QoS-aware bandwidth provisioning for IP network links

Transcription

1 Comuter Networks 5 (26) QoS-aware bandwidth rovisioning for IP network links Hans van den Berg a,b, Michel Mandjes c,b,1, Remco van de Meent b, Aiko Pras b, Frank Roijers a,c, *, Pieter Venemans a a TNO Information and Communication Technology, P.O. Box 55, 26 GB Delft, The Netherlands b University of Twente, Enschede, The Netherlands c CWI, Amsterdam, The Netherlands Received 23 February 24; received in revised form 1 February 25; acceted 8 May 25 Available online 2 August 25 Resonsible Editor: E. Altman Abstract Current bandwidth rovisioning rocedures for IP network links are mostly based on simle rules of thumb, using coarse traffic measurements made on a time scale of e.g., 5 or 15 min. A crucial question, however, is whether such coarse measurements give any useful insight into the caacity actually needed: QoS degradation exerienced by the users is strongly affected by traffic rate fluctuations on a much smaller time scale. The resent aer addresses this question. The goal is to develo rovisioning rocedures that require a minimal measurement effort. The bandwidth rovisioning formula that we roose (and which we justify under minimal model assumtions) is of the form q þ a ffiffiffi q. Here q (in Mbit/s) is the load of the system, which can evidently be estimated by coarse traffic measurements (e.g., 5 or 15 min measurements). The a deends on the characteristics of the individual flows and the QoS requirements. The QoS measure used is the robability that the traffic suly exceeds the available bandwidth, over some redefined (small) interval T, is below some small fixed number e. The imact of changing the ÔQoS arametersõ, i.e., T and e, on the coefficient a is exlicitly given. The validity of the bandwidth rovisioning rule is assessed through extensive measurements erformed in several oerational network environments. Ó 25 Elsevier B.V. All rights reserved. Keywords: Bandwidth rovisioning; Quality of Service; Traffic measurements; Gaussian traffic; M/G/1 inut * Corresonding author. Address: TNO Information and Communication Technology, P.O. Box 55, 26 GB Delft, The Netherlands. Tel.: ; fax: addresses: j.l.vandenberg@telecom.tno.nl (H. van den Berg), michel@cwi.nl (M. Mandjes), r.vandemeent@utwente.nl (R. van de Meent), a.ras@utwente.nl (A. Pras), f.roijers@telecom.tno.nl (F. Roijers),.h.a.venemans@telecom.tno.nl (P. Venemans). 1 Michel Mandjes is currently affiliated with the CWI and the University of Amsterdam /$ - see front matter Ó 25 Elsevier B.V. All rights reserved. doi:1.116/j.comnet

2 632 H. van den Berg et al. / Comuter Networks 5 (26) Introduction Bandwidth rovisioning rocedures require a thorough understanding of the relation between the characteristics of the offered traffic, the link seed and the resulting Quality of Service (QoS). The availability of such a relation enables the selection of a link caacity that guarantees that the aggregate rate of the offered traffic exceeds the link caacity less than some redefined (small) fraction of time. In such a ÔQoS by rovisioningõ aroach (i.e., allocating enough bandwidth to meet the QoS requirements of all alications resent), all traffic streams are treated in the same way. An alternative is to use traffic differentiation mechanisms, such as those of the IntServ aroach develoed within the Internet Engineering Task Force (IETF). IntServ enables stringent QoS guarantees, by relying on er flow admission control, but suffers from the inherent scalability roblems. Therefore it may be alied in the edge of the network (where the number of flows is relatively low), but not likely in the core. The DiffServ architecture can be seen as a hybrid aroach between ure rovisioning and Int- Serv: agreements are made for aggregates of flows rather than micro-flows, thus solving the scalability roblems. However, then the lack of admission control demands adequate bandwidth rovisioning, in order to actually realize the QoS requirements. Thus both in the ure rovisioning aroach and in DiffServ, i.e., the most romising QoS-enabling mechanisms, a rominent role is layed by bandwidth rovisioning rocedures. Bandwidth rovisioning has several other advantages over traffic differentiation mechanisms, see also, e.g., [9]. In the first lace, the comlexity of the network routers can be ket relatively low, as no advanced scheduling and rioritization caabilities are needed. Secondly, traffic differentiation mechanisms require that the arameters involved are Ôtuned wellõ, in order to meet the QoS needs of the different classes this usually requires the selection of various arameters (for instance: weights in weighted fair queueing algorithms, token bucket arameters, etc.). Bandwidth rovisioning has drawbacks as well: the lack of any QoS differentiation mechanism dictates that all flows should be given the most stringent QoS requirement, thus reducing the efficiency of the network (in terms of maximum achievable utilization). However, it is exected that this effect is mitigated if there is a high degree of aggregation, even in the resence of heterogeneous QoS requirements across users, as argued in, e.g., the introduction of [9] and in [12]. Clearly, the challenge for a network oerator is to rovision bandwidth such that an aroriate trade-off between efficiency and QoS is achieved: Without sufficient bandwidth rovisioning, the erformance of the network will dro below tolerable levels, whereas by rovisioning too much the erformance hardly imroves and is otentially already better than needed to meet the usersõ QoS requirements, thus leading to inefficient use of resources. The bandwidth rovisioning rocedures currently used in ractice are usually very crude. A common rocedure is to (i) use MRTG [17] to get coarse measurement data (e.g., 5 min intervals), (ii) determine the average traffic rate during these 5 min intervals, and (iii) estimate the required caacity by some quantile of the 5 min measurement data a commonly used value is the 95% quantile. This rocedure is sometimes ÔrefinedÕ by focusing on certain arts of the day (for instance office hours, in the case of business customers), or by adding safety and growth margins. The main shortcoming of this aroach is that it is not clear how the coarse measurement data relates to the traffic behavior at time scales relevant for QoS. More recisely, a crucial question is whether the coarse measurements give any useful information on the caacity needed: QoS degradation exerienced by the users may be caused by fluctuations of the offered traffic on a much smaller time scale, e.g., seconds (file transfers, web browsing) or even less (interactive, real-time alications) Contribution The goal of our work is to develo accurate and reliable rovisioning rocedures that require a minimal measurement effort. In articular, we derive an ÔinterolationÕ formula that redicts the bandwidth requirement on relatively short time

