classical limit theorem stating thatsuperpositions of arrival processes, suitably scaled, converge to a Poissonprocess as the number of component arrival processes increases; e.g., seeC, inlar [19]. See Heffes and Lucantoni [30], Sriram and Whitt [43] andFendick, Saksena and Whitt [25] for related performance studies. In contrast,with the effective-bandwidth approximation, the burstiness of n superposedindependent and identically distributed sources is the same as for a singlesource (e.g., see p. 76 of [45]). This implies that the effective-bandwidthapproximation will predict greater congestion for any fixed arrival rate than itshould.

Nevertheless, there is a case for the effective bandwidths, because previousteletraffic analysis (such as in [25], [30], [43]) did not focus on extremely small

loss probabilities such as 10 − 9. Such very small loss probabilities naturallysuggest that appropriate asymptotics should provide what we want. And theasymptotic analysis associated with effective bandwidths indicates no trafficsmoothing.

The question, then, is what will actually happen in real systems? Will therebe significant traffic smoothing or not? Is the asymptotic analysis supportingeffective bandwidths sufficiently accurate or is it not?

These questions have plagued researchers in recent years because thenatural models are very difficult to analyze. It is difficult to calculate verysmall tail probabilities in models with many independent bursty sources, byeither exact analytical formulas or by computer simulation.

Our main contribution is a new algorithm for a queueing model thatenables us to compute the desired small tail probabilities exactly when thereare many bursty sources. From this exact analysis, we find that in some casesthe effective-bandwidth approximation is excellent, but in other cases (whichwe think are realistic), it is not. Thus, we conclude that the notion of effectivebandwidths based directly on large-deviation asymptotics may not be effectivefor connection admission control.

However, from an engineering perspective, the notion of effectivebandwidths remains very appealing. Thus it is natural to look formodifications of the proposed effective bandwidth approximation which willwork well; e.g., as in Guerin and Gun [29]. It is important to note that ouranalysis does not rule out the general notion of effective bandwidths. Instead,our analysis only indicates that there may be serious difficulties with theimplementation based directly on a particular kind of asymptotics, withoutrefinements.

A second main contribution of this paper is to develop modifications of thebasic effective-bandwidth approximation for tail probabilities that are moreaccurate (namely, (1.1), (1.5) and (1.7) below). However, these newapproximations do not support the simple notion of effective bandwidthsdiscussed above. Nevertheless, the approximate tail probabilities can be usedfor admission control.

The key to our analysis here is the exact numerical solution of the

i = 1Σn

G i/G/1 queueing model, which has a single server, unlimited waiting room,

the first-in first-out service discipline and i.i.d. (independent and identicallydistributed) service times that are independent of a superposition arrivalprocess. The arrival process is the superposition of n independent componentgeneral arrival processes. Our algorithm permits these component arrivalprocesses to be heterogeneous Markovian arrival processes (MAPs), as inLucantoni [35,36]; see Section 8 here. (The general theory also allows batcharrivals, but in this paper we consider only single arrivals.) Sincesuperpositions of independent MAPs are again MAPs, it suffices to considerthe MAP/G/1 queue. However, the computational effort increases when thenumber of sources increases. Our algorithm is based on Lucantoni [35]. Amajor new idea is the combination of the matrix-analytic results in [35] withnumerical transform inversion. In particular, we use the Fourier-series methodin Abate and Whitt [5]. We also use a scheme for accuracy enhancement forcomputing small probabilities as described in [15]. Moreover, we use anumber of techniques for speeding up computation, in order to treat a largenumber of sources. These techniques are described in Section 8. For thecomputation of the asymptotic parameters we use algorithms in [1], [2], [14].

We have also extended the numerical transform inversion algorithms tomultidimensional transforms [15] and applied the multidimensional inversionto calculate transient performance measures in the MAP/G/1 queue [37].Thus, we are in a position to study the transient behavior as well as thesteady-state behavior. With the ability to calculate time-dependent tailprobabilities, we can study whether or not transient analysis is needed in theadmission control problem. Here, however, we only discuss steady-statebehavior.

In our examples here, we consider only homogeneous sources, but inSection 7 we also indicate how to treat large systems with heterogeneoussources approximately. We primarily consider sources that are more burstythan a Poisson process, in particular, Markov modulated Poisson processes(MMPPs) with two-state Markov environment processes. (The MMPP is aPoisson process whose rate is itself a continuous-time Markov chain.) For the

main example, we consider on-off sources in which the arrival rate in oneenvironment state is zero, i.e., interrupted Poisson processes, which are knownto be renewal processes having hyperexponential interarrival-timedistributions [34]. These bursty source models are often used to model ATMtraffic. They can represent quite bursty traffic. We have also obtained similarresults for other (non-renewal) MMPP sources and multiple numbers of morethan one kind (two or three) of MMPP sources.

We also consider sources that are less bursty than a Poisson process. Inparticular, we consider renewal processes with E 2 (Erlang of order 2)interarrival times. Such less bursty sources might appear in ATM networks asa consequence of traffic shaping at the edge of the network. Contrary toconventional wisdom, we show that the effective-bandwidth approximationunderestimates the exact tail probabilities with these less bursty sources, sothat the effective bandwidth approximation need not be conservative.

As a surrogate for the steady-state blocking probability with a finite buffer,we consider the tail probability of the steady-state waiting time in our infinite-capacity queue. When the service times are deterministic with mean 1, as isthe case with ATM cells (with the appropriate time units), the least integerabove the steady-state waiting time coincides with the steady-state queuelength or buffer content seen by an arrival. (We can also directly compute thequeue-length distribution in the MAP/G/1 queue.) We are thus approximatingthe steady-state blocking probability in the finite-capacity system by thesteady-state probability that the buffer content exceeds that capacity level inthe infinite-capacity system at an arrival epoch.

Hence, we let W be the steady-state waiting time until beginning serviceand we focus on the tail probabilities P(W > x). It turns out that inconsiderable generality these waiting-time tail probabilities are asymptoticallyexponential, i.e.,

P(W > x) ∼ αe − ηx as x → ∞ , (1.1)

where η is a positive constant called the asymptotic decay rate, α is a positiveconstant called the asymptotic constant, and f (x) ∼ g(x) as x → ∞ means thatf (x)/ g(x) → 1 as x → ∞; see Abate, Choudhury and Whitt [2]. In standardsingle-source examples the approximation provided by (1.1) is oftenremarkably accurate [2]–[4]. Moreover, numerical experience indicates thatfor bursty sources the approximation provided by (1.1) with α replaced by 1often tends to be conservative, i.e.,

P(W > x) ≤ e − ηx for all x . (1.2)

This is a basis for the simple one-parameter approximation

P(W > x) ∼∼ e − ηx (1.3)

The asymptotic decay rate η in (1.1)–(1.3) is the basis for the concept ofeffective bandwidths. The simple one-parameter approximation (1.3) isappealing because the key parameter η in (1.1)–(1.3) is relatively easy todetermine, exactly or approximately. Indeed, it is typically much easier toobtain than the asymptotic constant α. Moreover, the resulting admission-control algorithm using effective bandwidths based on (1.3) is remarkablysimple. Hence, we call (1.3) the effective-bandwidth approximation.

