Communications, IEEE Transactions on - Semantic Scholar

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 6, JUNE 2001

Rate Control of Elastic Connections Competing with Long-Range Dependent Network Traffic Sven A. M. Östring, Student Member, IEEE, Harsha R. Sirisena, Member, IEEE, and Irene Hudson

Abstract—Long-range dependence is regarded as a fundamental property of network traffic. Using an original approach, we incorporate this property in a traffic control mechanism for elastic connections that can adapt to the instantaneous network load in a differentiated services-type framework. In this scenario, the network makes predictions of bandwidth requirements of the high-priority traffic and returns feedback information to the elastic source. We include a prediction compensation algorithm that compensates for the larger prediction errors for connections with longer roundtrip delay, and analyze the performance of this algorithm. The specific topology involved in traffic control for differentiated services is thus harnessed, together with the long-range dependence, to improve network performance, thereby counteracting the undesirable characteristics of self-similarity. Furthermore, an adaptive version of the rate-based control algorithm is studied, based on the use of real-time estimates of traffic parameters, including the mean, variance, and Hurst parameter. Index Terms—Adaptive control, available bandwidth utilization, differentiated services, long-range dependence, traffic control.

I. INTRODUCTION

T

RAFFIC control is an important aspect of effectively managing elastic network traffic so that available bandwidth is utilized and degradation of the quality of higher priority connections is avoided. Examples of elastic traffic include available bit rate (ABR) traffic in asynchronous transfer mode (ATM) networks and TCP/IP traffic using the best effort service in the IP DiffServ framework. Conceptually, the network returns feedback information to elastic sources regarding the state of the network, and the sources adapt their transmission rates accordingly. Due to the roundtrip delay that is experienced by elastic connections, the traffic control mechanisms need to predict the future bandwidth requirements of the high-priority traffic. To do this well, optimal predictors must be developed using accurate models of the background network traffic. There is significant evidence that a wide range of network traffic is long-range dependent, including data traffic like Ethernet and World Wide Web traffic, and real-time video Paper approved by G. P. O’Reilly, the Editor for Communications Switching of the IEEE Communications Society. Manuscript received March 10, 2000; revised November 1, 2000. This work was supported by the Royal Society of New Zealand RHT Bates Postgraduate Scholarship and Telecom New Zealand under Research Grant E4567. This work was presented in part at the International Conference on Communications (ICC), Vancouver, BC, Canada, June 1999. S. A. M. Östring and H. R. Sirisena are with the Department of Electrical and Electronic Engineering, University of Canterbury, Christchurch, New Zealand (e-mail: [email protected]; [email protected]). I. Hudson is with the Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand (e-mail: [email protected]). Publisher Item Identifier S 0090-6778(01)04883-8.

traffic like video-streaming and video conferencing traffic [1]–[3]. Most research into control mechanisms for elastic traffic has ignored the long-range dependence of network traffic [4]–[8], and the transmission rates were determined without reference to the correlation structure of the traffic. Tuan and Park’s multiple-time scale congestion control algorithm [9] is a notable exception. Their proposal modifies the well-known linear increase/multiplicative decrease concept used in the TCP algorithm so that long-term trends in network traffic are accounted for. However, they did not explicitly use the correlation structure of the long-range dependent traffic. When the Hurst parameter, which is the index of self-similarity, of network , this signifies that the traffic is in the interval traffic is long-range dependent. The correlations within the data decay hyperbolically rather than exponentially, so that there is significantly more information in past traffic states, regarding the current traffic level, than for a short-range dependent process. This property of long-range dependence implies that self-similar processes can be predicted more accurately, which Gripenberg and Norros demonstrated in their development of a predictor for fractional Brownian motion [10]. We have used predictions based on discrete-time traffic measurements for developing rate-based congestion control algorithms for the ABR service [11] and shown the performance advantages of this type of controller. In this paper, we define a rate-based control approach for elastic traffic which incorporates the structure of long-range dependent traffic, and analyze the approach thoroughly. Thus, it is demonstrated that the long-range dependent structure can be utilized to improve the performance of the network. This is an area that has not been addressed as extensively as studies that have reported the undesirable effects of the burstiness of self-similar traffic. In Section II, the model for the traffic control algorithm is introduced and the concepts of self-similarity and its prediction are summarized. Our implementation of the algorithm is described in Section III, together with the analysis of a prediction compensation algorithm which we developed. An adaptive version of our controller is considered in Section IV, which uses real-time estimates of the traffic parameters, and our conclusions are drawn in Section V. II. SYSTEM AND BACKGROUND TRAFFIC MODELING A. Modeling Reactive Traffic Control We consider a set of controlled sources that each receive feedback information from the network regarding the explicit bandwidth available for that particular elastic connection. This model