3 H. van den Berg et al. / Comuter Networks 5 (26) scales (say the order of 1 s), by using large time scale measurements (e.g., in the order of 5 min). In our aroach we exress QoS in terms of the robability (to be interreted as fraction of time) that, on a redefined time scale T, the traffic suly exceeds the available bandwidth. The bandwidth C should be chosen such that this robability does not exceed some given bound e. The time scale T and erformance target e are case-secific: they are arameters of our model, and can be chosen on the basis of the secific needs of the most demanding alication involved. We remark that in this setting buffers are not exlicitly taken into account; evidently, there is a relation between the time scale T in which the traffic rate exceeds the link rate and the buffer size needed to absorb the excess traffic. Our aroach relies on minimal modeling assumtions. Notably, we assume that the underlying traffic model is Gaussian emirical evidence for this assumtion can be found in e.g., [9,13]. For the secial case of eak-rate constrained traffic (eak rate r), we can use (the Gaussian counterart of) M/G/1 tye of inut rocesses [1], leading to an elegant, exlicit formula for the required bandwidth; M/G/1 corresonds to a flow arrival rocess that is Poisson with rate k and flow durations that are i.i.d. as some random variable D (with d := ED). We find that, measuring a load q kdr (in Mbit/s), the required bandwidth (to meet the QoS criterion) has the form q þ a ffiffiffi q. It is clear that the q can be estimated by coarse traffic measurements (e.g., 5 or 15 min measurements). The a deends on the characteristics of the individual flows, and its estimation requires detailed (i.e., on time scale T) measurements. In many situations, however, there are reasons to believe that the a is fairly constant in time; the estimate needs to be udated only when one exects that the flow characteristics have changed (for instance due to the introduction of new alications). We exect that the rovisioning aroach advocated in this aer extends to several other networking environments. In situations with large numbers of more or less i.i.d. users, the Gaussian assumtion will aly due to Central-Limit tye of arguments, and hence the rocedure followed goes through. In this aer we have tested the aroach in an IP setting; future work includes an assessment of the rovisioning guidelines in other tyes of networks. Aart from its simlicity, our bandwidth rovisioning formula q þ a ffiffiffi q has a number of attractive features. In the first lace it is transarent, in that the imact of changing the ÔQoS arametersõ (i.e., T and e)ona is exlicitly given. Secondly, the rovisioning rule is to some extent insensitive: a does not deend on k, but just on characteristics of the individual flows, i.e., the flow duration D and the eak rate r. This roerty enables a simle estimate of the additionally required bandwidth if in a future scenario traffic growth is mainly due to a change in k (e.g., due to growth of the number of subscribers). Furthermore, the analytical exression for a rovides valuable insight into the imact of changes in D and r. Our bandwidth rovisioning rule has been emirically investigated through the analysis of extensive traffic measurements in various network environments with different aggregation levels, user oulations, etc Literature There is a vast body of literature on bandwidth rovisioning, see for instance [2]. With resect to traffic modeling, it was emirically shown that Poisson acket arrivals do not accurately cature the deendencies resent in network traffic [19]. Gaussian aroximations do incororate these deendencies; their use was advocated in several aers, e.g., [1,9,13,16]. The use of flow level traffic models, like the M/G/1 model (in which flows arrive according to a Poisson rocess), is justified in, e.g., [3,4,18]. In[18] it is ointed out that the M/G/1 traffic model is extremely flexible, in that it allows all tyes of deendence structures: by choosing the flow durations Pareto-tye one can construct long-range deendent traffic, whereas exonential-tye flows lead to short-range deendent traffic. The use of M/G/1 inut is also investigated extensively in [2]; this aer also includes the analysis of a number of dimensioning rules. The study by Fraleigh et al. [9] is related to ours, in that it uses bandwidth rovisioning based

4 634 H. van den Berg et al. / Comuter Networks 5 (26) on traffic measurements to deliver QoS. An imortant difference, however, is that in their case the erformance metric is acket delay (rather than our link rate exceedance criterion). Also, in [9] measurements are used to fit the Gaussian model, and subsequently this model is used to estimate the bandwidth needed; this is an essential difference with our work, where our objective is to minimize the required measurement inut/effort, and bandwidth rovisioning is done on the basis of only coarse measurements. Another closely related aer is [8], where several bandwidth rovisioning rules are emirically validated Organization The remainder of this aer is organized as follows. In Section 2 we describe in detail the objectives of this study and the roosed modeling aroach; next, we rovide the analysis leading to our bandwidth rovisioning rule. Numerical results of our modeling and analysis are resented and discussed in Section 3. In Section 4, the bandwidth rovisioning rule is assessed through extensive measurements erformed in several oerational network environments. Finally, conclusions and toics for further research are given in Section Objectives, modeling and analysis The tyical network environment we focus on in this aer corresonds to an IP network with a considerable number of users generating mostly TCP traffic (from, e.g., web browsing, downloading music and video, etc.). Then the main objective of bandwidth rovisioning is to take care that the links are more or less ÔtransarentÕ to the users, in that the users should not (or almost never) erceive any degradation of their QoS due to a lack of bandwidth. Clearly, this objective will be achieved when the link rate is chosen such that only during a small fraction of time e the aggregate rate of the offered traffic (measured on a sufficiently small time scale T) exceeds the link rate. The values to be chosen for the QoS arameters T and e tyically deend on the secific needs of the alication(s) involved. Clearly, the more interactive the alication, the smaller T and e should be chosen. In more formal terms our objective can be stated as follows: The fraction (ÔrobabilityÕ) of samle intervals of length T in which the aggregate offered traffic exceeds the available link caacity C should be below e, for resecified values of T and e. In other words, with A(t) denoting the amount of traffic offered in [, t], PrðAðT Þ P CT Þ 6 e: ð1þ For rovisioning uroses, the crucial question is: for given T and e, find the minimally required bandwidth C(T, e) to meet the target. In the remainder of this section we derive exlicit, tractable exressions for our target robability Pr(A(T) P CT), see (1). We do this for a given traffic inut rocess {A(t), t P }; the only exlicit assumtion imosed is that {A(t), t P } has stationary increments, i.e., for any s, t and u > we require that the amount of traffic A(s + u) A(s) arrived in [s, s + u] has the same distribution as the amount of traffic A(t + u) A(t) arrived in [t,t + u]. In other words: the amount of traffic offered in a certain window deends on the window length only, and does not deend on the ÔositionÕ of the window. This stationarity will likely hold on time-scales that are not too long (u to, say, hours); on longer time-scales there is no stationarity due to day-atterns, and growth (or decline) of the number of subscritions (time-scale of weeks, months,...). Once we have an exression for (an uer bound to) Pr(A(T) P CT), we can find the minimal C required to make sure that this robability is ket below e. We thus find the required bandwidth C(T, e) it is exected that this function decreases in both T and e (as increasing T or e makes the service requirement less stringent) General traffic Based on the classical Markov inequality PrðX P aþ 6 ðex Þ=a for non-negative random variables X, we have, by utting X ¼ exðhaðt ÞÞ, for h P, the uer bound