It is important to note that much of the effective-bandwidth literature hasnot arrived at (1.3) via (1.1). The large-deviation result supporting (1.3) is theweaker limit

x − 1 log P(W > x) → − η as x → ∞ . (1.4)

General sufficient conditions for (1.4) are given in Chang [10] and Glynn andWhitt [27]. Clearly (1.1) implies (1.4), but not conversely. Indeed, the weakerlimit (1.4) yields no information about the asymptotic constant α in (1.1).

We have indicated that most of the effective-bandwidth literature hasarrived at approximation (1.3) via the limit (1.4) or related bounds such as(1.2). However, several people have focused on (1.1) too, namely, Abate etal. [2], Baiocchi [7], Elwalid and Mitra [23,24] and Neuts [38]. Given (1.1),(1.3) is a convenient simple approximation because α is much harder to obtainthan η, and because (1.3) is consistent with the notion of effective bandwidths.

As discussed in [3], for standard single-source models approximation (1.3)is often a reasonable substitute for approximation (1.1)–(1.4) when we areprimarily interested in percentiles of the distribution, rather than theprobabilities themselves. If the asymptotic constant α in (1.1) is not too

different from 1, then it plays a relatively small role in higher percentiles.

It should be noted that the asymptotic decay rate η in (1.1) is identical tothe asymptotic decay rate for the blocking probability in the correspondingfinite-buffer model as the buffer size gets large; see Baiocchi [7]. Hence, ouranalysis of the infinite-capacity model is directly relevant to the finite-capacitymodel with large buffer sizes.

In order to do better than (1.1), without calculating the exact tailprobabilities, we have also developed a refined three-term approximation ofthe form

P(W > x) ∼∼ α 1 e− η1 x + α 2 e

− η2 x + α 3 e− η3 x , (1.5)

where α 1 and η 1 are the asymptotic constant and asymptotic decay rate in(1.1), while α 2 , α 3 η 2 and η 3 are chosen to match the probability of delayP(W > 0 ) and the first three moments EW, E(W 2 ) and E(W 3 ), with η 2 andη 3 required to satisfy η 1 ≤ min {η 2 ,η 3 }; see Choudhury, Lucantoni andWhitt [17].

We thus have four ways to ‘‘calculate’’ the tail probability P(W > x) :(1.3), (1.1), (1.5) and the full numerical algorithm. Each successive methodtends to be more accurate, but each successive method requires substantialmore computational effort. The effective-bandwidth approximation (1.3) is byfar the easiest to compute, with there being virtually no limit to the size of themodel; e.g., see [45].

Using our exact numerical algorithm for the MAP/G/1 queue, we haveinvestigated the three approximations in (1.3), (1.1) and (1.5). For manystandard single-source models, the refinement in (1.1) is remarkably accurateeven for x not too large, e.g., for the 90th percentile of the distribution. Forstandard single-source models for which (1.1) is excellent at the 90th percentileand beyond, (1.5) typically is excellent for the entire distribution.

However, we have found that the story changes dramatically when thearrival process is the superposition of many component processes. Then theeffective bandwidth approximation (1.3) can perform very badly. It is evenpossible for (1.1) and (1.5) to perform badly, but there is a substantial regionwhere (1.3) performs badly and (1.1) performs well. This is the basis for ournew improved approximations beyond (1.3).

We emphasize that, for the model being considered, the limits in (1.1) and(1.4) do indeed hold, and we are indeed interested in very small tailprobabilities, and relatively large x. However, there is a serious difficulty(evidently pointed out for the first time here): The asymptotic constant α in(1.1) can be very different from 1 when the number of component arrivalprocesses is large.

We also explain why the asymptotic constant α can be very different from1 with many sources. To do so, we consider the case of n identical sourceswith fixed total rate. As n increases, the total rate is kept fixed by properlyscaling the individual streams. With this structure, it is known that theasymptotic decay rate η in (1.1) is actually independent of n. We givenumerical examples showing that the asymptotic constant with n sources (andthis scaling), α n, is itself asymptotically exponential in n, i.e.,

α n ∼ βe − nγ as n → ∞ , (1.6)

where β > 0 and, for sources more bursty than Poisson, γ ≥ 0, while forsources less bursty than Poisson, γ ≤ 0. Moreover, γ in (1.6) tends to belarger when the burstiness gets further from Poisson and the traffic intensitydecreases, which is a likely operating condition for ATM networks.

Combining (1.1) and (1.6), we obtain the refined asymptotic approximation(with the scaling for (1.6))

P(W > x) ∼∼ βe − nγ e − ηx . (1.7)

Our numerical results supporting (1.6) and (1.7) naturally suggest a limittheorem, on which we were working but others have delivered. The idea isstill to consider large-deviation asymptotics, but now letting both n and x go toinfinity. Since this paper was originally submitted, large deviation limitssupporting (1.6) and (1.7) have been reported by several authors, the firstknown to us being A. Dembo and O. Zeitouni (talk at National Meeting of theOperations Research Society of America, Boston, April 1994). Large-deviation papers supporting (1.6) and (1.7) have been written by Botvich andDuffield [9], Courcoubetis and Weber [21], Simonian [40] and Tse, Gallager

and Tsitsiklis [44]. To state their result, let W n be the steady-state waiting timewith n sources. Paralleling (1.4), their result is

n − 1 logP(W n > nx) = I(x) as n → ∞ , (1.8)

for an appropriate function I(x), yielding the approximation

P(W n > x) ∼∼ e − nI(x / n) . (1.9)

Clearly (1.8) provides strong theoretical support for (1.6) and (1.7), but (1.8) issubstantially weaker than (1.1) and does not yield the asymptotic constant β in(1.6) and (1.7). Nevertheless, the form of the limit function I(x) providesuseful insights, as can be seen from the references.