0090–6778/01$10.00 © 2001 IEEE

ÖSTRING et al.: RATE CONTROL OF ELASTIC CONNECTIONS COMPETING WITH LONG-RANGE DEPENDENT NETWORK TRAFFIC

is primarily applicable to the ABR service in ATM networks. The approach adopted in our research is to determine optimal predictions of future traffic levels, and calculate the transmission rates for the controlled elastic sources from these predictions. Our state variable is the deviation of the total input workload around a target utilization of the outgoing link, similarly as in Zhao et al. [8]. The link bandwidth is utilized by the elastic connections, which consume the rate-controlled band, and the background traffic . width is made up of the The total rate-controlled bandwidth ratesummation of the individual bandwidths used by the controlled connections. Each connection has its own roundtrip through the network, so we can now write the state delay equation as (1) is defined as Furthermore, the variable the available bandwidth that is allocated to the rate-controlled sources, thus resulting in the equation (2) Equation (2) is in the form of a multi-input single-output (MISO) system and the aim of the controller is to determine the so that the variance is minimized. Howinputs ever, a control system for an MISO system is computationally too expensive. We simplify the system by sharing the available bandwidth among the rate-controlled connections using weights , where . The weights are arbitrary and specified when a new connection is admitted, but it should be noted that this framework includes the case where the available bandwidth ) at the congested link. Thus, is shared equally (i.e., we define the available bandwidth for a particular elastic source (3) The system now becomes a collection of with their own controller

subsystems, each

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B. Self-Similar Models for Background Traffic There is significant statistical evidence that a wide range of classes of network traffic is self-similar in nature. This means that there is no natural length of the bursts in the traffic, and the traffic remains bursty over a range of time scales (hence the term “self-similar”). The Hurst parameter is the index of self-similarity of a process, and a process is categorized as long-range dependent (LRD) when the parameter lies in the in. Long-range dependence means that the terval correlations within a process decrease hyperbolically rather than exponentially, so that the autocovariance function for an LRD process is nonsummable. While the burstiness of LRD traffic can cause buffer overflows and losses, long-range dependence can be used to one’s advantage in terms of prediction [10], [11]. These large long-term correlations mean that there is significantly more information within previous states regarding the current state in LRD processes than in short-range dependent (SRD) processes, and more accurate predictions can be achieved from appropriately filtering stored measurements of the process in the past. There are a number of well-known models used for processes which display self-similarity. Fractional Brownian motion is the canonical example of a self-similar process, and its incremental process [called fractional Gaussian noise (fGn)] has the autocovariance function

as

(6)

where the asymptotic behavior of the autocovariance function shows that the process is long-range dependent. A self-similar process can be parsimoniously represented by an fGn model, if the mean, variance, and Hurst parameter of the process are known. Another important class of self-similar models is the fractional ARIMA family of models, which are a natural exmodels where the differencing tension of the ARIMA parameter is allowed to assume fractional values. Fractional ARIMA processes have the particular advantage of modeling both short-range and long-range dependence. C. Optimal Prediction of Self-Similar Processes

(4) where the roundtrip delays have been ordered such that without loss of generality. . This is equivOur control aim now is to minimize for all . Taking the alent (refer to [12]) to requiring , we have the following expectation of (4) and setting general control law for each subsystem: (5) Thus, we require the allowed rates of the individual source rates to be equal to the predicted values of the system parame, which are determined by the amount of bandters width available in the outgoing link. Using the long-range dependence of the network traffic, we can predict the available bandwidth more accurately.