5 H. van den Berg et al. / Comuter Networks 5 (26) PrðAðT Þ P CT Þ¼Pr e haðt Þ P e hct 6 Ee haðt Þ hct. Because this holds for all non-negative h, we can choose the tightest uer bound PrðAðT Þ P CT Þ 6 min hp haðt Þ hct Ee. ð2þ This bound, also known as the Chernoff bound, is unfortunately rather imlicit, as it involves both the comutation of the moment generating function E exðhaðt ÞÞ and an otimization over h. Note that C(T, e) could be chosen as the smallest number C such that the right-hand side of (2) is smaller than e: CðT ; eþ :¼ min C : min Ee hp haðt Þ hct 6 e. Rearranging terms, we find that equivalently we are looking for the smallest C such that there is a h P such that C P log EehAðT Þ log e. ht This C is obviously equal to the infimum of the right-hand side over h P : CðT ; eþ ¼min hp log Ee haðt Þ log e. ð3þ ht 2.2. Exlicit formula for Gaussian traffic Assuming that A(T) contains the contributions of many individual users, it is justified (based on the Central-Limit theorem) to assume that A(T) is Gaussian if T is not too small, see e.g., [9,13]. In other words A(T) Norm(qT, v(t)), for some load q (in Mbit/s), and variance v(t) (in Mbit 2 ). For this Gaussian case we now show that we can determine the right-hand side of (3) exlicitly. The first ste is to comute the moment generating function involved (this is done by isolating the square): Ee haðt Þ ¼ ex hqt þ 1 2 h2 vðt Þ. The calculation of the minimum in (3) is now straightforward 1 2 CðT ; eþ ¼q þ min hvðt Þ log e hp T ht ¼ q þ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 2 log eþvðt Þ; ð4þ T the minimum is attained at h ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 2 log eþ=vðt Þ. Evidently, C(T, e) can also be found by first comuting the Chernoff bound for Gaussian traffic! PrðAðT Þ P CT Þ 6 ex 1 ðc qþ 2 T 2 ; ð5þ 2 vðt Þ then it is easily checked that (4) is the smallest C such that (5) is below e. As for any inut rocess with stationary increments v(æ) cannot increase faster than quadratically (in fact, a quadratic function v(æ) corresonds to erfect ositive correlation), ffiffiffiffiffiffiffiffiffi vðt Þ =T is decreasing in T, and hence also the function C(T,e) the longer T, the easier it is to meet the QoS requirement. Also, the higher e, the easier it is to meet the requirement, which is reflected by the fact that the function decreases in e. Remark 1 (Effective bandwidth). There is some reminiscence between formula (4) and the effective bandwidth concet roosed earlier in the literature, see, e.g. [6,7,11], but there are major differences as well. One of the key attractive roerties of effective bandwidths is their ÔadditivityÕ: if there are two sources, both are assigned a bandwidth, arametrized by the QoS-criterion, such that their sum reresents the bandwidth needed by their suerosition: C 1þ2 ðeþ ¼C 1 ðeþþc 2 ðeþ. Imortantly, it can be argued that interreting (4) as an effective bandwidth would lead to a bandwidth allocation too essimistic: noting that ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v 1 ðt Þþv 2 ðt Þ 6 ffiffiffiffiffiffiffiffiffiffiffi v 1 ðt Þ þ ffiffiffiffiffiffiffiffiffiffiffi v 2 ðt Þ; the amount of bandwidth to be rovisioned for the aggregate inut could be substantially less than the sum of the individually required bandwidths.

6 636 H. van den Berg et al. / Comuter Networks 5 (26) Remark 2 (Equivalent caacity from Guérin et al. [1]). We remark that exression (4) is of the same sirit as the ÔGaussianÕ equivalent caacity formula advocated in [1], but some remarks need to be made. Time scale. The (classical) formula roosed in [1] is of the form CðeÞ ¼q þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 log e logð2þ r; ð6þ where r 2 is the variance of the Ôinstantaneous traffic rateõ R r 2 :¼ VarR ¼ lim Var AðT Þ vðt Þ ¼ lim T # T T # T. 2 Hence, C(e) as derived in [1] relates to the time scale T =, and is in this sense less general than our C(T, e). We remark that for many Gaussian rocesses r does not exist; think of fractional Brownian motion with H < 1. Exceedance robability: Guérin et al. s [1] aroximation vs. Chernoff bound. It is easily verified that (6) essentially relies on the aroximation! PrðR > CÞ 1 ffiffiffiffiffi ex 1 ðc qþ 2 ; ð7þ 2 2 r 2 the actual value of Pr(R > C) is the (comlementary) Gaussian distribution function Z! 1 1 PrðR > CÞ¼ ffiffiffiffiffi ex 1 ðx qþ 2 dx. C 2 r 2 r 2 In [1] it is reorted that C(e) based on the aroximation (7) of the exceedance robability Pr(R > C) is Ôa good aroximationõ of the equivalent caacity, but no (mathematical) motivation was given. Relying on the aroximation PrðN > xþ x 1 ð2þ 1=2 exð x 2 =2Þ, where N Norm(, 1), we find that Pr(R > C) reads Z! 1 1 ffiffiffiffiffi ex 1 ðx qþ 2 dx C 2 r 2 r 2! r 1ffiffiffiffiffi ex 1 ðc qþ 2. C q 2 2 r 2 Hence, if C q r, then we indeed find (7). However, we could not find a rationale for C q being of the same order as r. In fact, we could construct cases in which (7) is extremely otimistic, in that it substantially underestimates Pr(R > C). Hence, its use is not aroriate for rovisioning uroses.this motivates why our aroach above uses the (rovably conservative) Chernoff bound (5) rather than aroximations of the tye of (7). Also, numerical exeriments indicated that there is usually just a modest difference between the caacities based on the Chernoff bound and caacities based on inversion of the (comlementary) Gaussian distribution function, where, evidently, the former are always conservative. The formula for C(T, e) indicates that, given that we are able to estimate the load q and the variance v(t) on the ÔadvertisedÕ time scale T, we have found a straightforward rovisioning rule. In the next subsection, we focus on the secial case of (the Gaussian counterart of) M/G/1 inut; in that (still quite general) case the exressions simlify further M/G/1 traffic Whereas the above rovisioning formula holds for general Gaussian traffic, we now focus on an imortant sub-class: Gaussian traffic that has the variance function of the M/G/1 inut rocess, also called the Gaussian counterart of the M/G/1 inut rocess [1]. In the M/G/1 inut model, jobs arrive according to a Poisson rocess with rate k, and stay in the system during a eriod that is distributed as the random variable D (i.i.d.). While in the system they generate traffic at rate r. Hence, q = kdr, with d =ED. Notice that the M/G/1 traffic model is articularly aroriate in scenarios in which a eak-rate limitation is imosed, see also [1,18]. As we will see later, by choosing D aroriately, it covers a broad range of correlation structures. Denote by f X and F X the density and the distribution function, resectively, of the random variable X. Let D r be the residual distribution of D, i.e., 1 F D ðxþ ¼df D rðxþ. Now the variance of