We propose (1.7) as a basis for developing useful approximations for thetail probabilities when there are large numbers of each of a few types ofsources. In particular, we suggest using (1.7) to estimate β and γ byextrapolating using (1.6) and exact calculations for small n; see Section 7 forfurther discussion.

Approximation (1.7) can be regarded as an approximation to the trueasymptote (1.1). Our numerical experience indicates that for sources morebursty than Poisson the true asymptote (1.1) is a good approximation at 10 − 9 ifα is not too small, e.g., α ≥ 10 − 4. For a wide range of problems, this is theregion of interest. Our experience also indicates that the true asymptote (1.1) isa poor approximation at 10 − 9 if α is too small, e.g., α ≤ 10 − 8. For estimatedvalues of α in between, e.g., 10 − 8 < α < 10 − 4, the true asymptote is onlymoderately good as an approximation. In this region, we propose using thetrue asymptote with α replaced by 10 − 4 as a rough conservative heuristicapproximation. For sources less bursty than Poisson, (1.1) tends to be a good(bad) approximation at a target blocking probability 10 − 9 ifα < 104 (α > 108 ).

Here is how the rest of the paper is organized. In Section 2 we consider aspecific example with bursty sources to demonstrate that the effective-bandwidth approximation (1.3) can perform poorly. In Section 3 we considerthe case of n identical sources scaled so that the total rate is independent of n,and give examples supporting (1.6). Next in Section 4 we presentmodifications of the example in Section 2 with different numbers of sources inwhich the two asymptotic approximations in (1.1) and (1.3) are, first, bothgood and, second, both bad. These cases seem identifiable by looking at theobserved value of the asymptotic constant α. In Section 5 we investigate theimpact of changing other variables. We show that the asymptoticapproximations tend to get worse as the number of sources increases, the buffersize decreases, the channel utilization decreases, the target blocking probabilityincreases, and the source gets further from Poisson, either more bursty or lessbursty.

In Section 6 we consider sources less variable than Poisson. For thesesources, we show that the effective-bandwidth approximation need not beconservative; i.e., (1.2) does not hold. In Section 7 we show how to doapproximations in large systems with heterogeneous sources. In Section 8 webriefly describe the computational algorithm. Finally, in Section 9 we state ourconclusions.

II. An Example where the Effective-Bandwidth Approximation Fails

In this section we consider an example with homogeneous on-off sources.(The on-off property is not essential for our model or for the conclusions wedraw from it. A source can have multiple states, each characterized by adifferent arrival rate. We use such sources in later examples.) Our primarypurpose is to show that the effective-bandwidth approximation (1.3) can be abad approximation for a realistic example and can lead to seriouslyunderestimating the number of sources the queue can accommodate.

We let the service times be i.i.d. with mean 1. Thus the unit of time is thetime required to transmit one ATM cell. We let the service time have anErlang distribution of order 16 (E 16 ), which has squared coefficient ofvariation (SCV, variance divided by the square of the mean) cs

2 = 1/16. Thisdistribution is similar to the deterministic (D) distribution of ATM cells, butbetter behaved for numerical calculations. (It is possible to do calculations forD distributions, but it requires more computational effort to achievecomparable precision.) In fact, we have computed using the E 1024 distributionand have found for the examples in this paper with MMPP sources that there is

little difference between E 16 and E 1024, supporting our hypothesis that D iswell approximated by E 16.

The on-off source has exponentially distributed on and off periods. Duringthe on periods, cells arrive according to a Poisson process; during the offperiods there are no arrivals. Each on-off source is characterized by threeparameters, the mean on period ω, the mean off period ζ and the (peak) rateduring the on period, p. Here we let the on and off periods have meansω = 436. 36 and ζ = 4363. 63, respectively. We let the peak rate bep = 0. 1375. The overall average rate is thus λ = pω/(ω + ζ) = 0. 0125.The ratio of peak to mean rates is thus p /λ = 11. 0. The mean number of cellsduring an on period is pω = 60. 0.

We feel that the level of source burstiness considered above can happenquite naturally in bursty data sources. Furthermore, the example is chosenmainly to illustrate that bad behavior can occur. Later we discuss how thebasic behavior of superposed sources changes with change in burstiness, buffersize and other parameters.

The above on-off source (an interrupted Poisson process) can also becharacterized as a renewal process with a hyperexponential (H 2, mixture oftwo exponentials) interarrival time distribution [34]. The mean interarrivaltime for one source is thus 1/λ = 80 and the SCV is ca

2 = 100. 2. Scaled tohave mean 1, the first four moments of an interarrival time are: m 1 = 1,m 2 = 101. 2, m 3 = 1. 68 × 104 and m 4 = 3. 73 × 106. This is another wayto see that the individual sources are quite bursty. (For a simple exponentialrandom variable with mean 1, the n th moment is n!)

Since the arrival process is a renewal process, The SCV coincides with thenormalized asymptotic variance or limiting index of dispersion for counts forthe counting process N(t), i.e.,

I c (∞) ≡t→ ∞lim

EN(t)Var N(t)_ _______ = ca

2 = 100. 2 , (2.1)

see [25,43]. The value I c (∞) = 100. 2 is large, so that each source is indeedquite bursty.

We let the buffer size be 600 cells, which falls in the range considered byswitch manufacturers. The buffer size 600 is ten times the mean number ofcells in an on period; i.e., the buffer contains 10 bursts. The quality of theapproximations depends strongly on the buffer size. (We discuss this further inSection 5.) Currently buffer size is limited by the availability of fast memory.We regard 600 as a representative moderate buffer size.

As is customary in ATM, we let the target blocking probability be 10 − 9.Hence, we choose the number of sources so that

P(W > 600 ) ∼∼ 10 − 9 . (2.2)

Our exact numerical results indicate that (2.2) is attained with n = 24 sources,yielding a utilization of 30%.

Figure 1 displays the exact tail probabilities P(W > x) and theapproximations (1.1), (1.3) and (1.5). In addition, Figure 1 displays thePoisson approximation, which is the exact tail probability in the M/E 16/1queue, computed by numerical transform inversion. The tail probabilityP(W > x) is our approximate probability of buffer overflow with x being thebuffer size.

___________________________Insert Figure 1 about here.

___________________________

The (exact) asymptotic parameters in (1.1) are η = 0. 01809 andα = 1. 453 × 10 − 5. The three-term approximation is also fully determined,based on the exact probability of delay P(W > 0 ) = 0. 4242 and first threemoments EW = 0. 6924, E(W 2 ) = 8. 573 and E(W 3 ) = 579. 3.