As we have defined our system model in Section II-A, we require the prediction of the background network traffic at a particular node to determine the desired rates of the controlled sources. This information experiences a delay in the network before the effects can be observed at the same node, so we require a -step predictor. Using the correlation structure of the can be formulated as traffic, the optimal linear predictor (7) is a covariance stationary stochastic process where , and autocovariance function with zero mean, variance , and is the vector of stored traffic measurements . is the memory-length of the predictor. The solution is given by [13] (8)

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where


is the

covariance matrix, is -values of the autocovariance function starting at lag . The variance of the prediction errors is given by (9) III. IMPLEMENTATION OF THE RATE CONTROL ALGORITHM FOR SELF-SIMILAR TRAFFIC The rate-based control algorithm is implemented as follows. At the kernel of the algorithm is a discrete-time predictor that uses measurements of the background traffic to predict future bandwidth requirements of the high-priority traffic. The predicted background traffic level is given by

Since we are predicting the background traffic level based on the delay, the rate calculations for the low latency connections will be more accurate, especially as we know the rates that were previously allocated to the high latency connections. Assuming that the desired rates are stored for policing purposes, a prediction compensation algorithm can be developed which adjusts the rates for the short latency connections based on previous rate calculations for the long latency connections. In the derivation which follows, the focus is on the prediction compensation component for source when its desired rate is calculated at time . The partial controlled rate from the elastic sources (the proportion of the total controlled rate known at time ) arriving at the node after time slots is determined from the stored explicit rate values

(10)

(12)

is the vector of samples of the level of where background traffic updated at each calculation instant and . is assumed to be a stationary stochastic process with mean , variance , and autocovariance . The desired rates for the controlled sources are function calculated using

is the partial controlled rate. This is the total rate from , which have elastic sources with . It is previously been allocated bandwidth for time slot assumed that the transmission of these sources is at their specified allowed rate. In reality, the source rates may transmit below their allowed rate, so (12) is the worst-case scenario. for source is also calculated at time The optimal rate using the predicted available bandwidth. Since this prediction of is more accurate the available bandwidth for the time slot than the predicted values used for sources with longer roundtrip delay, we can determine what the optimal partial controlled rate should be

(11) where we have assumed that the available bandwidth is shared debetween the controlled connections using the weights fined previously. We further introduce a prediction compensation algorithm which compensates for larger prediction errors in calculating the rates for connections with longer roundtrip delay by adjusting the rates for the connections with shorter delay, which is described in the following section. The assumption of stationarity deserves further attention at this point. By definition, long-range dependence is an asymptotic property which is not affected by short-term time variations but rather describes the trend in the relationship between correlations as the lag increases indefinitely. There are valid arguments for modeling data either as a long-range dependent process or using a short-memory model with nonstationary mean [14]. However, the modeling approaches, from a practical perspective, appear to differ in philosophy rather than in functionality. A key feature of long-range dependent models is their practical value in parsimoniously modeling a process. In addition, estimators of the Hurst parameter based on the wavelet transform can disassociate nonstationarities in the mean and variance from estimating the long-range dependence [15], thus identifying long-range dependent processes with more precision. Thus, there are ways of isolating the nonstationarities that occur within the data, and assuming stationarity is a valid starting point. Later in this paper, we relax the assumption of stationarity and investigate adaptive methods of handling variations in the traffic parameters. A. Prediction Compensation Algorithm For a specific time slot, the desired rate for connections with longer roundtrip delay (high latency) will have been calculated prior to the rates for connections for shorter delay (low latency).

(13) where we have introduced the notation . It should be noted that since the summation of all the weights is equal to one. We can compensate for the deviation of the partial controlled rate from its optimal by providing a prediction compensation component for source rate , calculated as

(14) . This formula assumes that where sources have been ordered with decreasing network delay, and that the predicted rates are calculated following that order. In addition, the deviation from the optimum has been shared between the source and the elastic sources which have not yet been al, based on the weights located bandwidth for time slot for the remaining sources. This concept allocates the higher frequency proportion of dynamically available bandwidth to the elastic connections with smaller roundtrip delay. This is similar to the principle used by


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Hu [16], which is that dynamically available bandwidth should be allocated only to those connections that can use it in time. We introduce a higher level of granularity, with the explicit rate (ER) value for elastic source calculated as

where the entries of the matrix are given by , where is the autocorrelation function. Next, the system using compensated explicate rate information is considered. The prediction compensation algorithm requires the sources to be ordered in decreasing roundtrip delay, so that rate-controlled sources with smaller roundtrip delay can compensate for connections with larger delay. In this case, the explicit rate information can be derived from (15) in terms of the predicted values of