7 H. van den Berg et al. / Comuter Networks 5 (26) AðT Þ¼r R T NðtÞdt (with N(t) denoting the number of flows resent at time t) can be exlicitly calculated, as follows. As the number of flows resent at both time s and time t has a Poisson distribution with mean k ED Pr(D r > jt sj), we obtain that CovðNðtÞ; NðsÞÞ ¼ k ED PrðD r > jt sjþ. Hence vðt Þ¼r 2 Var ¼ r 2 Z T ¼ qr Z T Z T Z T Z T NðtÞdt CovðNðtÞ; NðsÞÞdsdt PrðD r > jt sjþdsdt; which can be simlified to the sum of three single integrals, see also [14,15]: Z T Z T kr 2 2T xð1 F D ðxþþdx d x 2 f D rðxþdx þdt 2 ð1 F D rðt ÞÞ. ð8þ Hence, cf. (4), the required bandwidth C(T, e) can be exressed as CðT ; eþ ¼q þ a ffiffiffi q. ð9þ Imortantly, a deends exclusively on the characteristics of the individual flows, i.e., the distribution of the flow duration D and the eak rate r (and QoS requirements T and e), but does not deend on the flow arrival rate k this will turn out to be a key roerty in our exerimental investigations on rovisioning that are resented in Section 4. Now we evaluate (8) for different distributions D, covering both the long-range deendent and short-range deendent case Exonential flow durations For exonentially distributed flow lengths D, the variance v(t) reads vðt Þ¼2qd 2 rðe T =d 1 þ T =dþ; such that a ¼ T 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 2 log eþ2rðe d T =d 1 þ T =dþ : Observe that v(t) is, for T large, linear, corresonding to short-range deendent inut. Also observe, that a deends on T only through the ratio T/d Pareto flow durations For Pareto-distributed flow lengths D, i.e., obeying a b F D ðxþ ¼1 ; x P ; ð1þ x þ b and d ¼ b=ða 1Þ (where a > 1 and b > ), substantial calculus gives (assume for ease a 5 2, a 5 3) 2qr vðt Þ¼ ð3 aþð2 aþ ðba 1 T þ bþ 3 a ð3 aþbt b 2 ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ 1 2r ð 2logeÞ T ð3 aþð2 aþ ðba 1 ðt þ bþ 3 a ð3 aþbt b 2 Þ. (Notice that [1] uses F D ðxþ ¼1 ðb=xþ a for x P b, but, as flow sizes do not obey some natural lower bound b, we have chosen to use the more natural Ôshifted versionõ (1) instead.) If a < 2, v(t) grows ÔsuerlinearlyÕ for large T (in fact, it grows as T 3 a ), corresonding to long-range deendent inut; for a > 2, we see that v(t) is essentially linear, cf. [5] Discussion on the M/G/1 inut model 1. If T is small (i.e., small comared to d), then a becomes insensitive in the flow duration D. This can be seen as follows. From (8) it can be derived that v(t)/t 2! qr T #. Then (4) yields CðT ; eþ q þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 2 log eþqr; exclusively deending on q, for T small. This result can be derived differently, by noting that for T #, the erformance criterion boils down to requiring that the number of active users does not exceed C/r. It is well-known that the number of active users has a Poisson distribution with mean kd; this exlains the insensitivity. 2. The case of exonential flow lengths can be easily extended to, e.g., hyerexonentially distributed flows; a random variable X is hyerexonentially distributed [21,. 446] if with robability 2 (, 1) it is distributed exonentially with mean d 1, and else exonentially with

8 638 H. van den Berg et al. / Comuter Networks 5 (26) mean d 2. Then the hyerexonential case is just the situation with two flow tyes feeding indeendently into the link (each tye has its own exonential flow length distribution); note that the variance of the total traffic is equal to the sum of the variances of the traffic generated by each of the different exonential flow tyes. 3. The above aroach assumes that traffic arrives as ÔfluidÕ: it is generated at a constant rate r. Itis erhas more realistic to assume that, during the flowõs Ôlife timeõ, traffic arrives as a Poisson stream of ackets (of size s); the rate of the Poisson rocess is c, where cs is equal to r. Denoting the above, fluid-based, variance function by v f (Tjr), and the acket-based variance function by v (Tjc,s), it can be verified that v ðt j c; sþ ¼v f ðt j rþþqst ; ð11þ irresective of the flow duration distribution. Imortantly, the rovisioning formula CðT ; eþ ¼q þ a ffiffiffi q remains valid (for an a that does not deend on k). 3. Numerical results This section resents numerical results obtained by using the analytical model of Section 2. The goal is to illustrate a few key features of our bandwidth rovisioning formula. We use the traffic Table 1 Default arameter settings for the numerical results arameters and QoS arameters dislayed in Table 1, unless secified otherwise Exeriment 1: Fluid model vs. acket-level model Fig. 1 shows the required caacity obtained by the acket-level and fluid model as a function of T, for various mean flow durations d. It is seen that for large values of the time scale T, both models obtain the same required caacity. This can be understood by looking at the extra term qst of (11), which influence on C(T, e) becomes negligible for increasing T, cf. (4). For T # the required caacity obtained by the acket-level model behaves like CðT ; eþ q þ ffiffiffiffiffi 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qs ffiffiffi 2 log e; T 3 REQUIRED CAPACITY Flow delta=1 3 REQUIRED CAPACITY C (T,.1) T (in sec) Packet delta=1 Flow delta=1 Packet delta=1 Flow delta=.1 Packet delta=.1 Flow delta=.1 Packet delta=.1 C (T,.1) T (in sec) Fig. 1. Comarison of the required caacity for the flow-level and acket-level model as a function of the time scale T. Left: logarithmic axis. Right: linear axis.