Of course, there is the question of numerical accuracy of the exact resultsreported in Figure 1. We verified them using a built-in accuracy check in ournumerical procedures, as explained in [15, 37] and Section 8 here. In therelated case of 64 sources with exponential service times, numerical results byElwalid and Mitra [24] confirmed our results.

Figure 1 indicates that (1.1) and (1.5) are still in error by a factor of 3.9 atx = 600. This error is relatively small, though, compared to the error in (1.3),which is by a factor of 105. The Poisson approximation obviously is even

worse than (1.3).

It is interesting to consider Figure 1 in relation to previous discussions of‘‘cell-level congestion’’ and ‘‘burst-level congestion,’’ e.g., in [39]. To a largeextent, Figure 1 indicates two nearly linear regions for the exact curve. Asdiscussed on p. 19 of [39], there is an initial period of steep nearly lineardecline corresponding to ‘‘cell-level congestion’’ and a later period of moregradual nearly linear decline corresponding to ‘‘burst-level congestion.’’ Thisburst level congestion corresponds to the true asymptote (1.1). However, thetransition between these two regimes is smooth and fairly long. As in [39],much of the ATM literature considers separate models to represent cell-leveland burst-level congestion. In contrast, we represent both within a singlemodel. Indeed, we consider precisely the multiple-MMPP-source modeldescribed as difficult on p. 185 of [39]. In this context, a major contributionhere is to point out that the intercept with the y-axis of the asymptote, which isthe asymptotic constant α in (1.1), may well be very small.

Now suppose, instead, that we use approximation (1.3). Let η n be η in(1.1) as a function of n (without any rescaling of individual sources). Asindicated above, η 24 = 0. 01809, but η increases as n (and ρ) decreases.Figure 2 displays η n as a function of n (without any rescaling of individualsources). It turns out that

e− 600ηn ≤ 10 − 9 for all n ≤ 12 , (2.3)

but not for n ≥ 13; η 12 = 0. 03749 and η 13 = 0. 03354. Hence, using theeffective-bandwidth approximation (1.3), we conclude that the queue canaccommodate only 12 sources with criterion (2.2). Hence, (1.3) indeedsignificantly underestimates the capacity. The actual capacity is two times thatpredicted by (1.3).


___________________________

Table 1 compares seven different procedures for determining the number ofsources that can be supported:

(i) The exact tail probabilities

(ii) the effective-bandwidth approximation (1.3)

(iii) the full asymptotic approximation (1.1)

(iv) the three-term approximation (1.5)

(v) the Poisson approximation

(vi) average-rate engineering

(vii) peak-rate engineering.

___________________________Insert Table 1 about here.

___________________________

From Figure 1 and Table 1, we see that even though the true asymptote isnot too accurate for the tail probability (being off by a factor of 4), it producesa good estimate for the number of sources (being off by only one).

Since the average rate is λ = 0. 0125, if we could size by average rate,then the queue could accommodate 80 sources. Since the peak rate isp = 0. 1375, if we sized by peak rate, then the queue can accommodate 7sources. The numbers 7 and 80 help put the realized improvement from 12 to24 going from approximation (1.3) to the exact numerical result in perspective.

There is a simple explanation for the poor performance of approximation(1.3). For n = 24, the asymptotic constant in (1.1) is α 24 = 1. 453 × 10 − 5.As can be seen from Figure 1, in this case the one-term asymptoticapproximation αe − ηx based on (1.1) is a good approximation, but replacing αby 1 introduces a large error.

In customary models with a single source, the asymptotic constant α isusually not too different from 1. Hence, it is interesting to see how α n

depends on n (again, without rescaling the individual sources). This is shownin Figure 3. There we see that α n declines from α 1 = 0. 340 to a minimumvalue of α 11 = 6 × 10 − 8 and then increases toward 1 again as the trafficintensity approaches 1. From heavy-traffic theory, we anticipate that α n → 1as ρ n → 1. Assuming that α n stays well above the target value 10 − 9 for all

n, as it does in Figure 3, we can anticipate that the asymptote (1.1) will be areasonably good approximation, but we clearly cannot simply replace theasymptotic constant α by 1.


___________________________

Instead of fixing the buffer size at 600 and asking how many sources thequeue can accommodate, we could instead fix the number of sources at 24 andask what size buffer is required. The exact numerical results indicate a buffersize of 600. The approximations (1.1) and (1.5) indicate that a buffer size of530 is required, approximation (1.3) indicates 1146 and the Poissonapproximation indicates only about 20. These results are also displayed inTable 1. This is another way to look at the weakness of the effective-bandwidth and Poisson approximations. This view also shows thatapproximations (1.1) and (1.5) are reasonably good.

As shown in [4], the asymptotics (1.1) for the steady-state waiting time areclosely related to corresponding asymptotics for other steady-state quantities,such as the workload, sojourn time and queue length (at arrivals and atarbitrary times). For this example, the queue length decay rate parameter σ in[4] is σ = 0. 9821. Although we consider only the waiting time here, ourexperience indicates that a detailed analysis of one of the other steady-statedistributions tells essentially the same story. For the example considered inthis section, if we look at the queue length at an arbitrary time (instead of at thearrival instant) then both the exact value and approximation (1.1) drop roughlyby a factor of 3, but the effective-bandwidth approximation (1.3) remainsunchanged (another weakness of (1.3)). However this further difference by afactor of 3 is small compared to the already-established difference by a factorof 105.

III. Asymptotics for the Asymptotic Constant in Scaled Superposition Processes

As shown in [10], [18], [23], [24], [27], [45], the asymptotic decay rate η n

in thei = 1Σn

G i/G/1 model with the superposition of n i.i.d. arrival processes is

independent of n when we fix the total arrival rate by scaling the componentarrival processes. In particular, let {N(t) : t ≥ 0 } be the arrival countingprocess with only one source. The proper scaling is achieved with n sourcesby letting each component arrival process be distributed as {N(t / n) : t ≥ 0 }.If the arrival rate of N(t) is λ, then the arrival rate of N(t / n) is λ/ n, so that thetotal arrival rate with n sources is λ for all n. (This scaling is also discussedpreviously in [43] and 25].)

Of course, when we add sources in a real network, we do not do anyrescaling, so we did not do any rescaling in Section 2. However, since η n isindependent of n with this rescaling, the rescaling helps us understand what ishappening with the various approximations. As discussed in [43], a keytheoretical reference point is the fact that, with the scaling, the superpositionprocess approaches a Poisson process as n → ∞. Since η n does not changewith n, we see that the two limits x → ∞ and n → ∞ do not interchange. Thisis a source of our difficulties.