(15)

(21)

B. Analytical Performance of the Prediction Compensation Algorithm

where the superscript denotes the system using prediction compensation. The total input to the network node is now

The performance of the prediction compensation algorithm is analyzed as follows. The background traffic is represented as (16) (22)

has the same correlation structure of the backwhere ground traffic but with zero mean. As a result, future predictions of the traffic are given by

The proofs for (21) and (22) are found in the Appendix. The , as before, but the variance is now mean of the input is

(17)

(23)

and is the vector of where the predictor stored traffic level measurements. Firstly, we consider the system with uncompensated explicit rate calculations. The system nodes return explicit rate information to the rate-controlled sources using (11)

This is equivalent to the variance of the prediction error of the connection with the smallest roundtrip delay . From these results, we can define the performance of the prediction compensation algorithm using the relative variance from (20) and (23): Theorem 1: The relative variance of the input to the system using the prediction compensation algorithm compared with the system without prediction compensation is

(18) The superscript refers to the uncompensated system that is being analyzed. Using this result, the total input rate to the network node can be determined as follows: (24)

(19) is the predicted value of with a -step prewhere diction horizon. Taking the expected value of the input rate gives , as desired, and the variance of the the mean input rate as input rate is (the proof is found in the Appendix)

(20)

Using fGn as a stochastic model for the background traffic, the surface of the relative variance as the number of rate-controlled sources and the Hurst parameter is varied is shown in Fig. 1. As additional rate-controlled sources are added to the system, the connection associated with the new source has a roundtrip delay which is proportional to the total number of sources. Thus, the range of roundtrip delays increases in proportion to the number of sources. Two main conclusions can be drawn from the figure. First, there is no performance gain from incorporating the prediction compensation algorithm with traffic which has Hurst parameter close to 0.5. This is due to the independence of future traffic levels on previous measurements (after any short-range dependence has been taken into account), and predictions with smaller horizons have no greater accuracy than predictions required for connections with greater roundtrip delay. The second conclusion is that the advantage of using the

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Fig. 1. The performance of the prediction compensation algorithm.

prediction compensation algorithm becomes more significant as the self-similarity of the background network traffic increases with a large number of rate-controlled sources having a wide range of roundtrip delays. C. Performance Results and Comparison We further investigate the performance of the proposed rate control algorithm using a simple simulation model. The system consists of six rate-controlled sources with roundtrip delays and a VBR source, representing the highpriority traffic, which are competing for the use of a congested link. The VBR source is a data trace formed by aggregating ten Star Wars MPEG video traces that have been randomly shifted in time, where the original MPEG stream was filtered to remove the correlations resulting from the GOP frame sequence. A fracmodel was identified for the resulting tional ARIMA data series, and the model parameters were estimated giving the following equation: (25) is a white noise process with zero mean and variance . It should be noted that this model is used for evaluating the performance of the predictive rate-control algorithm proposed in this paper and is not intended for use within an on-line algorithm. The switch has memory allocation for 64 samples of background traffic. The link capacity is set at 7.48 Mbits/s, which means that the background traffic is using, on average, 50% of the link capacity, and the target utilization of the system is 90%. The individual controlled source rates are shown in Fig. 2. The higher variability for short-latency connections is due both to the optimal filter which achieves better prediction of the background traffic, and also the prediction compensation algorithm

where

which adjusts these connections to compensate for larger prediction errors in the rate calculations for the long-latency connections. The performance of our rate control scheme is also compared in Fig. 3 with the ABR control scheme proposed by Zhao et al. [8], which basically low-pass filters (LPFs) the background network traffic without prediction, and the simulations show that our control scheme results in much lower buffer occupancy. It is well known that the effect of self-similar traffic on the performance of an uncontrolled finite buffer in a network system is that cell loss rate decreases hyperbolically with buffer size. This means that significant increases in the buffer size do not achieve the desired quality-of-service objectives. The comparison of the cell loss ratios of the two control schemes for different link utilizations is shown in Fig. 4. It can be observed from Fig. 4 that the LPF scheme of Zhao et al. does not adequately address this effect of self-similarity. Our rate control scheme, however, does compensate for the effects of the self-similar traffic by incorporating the underlying stochastic structure of the self-similar traffic into the prediction method. We also study the performance of the prediction compensation algorithm, which was shown in Section III-B to result in significant gain when used with large number of elastic sources in the presence of network traffic which was highly self-similar. The maximum queue length for the system is shown in Fig. 5 and compared with the equivalent system which does not use the prediction compensation algorithm. The algorithm effectively replaces the cumulative prediction errors of all the sources with the prediction error of the lowest latency connection acting alone, as can be seen from the variance of the total input traffic given in (23). This is also demonstrated in the simulation results, where there is only a minimal increase in the maximum buffer occupancy as the number of sources are increased. This is a significant performance advantage, because the maximum