9 H. van den Berg et al. / Comuter Networks 5 (26) and hence C(T,e) "1 as T #, whereas the required caacity of the fluid model converges to q þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 2 log eþqr; as was already argued in Section The fast increase in the required caacity in the acket-level model for a decreasing time scale T was also observed in e.g., [9]. Note that, in fact, the required caacity is not influenced by the absolute value of T, but rather by the ratio of T/d. The right grah of Fig. 1 shows the same results as the left grah, but now on a linear axis and only for T 2 [,1]. In the remainder of this section we restrict ourselves to the flow-level model, as we will focus on situations with values of T/d >.1 for which the required caacity is almost identical in both models Exeriment 2: Imact of the flow duration distribution Next we investigate the imact of the flow duration distribution on the required caacity. Fig. 2 contains four grahs with results for hyerexonentially distributed flow durations D. Each grah shows, for a articular value of the mean flow size d, the required caacity as a function of the offered load q, for different Coefficients of Variation (CoV) of D. These grahs show that the required caacity is almost insensitive to the CoV for the long (d = 1 s) and short-flow durations (d =.1 s). For the other cases (d 2 {.1,1} s) the required caacity is somewhat more sensitive to the CoV. The grahs show that for hyerexonentially distributed flow durations less caacity is required if the CoV increases. 1 δ = 1 REQUIRED CAPACITY T= 1 δ = 1 Required Caacity (Mbit/s) 8 T=1 CoV=1 8 T=1 CoV=2 6 6 T=1 CoV=4 4 4 T=1 CoV=8 2 T=1 CoV=16 2 X=Y Traffic load (Mbit/s) Traffic load (Mbit/s) Required Caacity (Mbit/s) 1 δ =.1 T= 1 δ =.1 Required Caacity (Mbit/s) 8 T=1 CoV=1 8 T=1 CoV=2 6 6 T=1 CoV=4 4 4 T=1 CoV=8 2 T=1 CoV=16 2 X=Y Traffic load (Mbit/s) Traffic load (Mbit/s) Required Caacity (Mbit/s) Fig. 2. Required caacity for hyer-exonential flow durations with different means and CoVs.

10 64 H. van den Berg et al. / Comuter Networks 5 (26) It should also be noticed that the required caacity for T =, also shown in Fig. 2, corresonds to the often used M/G/1 bandwidth rovisioning aroach, cf. the discussion in Section and the discussion on Exeriment 1. The numerical results show that articularly for short-flow durations significantly less caacity is required than suggested by the classical M/G/1 aroach; for longer flows this effect is less ronounced Exeriment 3: Imact of QoS arameter e Fig. 3 shows the required caacity as a function of the QoS requirement e, which secifies the fraction of intervals in which the offered traffic may exceed the link caacity. A larger value of e means relaxing the QoS requirement, and hence less caacity is needed. Obviously, for e! 1 the required caacity converges to the long term average load q = 1. For e # the required caacity increases ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi raidly to infinity (according to 2 log e) Exeriment 4: Imact of the CoV of the flow duration distribution To investigate the imact of the flow duration characteristics, we comuted the required caacity for exonential, hyerexonential, and Pareto distributed flow durations with different CoV values, see the left anel of Fig. 4. The grah shows that the required caacity is almost insensitive to the 3 REQUIRED CAPACITY 3 REQUIRED CAPACITY Flow delta=1 Required Caacity C(1,ε) Flow Flow delta=1 delta=.1 Required Caacity C(1,ε) Flow delta= E-4 1..E-3 1.E-2 1.E-1 1.E+ Esilon Esilon Fig. 3. Comarison of the required caacity for the flow-level and acket-level model as a function of the QoS requirement T. Left: logarithmic axis. Right: linear axis. Required Caacity (Mbit/s) REQUIRED CAPACITY Ex CoV= 1 Hy-ex CoV=2 Hy-ex CoV=8 Hy-ex CoV=16 Pareto CoV=2 Pareto CoV=8 Pareto CoV=16 REQUIRED CAPACITY X=Y Traffic load (Mbit/s) Traffic load (Mbit/s) Required Caacity (Mbit/s) Hy-ex delta=1 r=1 Hy-ex delta=2 r=.5 Hy-ex delta=1 r=.1 Pareto delta=1 r=1 Pareto delta=2 r=.5 Pareto delta=1 r=.1 X=Y Fig. 4. Left: required caacity for different flow duration distribution and CoVs. Right: required caacity for different access rates.

11 H. van den Berg et al. / Comuter Networks 5 (26) flow duration distribution. The left grah also confirms the earlier observations that the caacity is almost insensitive to the CoV of the flow duration distribution. Note, that for hyerexonentially distributed flow durations the required caacity slightly decreases for increasing CoV, while for Pareto distributed flow durations the required caacity slightly increases for increasing CoV Exeriment 5: Imact of the access rate Finally, the right anel of Fig. 4 studies the effect of the access rate r on the required caacity. Three values of the access rate r and the mean flow duration d are chosen such that the mean flow size d Æ r remains constant. As exected, the required caacity increases considerably when r becomes larger (i.e., the traffic burstiness grows). The results in this grah for hyerexonential and Pareto flow sizes confirm the conclusions from Exeriments 2 and 4 that the required caacity is almost insensitive to the flow duration distribution. 4. Exerimental verification and bandwidth rovisioning In this section we will analyze measurement results obtained in oerational network environments in order to validate the modeling aroach and bandwidth rovisioning rule resented in Section 2. In articular, we will investigate the relation between measured traffic load values ^q during 5 min eriods (long enough to assume stationarity) and the traffic fluctuations at a 1 s time scale within these eriods. Clearly, if (A) our M/G/1 traffic modeling assumtions of Section 2.3 aly and if (B) differences in the load q are caused by changes in the flow arrival rate k (i.e., the flow size characteristics remain unchanged during the measurement eriod), then, as a function of q, for given (T,e), the required bandwidth C q should satisfy C q ¼ q þ a ffiffiffi q ; ð12þ for some fixed value of a. To assess the validity of this relation, we have carried out measurements in three different network environments: (i) a national IP network roviding Internet access to residential ADSL users, (ii) a college network and (iii) a camus network. In the ADSL network environment the main assumtions made in Section 2.3 in order to justify use of the M/G/1 traffic model seem to be satisfied, i.e., the flow eak rates are limited due to the ADSL access rates (which are relatively small comared to the network link rates), and the traffic flows behave more or less indeendently of each other (the IP network links are generously rovisioned and, hence, there hardly is any interaction among the flows). The other network environments have essentially different characteristics. In articular, in the college and camus network the ratio of the access rate and link rate is relatively large, which, obviously, may lead to violation of our traffic modeling assumtions. In Section 4.3 we demonstrate how to estimate the a in (12) directly from measurements of the aggregate traffic at time scale T. An alternative to this aroach would be to fit the flow-size distribution, such that a can be comuted by inserting this into the exlicit formulae of Section 2.3. Recall that Exeriment 2 of Section 3 showed that the CoV of the flow-size has just a modest imact on a. Also, fitting the full flow-size distribution has the evident drawback that it requires er-flow measurements. Therefore, we refer direct estimation of a. The measurement scenarios and results will be described and discussed i more detail in the following subsections ADSL network environment We first focus on the ADSL network environment with residential users, see Fig. 5. An ADSL connection consists of an ADSL modem on both sides of the local loo between the subscriber and the local exchange. On the local exchange side, u to 5 modems are contained in Digital Subscriber Line Access Multilexers (DSLAM). The DSLAMs are connected to the core IP infrastructure by means of otical STM-1 (155 Mbit/s) links. The aggregated traffic of all the ADSL subscribers of a certain Internet Service