Our numerical experience indicates that with this rescaling as n → ∞ theasymptotic constant α n in (1.1) approaches 0 for sources more bursty thanPoisson and approaches infinity for sources less bursty than Poisson. Moreprecisely, α n appears to decay or grow exponentially as in (1.6) as n → ∞.Numerical evidence supporting (1.6) is given in Figures 4 and 5 for sourcesmore bursty than Poisson. There we display α n as a function of n in log scalefor different examples. Figure 4 displays α n as a function of n for the examplein Section 2, while Figure 5 displays α n as a function of n in four otherexamples. For Figure 4, the reference case is the case with n = 24 sources andρ = 0. 30. All other cases in Figure 4 are rescaled to have this same ρ. Ineach example we rescale the arrival processes as n changes, so that η n isindependent of n. The linearity in Figures 4 and 5 for n not too small providesstrong support for (1.6). (Figure 12 in Section 6 provides similar support forsources less bursty than Poisson.)

___________________________Insert Figures 4 and 5 about here.___________________________

The four examples depicted in Figure 5 represent all combinations of twodifferent traffic intensities, ρ = 0. 5 and ρ = 0. 8, and two MMPP arrivalprocesses. The service-time distributions always are E 16 with mean 1. Asbefore, these represent nearly deterministic service times. The two arrivalprocesses are two-state MMPPs with asymptotic variance constants in (2.1) of2.8 and 28. These represent moderately bursty and substantially more burstysources, respectively (but less bursty than the example in Section 2). Theseexamples have positive arrival rates in both environment states, so that thecomponent MMPPs are not renewal processes.

In particular, the MMPPs are characterized by four parameters, one ofwhich can be taken to be the arrival rate. The ratio of the arrival rates in thetwo environment states is fixed at 4. The expected numbers of arrivals duringvisits to the two environment states are equal, 5 in the less bursty example withI c (∞) = 2. 8 and 75 in the more bursty example with I c (∞) = 28.

From Figure 5, we see that the key decay rate parameter γ in (1.6) isdecreasing in the traffic intensity ρ but increasing in the burstiness. Thisappears to be a general tendency. This means that the phenomenon inSection 2 is most likely to occur with low ρ and high burstiness, which is whatwe anticipate for ATM. The phenomenon might have been missed by others,because they focused on higher ρ and less bursty sources. For instance, theexamples in Section 5 of Elwalid and Mitra [23] all have high ρ (above 0.75)and lower burstiness, so that α > 10 − 2.

IV. Three Regimes for the Asymptotic Approximations

The example in Section 2 was one for which the asymptotic approximationαe − ηx in (1.1) is pretty good, but the simple approximation e − ηx in (1.3) isbad (because α ∼∼ 10 − 5). In this section we present modifications of thisexample in which, first, both approximations (1.1) and (1.3) are good and,second, both approximations (1.1) and (1.3) are bad.

To obtain these alternative cases, we simply modify the number of sourcesin the example with n = 24 in Section 2, scaling the arrival processes as inSection 3 to keep the total arrival rate fixed at λ = 0. 3 and η fixed atη = 0. 01809. In particular, the two alternative regimes are obtained by lettingn = 2 and n = 60. The results are displayed in Figures 6 and 7.

Since η is independent of n, approximation (1.3) is the same for the threecases n = 2, n = 24 and n = 60. However, α 2 = 0. 505,α 24 = 1. 45 × 10 − 5 and α 60 = 2. 22 × 10 − 12. (For n = 1, α 1 = 0. 992. )From Figures 1, 6 and 7, it is evident that we have the three regimes asclaimed. In Figure 6 with n = 2, approximations (1.1), (1.3) and (1.5) are allvery close, while in Figure 7 it is evident that approximations (1.1), (1.3) and(1.5) are very far apart.


Since α 60 ∼∼ 10 − 12 < 10 − 9 when n = 60, it should come as little surprisethat the asymptotics (1.1) have not kicked in by the time the tail probabilitiesreach 10 − 9 in this case. As a rough rule of thumb, it appears thatapproximation (1.1) tends to be good only when α n is greater than the desiredtail probability P(W > x). To a large extent, it appears that we are able tounderstand when the various approximations will be sufficiently accurate bycomputing only the asymptotic constant α. (This computation is possible,even for very large n, by computing α n for small values of n and thenextrapolating using (1.6).)

From Figure 7, we also see that the three-term approximation (1.5) is notaccurate at 10 − 9. The performance of approximation (1.5) in Figure 7 withn = 60 is much worse than in Figure 1 with n = 24. It is reassuring that itspoor performance is signalled by the fact that it is not possible to findparameters exactly matching the third moment in this case; see [17].Moreover, the three-term approximation performs pretty well even in Figure 7for tail probabilities above 10 − 4, for which it was originally designed. Thepoor performance at 10 − 9 in Figure 7 suggests that alternative fittingprocedures should be considered for extremely small probabilities such as10 − 9. (This is being investigated.)

V. Changing Other Parameters

In Section 4 we saw what happens as we changed the number of sources,scaling them in the manner of Section 3 so that the total arrival rate remainsunchanged. We saw that the quality of the asymptotic approximations declineas n increases.

More generally, we conclude that the asymptotic approximations tend toget worse as the number of sources increases, the buffer size decreases, thechannel utilization decreases, the target blocking probability increases, and thesource gets further from Poisson. However, we would also like to point outthat there is significant difference in the accuracy and the region ofapplicability of the effective bandwidth asymptotic (1.3) and the otherasymptotic approximations (1.1) and (1.5). Roughly speaking, (1.3) is not badwhen 10 − 2 < α < 102, whereas (1.1) and (1.5) are not bad when10 − 8 < α < 108. We expect that in practical engineering situations withmultiple sources (1.1) and (1.5) will typically be a reasonably goodapproximation, while (1.3) may be quite bad.

In this section we consider the effects of buffer size and burstiness.Figures 8 and 9 compare the approximations with exact values when the buffersize is 6000 and 60, respectively, instead of 600 as in Section 2. Here we keepthe number of sources fixed at n = 24. We scale the sources in the manner ofSection 3 until the blocking probability at the indicated capacity is 10 − 9.


So far, we have focused on the steady-state waiting time at arrival epoch.Similar results hold for the steady-state workload at an arbitrary time (thevirtual waiting time). To demonstrate this, the remaining numerical results inthis paper, including those in Figures 8 and 9, are for the steady-stateworkload. (The differences between the waiting time and workload arenegligible compared to the main phenomena being discussed.)