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Fig. 2. Individual source rates.

Fig. 3. Comparison between the buffer requirements of the rate control algorithm designed for self-similar traffic and LPF schemes (

queue length increases rapidly without the prediction compensation algorithm. IV. ADAPTIVE CONTROL FOR NONSTATIONARITIES IN TRAFFIC We now consider a simple adaptive version of our predictive rate-controller which utilizes real-time estimates of the traffic parameters. Adaptive control is necessary in commercial implementations of traffic control mechanisms due to the time variations that occur within the traffic. These nonstationarities

= 0:9).

are caused by admission of new connections (or completion of existing connections) which results in a change in the mean traffic level. There is also a potential change in the burstiness of the traffic due to the type of data being transferred or the transmission control protocol that a new connection is utilizing, hence the need to update the traffic parameters within the controller. On-line estimation of the mean can be achieved using the standard sample mean. While long-range dependence increases the variance of the mean estimates, the performance of the

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Fig. 4.


Comparison between the cell loss ratios of the self-similar rate control scheme (solid) and LPF scheme (dashed) for different link utilizations .

Fig. 5. The performance of the prediction compensation algorithm as the number of sources is increased. The solid line shows maximum queue length of system which includes the prediction compensation algorithm, and the dashed line is the system which does not compensate for the prediction errors.

sample mean is close to optimal [17], and it is very simple to implement as an on-line estimator. With regard to estimating the Hurst parameter and the variance, we incorporate the real-time version of the Abry–Veitch estimator that uses the wavelet transform to decorrelate the long-range dependence within traffic data [15]. Since the wavelet transform can be computed very efficiently using the fast pyramid algorithm, the on-line formulation of the estimator can done with relative ease

[18], and the estimator can be used for jointly estimating the variance and the Hurst parameter. The performance of the adaptive algorithm which uses real-time measurements of the mean, variance, and Hurst parameter is compared with two other rate control approaches in Table I—the LPF algorithm considered previously in this paper and ERICA, an ABR traffic management algorithm being considered by a number of ATM switch manufacturers


PERFORMANCE PARAMETERS

OF THE

TABLE I ADAPTIVE SYSTEM USING REAL-TIME ESTIMATION THE LPF ALGORITHM AND ERICA

[19]. For our adaptive algorithm, the mean is estimated every 2 traffic samples, and the Hurst parameter and variance are updated every 2 samples. These block sizes were chosen after considering the performance of the adaptive system for a range of different block sizes. We also use two data sets—the Star Wars data set that we have already discussed in this paper, and the BC-pOct89 Ethernet trace that is one of the original data sets used to demonstrate the self-similarity of network traffic [1]. This second data set is of particular interest in this study, since it has been noted by other researchers that the trace has a level shift in mean. As we did previously with the Star Wars set, we aggregated ten copies of this trace that have been randomly shifted in time which means that there are ten shifts in the mean in the final trace. Two different simulation scenarios were considered, one with a buffer of infinite capacity and the other using a finite buffer with a maximum capacity of 100 cells. Comparing the performance of the adaptive algorithm with the other rate-control algorithms, we observe that our approach has a higher performance for all of the parameters of interest. Even though the performance of ERICA comes closer to our algorithm with respect to buffer occupancy, this comes at the cost of higher cell loss rates. Thus, we can conclude that our algorithm which incorporates simple real-time parameter estimation techniques will result in better control of elastic connections. V. CONCLUSIONS The self-similar nature of network traffic has been convincingly demonstrated in a number of research studies, and if it is ignored, this property affects the performance of networks resulting in higher delays and lower throughput. Our research is