12 642 H. van den Berg et al. / Comuter Networks 5 (26) ADSL subscribers Local loo DSLAM STM-1 CoreIP infrastructure STM-1 Gigabit Ethernet Fig. 5. Overview of ADSL infrastructure. ISP internet Provider (ISP) is carried over a high-caacity link between the core infrastructure and the ISP. Deending on the size of the ISP, this can vary between a single STM-1 link and multile Gigabit Ethernet links. At the time of the measurements, none of the network links were saturated, and hence the traffic was not affected by any shortage of caacity in the ADSL network. We choose the samle size T = 1 s, motivated by the fact that this can be assumed to be the time scale that is most relevant for the Quality of Service ercetion of end-users of tyical alications like web browsing. Elementary transactions, such as retrieving single web ages, are normally comleted in intervals roughly in the order of 1 s. If the network erformance is seriously degraded during one or several seconds, then this will affect the quality as erceived by the users. The method that was chosen to measure the traffic at the 1 s time scale was to use the internal traffic counters (interface MIBs) of the DSLAMs. These counters kee track of the accumulated number of bytes that are transorted on each ort in each direction. In this exeriment, the counters for the STM-1 orts in the downstream direction (toward the subscribers) were used. The counters were read-out using SNMP. The measurements were done during several evenings (between 5 PM and 11 PM), as this is the busiest eriod of the day for ADSL traffic. The measurements were erformed on a large number of DSLAMs, in locations ranging from small villages to major cities. Time was slit into 5 min chunks, over which the load q is determined for each STM-1 link. In addition, for each 5 min eriod, the 99% quantile of the 1 s measurements was determined. This quantile was assumed to indicate the minimum caacity C that is needed to fulfill the QoS requirement PrðAðT Þ P CT Þ 6 e, with e ¼ 1% and T =1s. The left grah in Fig. 6 results from measurements on 11 STM-1 links at various locations. Each location is reresented by a distinct color. For orientation uroses, the dotted line shows the unity (y = x) relation. It is remarkable how the 99% quantiles almost form a solid curve. We fitted a function q þ a ffiffiffi q, such that roughly 95% of the 99% quantiles are lower or equal to this 24 ADSL NETWORK 12 ADSL NETWORK 99% quantile load (in Mbit/s) % quantile measurement values ρ+ 1. ρ X=Y 99% quantile load (in Mbit/s) minute traffic load (Mbit/s) minute traffic load (Mbit/s) Fig. 6. Left: 99% maximum 1 s traffic as function of 5 min traffic mean. Right: synthesized traffic measurements for higher traffic volumes.

13 H. van den Berg et al. / Comuter Networks 5 (26) function. The reason for fitting an uer bound, instead of finding the function that gives the minimum least square deviation, is that eventually we intend to use this function for caacity lanning: then it is better to over estimate the required bandwidth than to underestimate it. The grah shows an extremely nice fit for the function C ¼ q þ 1: ffiffiffi q (with C and q exressed in Mbit/s). At the time of the measurements, the busiest STM-1Õs did not carry more traffic than 2 Mbit/ s during the busiest hours, so we could not verify that the found uer bound also holds for higher traffic volumes. To overcome this roblem at least artly, we synthesized artificial traffic measurements by taking the suerosition of the traffic measured on several (unrelated) STM-1Õs. The right grah of Fig. 6 shows the results of this exeriment. As exected on theoretical grounds, the fitted function C q ¼ q þ 1: ffiffiffi q remains valid College and camus network We have erformed similar exeriments in two other network environments, viz. a college network and a camus network, with essentially different characteristics than the ADSL network. In articular, in these alternative network environments the ratio of the access rate and link rate is relatively small, and, hence, one would exect that the M/G/1 modeling assumtion underlying the analysis in Section 2 is not valid anymore. The question is whether (or u to what extent) the bandwidth requirement formula (9) still alies. In the first scenario, we have measured a 1 Gbit/s link connecting a college network to the Internet. This link is shared by about 1 students and teachers, each having a 1 Mbit/s FastEthernet connection (a ratio of 1:1). In the second scenario, we have measured a 3 Mbit/s (trunked) link connecting an university camus (residential) network to the Internet. This link is shared by some 2 students, each of them having a 1 Mbit/s connection (a ratio of 1:3). Thus, theoretically, it takes only 1 or 3 users, resectively, to saturate the observed network links. The left grah of Fig. 7 shows the measurement results for the college network. As exected, the cloud of 99% quantiles of the 1 s traffic rate samles within 5 min intervals does not form such a nice ÔcurveÕ as in the revious (ADSL) scenario, but the tyical square-root behavior can still be recognized. For the university camus network, the Ôeak versus average loadõ is lotted in the right grah of Fig. 7. Note that the traffic in both directions has been aggregated during the measurements, which exlains that the link load as lotted in the grah is sometimes higher than the link caacity (which is one-way). Although the number of measurements available for the camus network is relatively low, we conclude from the grah that the relation between the average link loads and the 35 COLLEGE NETWORK 35 CAMPUS NETWORK 99% quantile load (Mbit/s) % quantile measurement values ρ+5. ρ X=Y 99% quantile load (Mbit/s) minute traffic load (Mbit/s) minute traffic load (in Mbit/s) Fig % maximum 1 s traffic as function of 5 min traffic mean. Left: college network. Right: camus residential network.