When the buffer size is increased to 6000, the buffer holds 100 burstsinstead of 10. Figure 8 shows that all the approximations perform very wellwith this larger buffer, just as in Figure 6. Now the utilization is 0.835. Theeffective-bandwidth approximation works very well; it predicts that the systemcan support 23 sources, which is within 1 of the exact value.

In contrast, Figure 9 shows that the approximations get even worse whenwe decrease the buffer size from 600 to 60, which corresponds to just oneburst. Figure 9 parallels Figure 7. Now the utilization is only 0.18. Theeffective-bandwidth approximation would now only admit 10 sources.

Suppose now we keep the buffer size at the high value of 6000 and thenumber of sources at n = 24, but increase the burstiness. Suppose that weincrease the burstiness by multiplying the mean number of bursts in an onperiod by 5. We keep the arrival rate fixed by multiplying the ratio of the offperiod to the on period by 5. This makes the mean number of arrivals in an onperiod 300, which means that the buffer now holds 6000/300 = 20 bursts.

Figure 10 compares the approximations with exact values in this case.Figure 10 shows that there is once again a big difference between theeffective-bandwidth approximation and the exact result, just as in Figure 1.For this example, the effective-bandwidth approximation would admit only 11sources. Hence, the advantage of the larger buffer of size 6000 is offset by thelarger burstiness. The actual performance of the approximations obviouslydepends on the combination of variables that actually prevails.


___________________________

VI. Sources Less Bursty than Poisson

In this section we consider sources that are less bursty than Poisson. Inparticular, we assume that each source is renewal with interarrival times thatare E 2. As before, we assume that the overall arrival process is thesuperposition of independent versions of the single source process. It seemsunlikely that the sources in an ATM network will actually be less bursty thanPoisson, but this is a possibility, due to traffic shaping at the network edge.

In the earlier bursty model we approximated the deterministic cell-lengthdistribution by an E 16 distribution and commented that E 16 and D give aboutthe same result. However, this is no longer true with less bursty arrivalprocesses. Therefore, in order to remain pretty close to D, in this section weassume an E 1024 service time distribution.

Figure 11 displays the approximations and exact tail probabilities for thiscase assuming 24 sources and a buffer size of 8. In order to meet the 10 − 9

buffer overflow probability, the channel utilization has to be 29%. In this casethe buffer overflow probability is greatly underestimated by the effective-bandwidth approximation (1.3). For this case of less bursty sources, theeffective-bandwidth approximation (1.3) would admit 39 sources in order tomeet the buffer overflow requirement with a buffer of size 8, instead of theproper number of 24.


___________________________

Figure 11 also shows the probability of buffer overflow for Poissonarrivals. For large tail probabilities, the exact result is close to the Poissonprediction and, for small tail probabilities, the exact result approaches the trueasymptote. It is also interesting to observe that the true asymptote is about tenorders of magnitude higher than the effective bandwidth approximation. Thisis in sharp contrast with the earlier highly bursty examples where the trueasymptote was always smaller than the effective bandwidth approximation.Also, note that there is a qualitative change in the shape of the exact curve. Itis concave in Figure 11 as opposed to being convex for the more burstysources.

The behavior of the tail probabilities in Figure 11 may be understood byplotting, in log scale, the asymptotic constant α n as a function of n, the numberof sources. This we do in Figure 12 for the E 2 sources at channel utilizations0.3 and 0.7 respectively. This is similar to what we did in Figure 5, and indeedFigures 5 and 12 look similar in the sense that in both cases the logarithm ofα n is asymptotically linear with n, i.e., α n changes exponentially with n.However, the striking difference is that with more bursty sources (Figure 5) α n

decays exponentially with n and approaches zero, while in the less bursty E 2

case, α n grows exponentially with n and approaches infinity. The growth rateincreases as the channel utilization decreases. We have also plotted how theα n changes with n for Poisson sources with channel utilizations of 0.3 and 0.7respectively. Of course, α n does not change with n in the Poisson case. ThePoisson case places in perspective the spectacular growth rate of α n with n inthe non-Poisson case.


___________________________

As before, the effective-bandwidth approximation (1.3) performs betterwith larger buffer sizes. This is shown in Figure 13 where the buffer size is100 instead of 8. As before, the number of sources is n = 24. The sources havebeen scaled in the manner of Section 3 so that P(W > 100 ) = 10 − 9. Thisdrives the channel utilization up to 0.949. Figure 13 shows that all theapproximations are close to the exact values in this case.


___________________________

VII. Approximations for Large Systems with Heterogeneous Sources

In this section we propose a method for obtaining useful approximationsfor tail probabilities in large systems. Approximations are needed, because ourexact algorithm cannot handle large numbers of sources, since the number ofphases of the superposed Markovian Arrival Process (MAP) grows rapidlywith the number of sources. Specifically, it can be shown that if there are Ltypes of sources, k i sources of type i, and each source of type i has m i phases,then the total number, P, of phases of the superposed MAP is given by

P =i = 1Π

L m i − 1k i + m i − 1

. (7.1)

Note that for Poisson sources m i = 1 and m i − 1k i + m i − 1

= 1, so that we can

add any number of them without increasing P. To run the exact model inreasonable time, we need P to be at most about 100. In all the numericalexamples in this paper we assumed L = 1 and m i = 2 so that we could treatup to about K 1 = 99 sources. (We actually considered up to 60 sources).However, with L = 2 and 3 (with m i still fixed at 2) we can treat only up toK i = 9 and K i = 3 sources of each type.

Now we specify our approximation procedure in terms of two examples.First, suppose that there are three classes, each with 100 homogeneous sources.We can calculate the asymptotic decay rate η exactly by considering the three-class system with one source in each class, with each source scaledappropriately, as in Section 3, so that the arrival rate of each single sourceequals the total arrival rate for all 100 sources, for each class, in the originalsystem.

In order to approximately determine the asymptotic constant α in theoriginal system, we calculate the exact asymptotic constants in the systemswith three classes and k sources in each class, again scaled to be consistentwith the original system according to Section 3, for several feasible k, e.g.,k = 1 , 2 and 3. (Note that here L = 3 and m i = 2, so that k i = 3 for all i isfeasible.) Then, assuming (1.6), we obtain an approximate asymptotic constantfor the original system by fitting β and γ in (1.6) to the data. If the resultingestimated asymptotic constant α is substantially greater than the target tailprobability, then we can apply approximation (1.1) with confidence. If theestimated asymptotic constant α is less than the target tail probability, then wenote that (1.1) is probably not appropriate. If the estimated asymptoticconstant is greater than the target tail probability, but not much greater, then wemight use the heuristic suggested in Section 1; i.e., we might use (1.1) with ahigher value of α as a rough conservative estimate.