14 644 H. van den Berg et al. / Comuter Networks 5 (26) % quantiles of 1 s samles shows a similar behavior as in the college network. From the above results it is concluded that, as exected, for these alternative scenarios our model develoed in Section 2 does clearly not aly as well as for the ADSL scenario. Indeed, it may be exected that this is caused by the high access link seed, which leads to a ossibly high variability in the rate at which traffic is generated by the users in the alternative scenarios (while the M/G/1 model assumes that sources generate traffic at a fixed rate). Aarently, under these highly variable traffic conditions the 5 min average traffic rate does not rovide sufficient information to estimate the traffic behavior on much smaller time scales (i.e., more detailed information than just q is needed), and, consequently, other underlying traffic models should be alied Bandwidth rovisioning rocedure Our formula (9) for the required bandwidth C(T,e) can be used to develo bandwidth rovisioning rocedures. Obviously, a first ste in this rocedure is to verify whether the main M/G/1 modeling assumtions are satisfied in the network environment under consideration, such that formula (9) can indeed be alied. A next ste is then to estimate q and a. Clearly, q can be estimated through coarse traffic measurements, as it is just the average load; a, however, contains (detailed) traffic characteristics on time scale T (viz., the variance v(t)). In articular, as noticed in Section 2.3, a deends on the flow eak rate r and on the arameters of the flow duration D, but, imortantly, a does not deend on the flow arrival rate k; a can be considered as a characteristic of the individual flows. This ÔdichotomyÕ between q and a gives rise to efficient rovisioning rocedures. Consider the following two tyical situations: Situations in which there is a set of links, that differ (redominantly) in the number of connected users; across the links, the individual users have essentially the same tye of behavior (in terms of the distribution D and the access rate r). Then the a can be estimated by erforming detailed measurements at (a art of) the existing links. When a new link is connected, one could obtain an estimate ^q of the load by erforming coarse measurements (e.g., every 5 min, by using the MRTG tool [17]). Then the rovisioning rule ^q þ ^a ffiffiffi ^q can be used. An examle is the ADSL scenario described above, in which one could use ^a 1: to dimension a new link. Growth scenarios in which it is exected that the increase of traffic is (mainly) due to a growing number of subscribers (i.e., the k), while the user behavior remains unchanged. Here it suffices to erform infrequent detailed measurements at time scale T, yielding an estimate ^a of a. If a future load ^q is envisaged, the required bandwidth can be estimated by the rovisioning rule ^q þ ^a ffiffiffi ^q. The estimate of a has to be udated after a certain eriod (erhas in the order of months). This should corresond to the time at which it is exected that the ÔnatureÕ of the use of resources changes (due to, e.g., new alications, etc.). The exlicit formulas for a derived in Section 2 are also useful when examining the imact of changes in the user behavior or the QoS arameters. For instance, the imact of an ugrade of the access seed r can be evaluated. Also one could assess the effect of imosing a stronger of weaker erformance criterion e: when relacing ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 1 by e 2, the a needs to be multilied by log e 2 = log e 1. The measurement eriod of 5 min mentioned above for estimating the load q is motivated by the fact that this is the time scale on which measurements in an oerational network can be (and are) erformed on a routine basis. A higher frequency would be desirable, but this would ut a high load on the rocessing caacity of routers, the transort caacity of management links, etc., articularly if there are many routers and orts involved. On the other hand, measurements erformed at lower frequencies (for instance 1 to several hours) are too coarse, as traffic is not likely to be stationary over such long eriods. Therefore, 5 min will usually be a suitable trade-off, as this is feasible to measure, and at the same time a reasonable eriod during which the traffic can still be assumed stationary.

15 H. van den Berg et al. / Comuter Networks 5 (26) Concluding remarks and further research In this aer we have considered bandwidth rovisioning for IP network links. Our goal was to develo accurate and reliable rovisioning rocedures that require a minimal measurement effort. First we derived a formula for the minimally required link bandwidth C(T, e), such that the aggregate traffic rate (measured on a time scale T) exceeds the link rate only during a small fraction e of time. In articular, for the situation that the traffic is generated by eak rate constrained flows that arrive according to a Poisson rocess and remain active for some random time D (i.e., M/G/1 inut traffic) the resulting bandwidth rovisioning rule is of the form CðT ; eþ ¼q þ a ffiffiffi q ; here q is the traffic load, which can be estimated easily from coarse traffic measurements (tyically in the order of 5 min). Imortantly, the coefficient a is determined by characteristics of the individual flows, and does not deend on the flow arrival rate k. We have shown that this roerty oens u the ossibility of elementary (but adequate) rovisioning rocedures. The exlicit exression of a shows the imact of the flow size, eak rate and other traffic and system arameters on the required link bandwidth. In articular, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia lies somewhere between and ð 2 log eþr; its exact value deends mainly on the ratio of the time scale of interest T and the mean flow duration. Extensive numerical results show that C(T,e) is quite insensitive to the flow size distribution (aart from its mean value). The above rovisioning rule has been emirically validated through the analysis of extensive traffic measurements in three ractical scenarios: (i) an IP network connecting rivate and small business ADSL users to the Internet, (ii) a college network and (iii) a camus network. A articularly good corresondence with our theoretical results was found from the measurements in the IP network scenario, where the flow rates are bounded by relatively small ADSL access rates. The measurement results for the other scenarios showed, as exected, less good corresondence: the M/G/ 1 modeling assumtions are not really satisfied there; in articular the flow rates may be strongly variable due to the relatively high access rates (comared to the network link rates) in these scenarios. It remains for further research whether other underlying traffic models could be used to imrove the results for network environments like the college and camus network. An attractive alternative traffic model is the fractional Brownian motion (fbm) model, as used in e.g., [9]. Another toic for further research is the investigation of the validity of the stationarity assumtion in our modeling aroach. In articular, u to which time scale ^t can the traffic arrival rocess be assumed stationary? It is clear that the estimates of the average rate q should be measured at a time scale smaller than ^t. It should also be investigated in more detail under which conditions (and to what extent) the Gaussian traffic assumtion is valid, cf. Section 2.2. As a last toic for further research we mention the QoS criterion used in this aer, i.e., the fraction of time e that the aggregate offered traffic rate (measured at time scale T) is restricted by the link rate. In articular, we could obtain more insight in the relation between this QoS criterion, which we used as link bandwidth rovisioning objective, and the actual QoS that the users are offered. In other words: to what extent does this criterion actually determine the duration of a congestion eriod (this will deend on the traffic characteristics, in articular the flow-level dynamics)? What are aroriate choices of T and e for different (TCP) alication tyes (file downloading, interactive web browsing, etc.)? Acknowledgement This work was (artly) carried out within the EQUANET roject, an ÔICT-doorbraakrojectÕ which is suorted by the Dutch Ministry of Economic Affairs via its agency Senter. References [1] R. Addie, P. Mannersalo, I. Norros, Most robable aths and erformance formulae for buffers with Gaussian inut traffic, Euroean Transactions on Telecommunications 13 (22)