The approach we have just described is satisfactory if all classes have manysources, as when there are 100 sources from each of the three classes.However, in actual applications, e.g., with video sources, there may be only afew sources from some classes. To illustrate, suppose that we have fourclasses, with 1 source in the first class, 2 sources in the second class and100 sources each in the last two classes. Let all sources have two states. Themethod we have just applied does not work for this example, but amodification does.

We have found that the asymptotic relation (1.6) still holds if we divide thesources into two groups, and hold one group of sources fixed, while wemultiply the number of sources in the second group by n. As before, we scalethe sources in the second group, so that the total arrival rate for each classremains fixed, independent of n.

As before, (1.1) holds for each n and the asymptotic decay rate η isindependent of n. Moreover, numerical experience indicates that theasymptotic relation (1.6) still holds, but now β in (1.6) is a function of the fixedsources. Now to estimate α in the original system we at first estimate β and γin (1.6) using n = 2 and 3 and then estimate α from those using n = 100.Note that here L = 4, m i = 2 for each i, k 1 = 1, k 2 = 2, k 3 = k 4 = n andhence n = 2 and 3 are feasible for the exact model.

More generally, the asymptotic result just described suggests anapproximation for m classes with multiplicities n 1 ,n 2 , . . . , n m of the form

α n1 , . . . , nm∼∼ βe − (γ 1 n1 + . . . + γm nm ) . (7.2)

For multiple classes, approximation (7.2) is convenient because we candetermine the asymptotic decay rates γ i by changing one class at a time.However, more work is needed on this.

VIII. The Algorithm

In this section we describe our algorithm for numerically computing theexact tail probabilities. To compute the tail probabilities, we draw heavily onLucantoni [35]; see [36] for a review. We model each source as a MarkovianArrival Process (MAP), which is a two-dimensional Markov process{N(t) , J(t) } on the state space { (i, j) :i ≥ 0 , 1 ≤ j ≤ m} with aninfinitesimal generator having the structure

Q =

...

0

0

D 0

...

0

D 0

D 1

...

D 0

D 1

0

...

D 1

0

0

. . .

. . .

. . .

(8.1)

where 0 , D 0 and D 1 in (8.1) are m×m matrices; D 0 has negative diagonalelements and nonnegative off-diagonal elements; and D ≡ D 0 + D 1 is anirreducible infinitesimal generator. We also assume that D ≠ D 0, whichensures that arrivals will occur. The variable N(t) counts the number ofarrivals in the interval [ 0 ,t], while J(t) gives the arrival ‘‘phase’’ at time t.

In this paper we consider the superposition of many sources. If theindividual sources are characterized by matrices D 0i and D 1i fori = 1 , 2 , . . . , K, where K is the number of sources, then it can be shown [35]that the superposed system is also a MAP characterized by matrices

D 0 = D 01 ⊕ D 02 ⊕ . . . ⊕ D 0K (8.2)

D 1 = D 11 ⊕ D 12 ⊕ . . . ⊕ D 1K (8.3)

where ⊕ is the Kronecker sum. We do all computations on the superposedMAP characterized by D 0 and D 1.

As explained in [35], the Laplace-Stieltjes transform (LST) of the steady-state waiting-time cumulative distribution function (CDF) is given by

W(s) = (πd) − 1 s( 1 − ρ) g[sI + D(h(s) ) ] − 1 e , (8.4)

where e is a vector of all 1’s, h(s) is service time LST, ρ is the traffic intensity(the mean arrival rate times the mean service time), d = D 1 e, π is thestationary probability vector of the Markov process with generator D (i.e., πsatisfies πD = 0 and πe = 1), g satisfies the equations gG = g and ge = 1,and G satisfies the equation

G =0∫∞

e (D0 + D1 G) xdH(x) , (8.5)

where H(x) is the service-time CDF and

D(h(s) ) = D 0 + D 1 h(s) . (8.6)

Our computation is based on, first, fast and accurate computation of W(s)using (8.4) and, second, fast and accurate transform inversion to get the waitingtime CDF from W(s). To do the inversion, we use the Fourier-series methodin [5] along with the round-off error control procedure in [15]. The procedurein [15] also provides a self-contained accuracy check by doing the calculationtwice with different round-off control parameters. This amounts to usingdifferent contours for complex inversion integration. With this procedure, weachieve high accuracy even at the tail probability of 10 − 9.

Next we describe two techniques for greatly speeding up the computationof the LST W(s). First, note that all individual sources are two-phase sources.The Kronecker sum operations in (8.2) and (8.3) makes the superposed sourcea 2K-phase source. This would prevent us from using large K, but we can takeadvantage of the fact that the sources are homogeneous to greatly reduce thedimensionality of the superposed source. We can define the phase of thesuperposed source as one plus the number of component sources in phase 1.Therefore, the total number of phases is K + 1 instead of 2K. The D 0 and D 1

matrices of the new superposed sources are given in terms of the componentD 01 and D 11 matrices of the (identical) component sources as follows:

(D 1 ) i ,i = (i − 1 ) (D 11 ) 1 , 1 + (K − i + 1 ) (D 11 ) 2 , 2 for i = 1 , 2 , . . . , K + 1

(D 1 ) i ,i + 1 = (K − i + 1 ) (D 11 ) 2 , 1 for i = 1 , 2 , . . . , K (8.7)

(D 1 ) i ,i − 1 = (i − 1 ) (D 11 ) 1 , 2 for i = 2 , 3 , . . . , K + 1

with (D 1 ) i , j = 0 for all other pairs (i, j),

(D 0 ) i ,i + 1 = (K − i + 1 ) (D 01 ) 2 , 1 for i = 1 , 2 , . . . , K

(D 0 ) i ,i − 1 = (i − 1 ) (D 01 ) 1 , 2 for i = 2 , 3 , . . . , K + 1 (8.8)

with (D 0 ) i , j = 0 for all other pairs (i, j) with i ≠ j, and

(D 0 ) i ,i = − [ (D 0 ) i ,i − 1 + (D 0 ) i ,i + 1 + (D 1 ) i ,i + (D 1 ) i ,i − 1 + (D 1 ) i ,i + 1 ] .(8.9)

For the service-time distribution, we approximate the deterministicdistribution by an Erlangian distribution of order k (E k ). A significantcomputational burden is the computation of G from the matrix integral

equation (8.5). The uniformization procedure recommended in [35] for generalservice-time distributions can be significantly improved for Erlangdistributions by noting that for the E k distribution with mean 1 (8.5) reduces to

G = [I − k − 1 (D 0 + D 1 G) ] − k . (8.10)

Equation (8.10) can easily be solved by successive substitution, i.e.,

G n + 1 = [I − k − 1 (D 0 + D 1 G n ) ] − k , (8.11)

where G 0 is chosen to be a stochastic matrix. Note that since we choosek = 2m (specifically 24 for bursty sources and 210 for smooth sources), eachiteration in (8.11) involves only a single matrix inversion and m matrixmultiplications.