16 646 H. van den Berg et al. / Comuter Networks 5 (26) [2] R. Addie, T. Neame, M. Zukerman, Performance evaluation of a queue fed by a Poisson Pareto burst rocess, Comuter Networks 4 (22) [3] S. Ben Fredj, T. Bonald, A. Proutière, G. Régnié, J.W. Roberts, Statistical bandwidth sharing: a study of congestion at flow-level, in: Proceedings SIGCOMM Õ1, San Diego CA, USA, 21. [4] T. Bonald, J.W. Roberts, Congestion at flow level and the imact of user behaviour, Comuter Networks 42 (23) [5] M. Crovella, A. Bestavros, Self-similarity in World Wide Web traffic: evidence and ossible causes, IEEE/ACM Transactions on Networking 5 (1997) [6] A. Elwalid, D. Mitra, Effective bandwidth of general Markovian traffic sources and admission control of high seed networks, IEEE/ACM Transactions on Networking 1 (1993) [7] A. Elwalid, D. Mitra, R. Wentworth, A new aroach for allocating buffers and bandwidth to heterogeneous, regulated traffic in an ATM Node, IEEE Journal on Selected Areas in Communications 13 (1995) [8] M. Fiedler, A. Arvidsson, A resource allocation law to satisfy QoS demands on ATM burst and connection level. Technical Reort 6, COST-257, [9] C. Fraleigh, F. Tobagi, C. Diot, Provisioning IP backbone networks to suort latency sensitive traffic, in: Proceedings IEEE INFOCOM 23, San Francisco CA, USA, 23. [1] R. Guérin, H. Ahmadi, M. Naghshineh, Equivalent caacity and its alication to bandwidth allocation in high-seed networks, IEEE Journal on Selected Areas in Communications 9 (1991) [11] F. Kelly. Notes on effective bandwidths, in: F. Kelly, S. Zachary, I. Ziedins (Eds.), Stochastic Networks: Theory and Alications, [12] F. Kelly, Mathematical modeling of the Internet, in: B Enquist, W Schmidt (Eds.), Mathematics Unlimited 21 and Beyond, Sringer-Verlag, Berlin, Germany, 21. [13] J. Kili, I. Norros, Testing the Gaussian aroximation of aggregate traffic. Proceedings Internet Measurement Worksho, Marseille, France, 22. Available from: <htt:// [14] M. Mandjes, I. Saniee, A. Stolyar, Load characterization, overload rediction, and load anomaly detection for voice over IP traffic. in: Proceedings 38th Allerton Conference, Urbana-Chamaign, US, 2, [15] M. Mandjes, M. Van Uitert, Samle-ath large deviations for tandem and riority queues with Gaussian inuts. CWI reort PNA-R221. Available from: <htt:// ft/cwireorts/pna/pna-r221.df>, 22. Acceted for ublication in Annals of Alied Probability. A short version aeared in Proceedings ITC, Berlin, Germany, 18, 23, [16] I. Norros, On the use of fractional Brownian motion in the theory of connectionless networks, IEEE Journal on Selected Areas in Communications 13 (1995) [17] T. Oetiker, MRTG: Multi Router Traffic Graher, Available from: <htt://eole.ee.ethz.ch/oetiker/webtools/ mrtg/>. [18] M. Parulekar, A. Makowski, M/G/1 inut rocesses: a versatile class of models for network traffic, Proceedings IEEE INFOCOM (1997) [19] V. Paxson, S. Floyd, Wide-area traffic: the failure of Poisson modeling, IEEE/ACM Transactions on Networking 3 (1995) [2] J. Roberts, U. Mocci, J. Virtamo, Broadband network teletraffic. Final reort of action COST 242, Sringer, Berlin, [21] H.C. Tijms, A First Course in Stochastic Models, Wiley, New York, 23. Hans van den Berg received the M.Sc. and Ph.D. degree in mathematics (stochastic oerations research) from the University of Utrecht, The Netherlands, in 1986 and 199 resectively. From 1986, he worked at the Centre for Mathematics and Comuter Science (CWI), Amsterdam. In 199, he joined KPN Research (now TNO Information and Communication Technology, since January 23). He is articularly working on erformance analysis, traffic management and QoS rovisioning for wired and wireless multi-service communication networks (ATM, IP, UMTS, WLAN, ad-hoc networks). Currently, he is a Senior Research Member and leader of the QoS grou within TNO ICT, Delft, The Netherlands. Since July 23 Hans van den Berg has a art-time osition as full rofessor within the faculty of Electrical Engineering, Mathematics and Comuter Science at the University of Twente. Michel Mandjes received M.Sc. (in both mathematics and econometrics) and Ph.D. degrees from the Vrije Universiteit Amsterdam (VU), the Netherlands. After having worked as a member of technical staff at KPN Research and Bell Laboratories/ Lucent Technologies (Murray Hill NJ, USA) and as a art-time full rofessor at the University of Twente, he currently has a joint osition as deartment head at the Centre for Mathematics and Comuter Science (CWI), Amsterdam, and as full rofessor at the University of Amsterdam, the Netherlands. His research interests include large deviations analysis of multilexing systems, queueing theory, Gaussian traffic models, traffic management and control in IP networks, and ricing in multi-service networks.

17 H. van den Berg et al. / Comuter Networks 5 (26) Remco van de Meent received his M.Sc. in Comuter Science from the University of Twente, The Netherlands, in 21. He currently is working towards his Ph.D. at the deartment of Electrical Engineering, Mathematics and Comuter Science at the University of Twente. His research interests include Internet traffic measurements and traffic modeling, network rovisioning and network management. Aiko Pras is associate rofessor at the University of Twente (UT), The Netherlands. From this university he received in 1995 a Ph.D. degree for the thesis: Ônetwork management architecturesõ. His current research interests include network management technologies, Web Services, network measurements and accounting. He is member of the IRTF NMRG, technical co-chair of the Ninth IFIP/IEEE International Symosium on Integrated Network Management (IM 25), and TPC member of many international conferences in the area of management. Frank Roijers received his M.Sc. degree in Business Mathematics and Informatics from the Vrije Universiteit Amsterdam (VU) in In 2 he joined TNO Information and Communication Technology (TNO ICT), the former KPN Research. His research interests include erformance analysis and traffic engineering of communication networks (IP, Wireless LAN, adhoc networks) and queueing theory. Since 24 he is working towards a Ph.D. in a joint roject of TNO ICT and the Centre for Mathematics and Comuter Science (CWI), Amsterdam. Pieter Venemans graduated in 1985 from Delft University of Technology, deartment of Electrical Engineering. From 1985 till 199 he worked as researcher at the same University in the field of formal methods for rotocol secification. In 199 he joined KPN Research (now TNO Information and Communication Technology). The ast 5 years he has been working in the field of erformance engineering and quality of service for ATM, IP, MPLS and mobile networks. He has a long exerience in measuring, analysing and modelling real-life traffic in oerational networks, as well as in the design of methods for caacity and erformance management.