In order to compute the asymptotic parameters α and η in (1.1), we use twoindependent algorithms (that cross-check each other): the algorithm in [2] andthe moment-based procedure in [14] and [1].

IX. Conclusions

Our first main conclusion here is that the effective-bandwidthapproximation (1.3) can break down when there is a large number ofindependent sources. Approximation (1.3) tends to get worse as the number ofsources increases, the channel utilization decreases, the buffer size decreasesand the source gets further from Poisson, either more bursty or less bursty.

If the sources are more bursty than Poisson, as is anticipated for ATMnetworks, then the effective-bandwidth approximation (1.3) is conservative.When the approximation is bad, there may be substantially more statisticalmultiplexing gain than approximation (1.3) predicts. On the other hand, if thesources are less bursty than Poisson, then the effective-bandwidthapproximation is no longer conservative, and may also be bad. In general,contrary to many statements, the effective-bandwidth approximation need notbe conservative.

Typically, the exact tail probabilities lie between the effective-bandwidthapproximation (1.3) and the true asymptote (1.1). If these two curves are veryclose to each other (differ by less than a factor of 10), then (1.3) is usuallyreasonably accurate. If the two curves are far from each other, but not too far(differ by less than 108), then (1.3) is bad, but (1.1) and (1.5) are reasonablyaccurate. Finally, if the two curves are extremely far apart (differ by more than108), then even (1.1) and (1.5) are not accurate. Our second main conclusionis that, even though (1.3) may not be a good approximation, the true asymptote(1.1) often is a good approximation. Moreover, from calculations of theasymptotic parameters α and η, we have a way to estimate whether or not theapproximations will be good.

A reason for the degradation of the effective-bandwidth approximation asthe number n of sources increases is that the asymptotic constant α n in (1.1) isitself asymptotically exponential in n. For sources more bursty than Poisson,α n decrease to 0 exponentially fast, as shown in Figures 4 and 5. For sourcesless bursty than Poisson, α n increases to infinity exponentially in n, as shownin Figure 12. The Poisson case is the reference case, because α n there does notchange with n. The asymptotically exponential form for α n in (1.6) allows usto compute it approximately for arbitrary n by extrapolating based oncomputed values for small n. For heterogeneous sources with differentmultiplicities, we exploit (7.1). Having a way to approximate the asymptoticconstant α is important, not only because we can use it in approximations (1.1)and (1.5), but also because the value of α indicates whether or not theapproximations will be good.

Finally, in addition to gaining a better understanding of the effective-bandwidth approximation (1.3), we have provided bases for more refinedanalysis tools via our exact MAP/G/1 numerical algorithm, the refinedapproximation (1.5) [17] and the exponential relation for α n in (1.6). We haveindicated how (1.6) and (7.1) can be combined with the true asymptote (1.1) toget an approximation that is almost as simple as the effective-bandwidthapproximation (1.3), but is applicable in a substantially wider region.Acknowledgments. We thank our colleagues Bharat Doshi, Ted Eckberg,Dave Houck and Pat Wirth for helpful comments.

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Figure 1. A comparison of approximations and exact values of the probability

of buffer overflow (approximated by the tail probability P(W > x)) as a

function of the buffer size x for the example in Section 2.

Figure 2. The decay rate η n in (1.1) as a function of the number n of sources

(without scaling the sources) for the example in Section 2.

Figure 3. The asymptotic constant α n in (1.1) as a function of the number n

of sources (without scaling the sources) for the example in Section 2.

Figure 4. The asymptotics constant α n as a function of the number n of

sources (with scaling) for the example in Section 2.

Figure 5. The asymptotic constant α n as a function of the number n of

sources (with scaling) for four examples in Section 3.


of buffer overflow (approximated by the tail probability P(W > x) as a

function of the buffer size x for the example with n = 2 in Section 4.


of buffer overflow (approximated by the tail probability P(W > x) as a

function of the buffer size x for the example with n = 60 in Section 4.


of buffer overflow (approximated by the workload-tail probability) as a

function of the buffer size x for the example with n = 24 in Section 2 and

higher buffer capacity 6000 in Section 5.


of buffer overflow (approximated by the workload-tail probability) n = 24 in

Section 2 and lower buffer capacity 60 in Section 5.

Figure 10. A comparison of approximations and exact values of the

probability of buffer overflow (approximated by the workload-tail probability)

n = 24 in Section 2, higher buffer capacity 6000 and higher burstiness in

Section 5.

Figure 11. A comparison of approximations with exact values of the

probability of buffer overflow (approximated by the workload tail probability)

as a function of the buffer size x for the example in Section 6 with E 2-

renewal-process sources, n = 24 and buffer size 8.

Figure 12. The asymptotic constant α n as a function of the number n of

sources (with scaling for the E 2 examples in Section 6.

Figure 13. A comparison of approximations with exact values of the

probability of buffer overflow (approximated by the workload tail probability)

as a function of the buffer size x for the example in Section 6 with E 2-

renewal-process sources, n = 24 and buffer size 100.

_ ____________________________________________________________________

Method of Number of Sources Allowed Buffer Size RequiredComputation for Buffer Size x = 600 to Support n = 24 Sources_ ____________________________________________________________________

exact 24 600_ ____________________________________________________________________effective-bandwidthapproximation (1.3) 12 1146_ ____________________________________________________________________(1.1) and (1.5) 25 530_ ____________________________________________________________________Poisson 78 20_ ____________________________________________________________________Average-Rateengineering 80 not applicable_ ____________________________________________________________________peak-rateengineering 7 not applicable_ ____________________________________________________________________

Table 1. A comparison of different methods for determining (i) the number ofsources for the fixed buffer size x = 600 and (ii) the buffer size required tosupport n = 24 sources for the example in Section 2.

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