OF

TRAFFIC PARAMETERS, COMPARED

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WITH

the first study addressing this issue for traffic control required within a differentiated services framework, where the rate control mechanisms have been developed that specifically utilize the significant long-term correlations of LRD traffic. The additional information from differentiating the latency of elastic connections has also been exploited, given that there is higher predictability for long-range dependent traffic when the prediction horizon is smaller. Thus, we have demonstrated that the characteristics of self-similarity, which have previously been considered undesirable, can be successfully included in traffic control mechanisms with an improvement in performance. Our rate-based control mechanism for elastic connections is summarized as follows. Control information is generated by predicting future traffic levels of the high-priority traffic using the correlation structure of long-range dependence, and returned to the elastic sources. Prediction errors for connections with larger roundtrip delays are compensated for by the connections with shorter roundtrip delays, and we have shown that the performance of this algorithm increases with the number of elastic sources and the self-similarity of the high-priority traffic. An adaptive version of this control algorithm has been investigated, using real-time estimates of the self-similar traffic parameters. The overall conclusion is that the properties of self-similarity can be successfully used within traffic control algorithms to enhance the overall performance of the network. Future work includes extending the control concepts presented to ensure the fairness of this algorithm when multiple links are involved. We are also interested in pursuing prediction based on the wavelet transform which can decorrelate the long-range dependent traffic data. The predictor would then predict the wavelet coefficients at each level based on simple short-range dependent models.

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For we assume that (21) holds for the explicit rate for . First, we source and derive the explicit rate for source . prove that the case for

APPENDIX A. Proof of the Analytical Performance of the Prediction Compensation Algorithm The total input rate without using the prediction compensation algorithm (19) (26) The variance of

is derived as

(29) as required. Now we assume that (27) as required, where the matrix is given in Section III-B. This concludes the proof of the results required for the system without prediction compensation. To derive the explicit rate values for the system with prediction compensation, we use an inductive method. The case where (the connection with the largest delay) is a special case, which needs to be handled separately

(28)

(30)

holds for source . The explicit rate value for the subsequent can be calculated using (15), as shown in (31), at source . the bottom of the page, since for any . This shows that (21) holds for all

(31)


The proof of result for the total input traffic for the prediction compensated system follows a similar reasoning

(32) since of

is

and . The expected value , and the variance is found using (9) (33) ACKNOWLEDGMENT

The authors would like to thank M. Roughan and D. Veitch for making their on-line Hurst parameter estimation software available during S. A. M. Östring’s visit to SERC, Royal Melbourne Institute of Technology. REFERENCES [1] W. Leland, M. Taqqu, W. Willinger, and D. Wilson, “On the self-similar nature of ethernet traffic (extended version),” IEEE/ACM Trans. Networking, vol. 2, pp. 1–15, Feb. 1994. [2] J. Beran, R. Sherman, M. Taqqu, and W. Willinger, “Long-range dependence in variable-bit-rate video traffic,” IEEE Trans. Commun., vol. 43, pp. 1566–1579, Feb/Mar/April 1995. [3] M. Crovella and A. Bestavros, “Self-similarity in world wide web traffic: Evidence and possible causes,” IEEE/ACM Trans. Networking, vol. 5, pp. 835–846, Dec. 1997. [4] L. Benmohamed and L. Meerkov, “Feedback control of congestion in packet switching networks: The case of a single congested node,” IEEE/ACM Trans. Networking, vol. 1, pp. 693–708, Dec. 1993. [5] E. Altman and T. Bas¸ar, “Multiuser rate-based flow control,” IEEE Trans. Commun., vol. 46, pp. 940–949, July 1998. [6] R. J. Gibbens and F. P. Kelly, “Resource pricing and the evolution of congestion control,” Automatica, vol. 35, pp. 1969–1985, 1999. [7] S. Low and D. Lapsley, “Optimization flow control I: Basic algorithm and convergence,” IEEE/ACM Trans. Networking, vol. 7, pp. 861–874, Dec. 1998. [8] Y. Zhao, S. Q. Li, and S. Sigarto, “A linear dynamic model for design of stable explicit-rate ABR control scheme,” in Proc. INFOCOM’97, Apr. 1997, pp. 283–292. [9] T. Tuan and K. Park, “Multiple time scale congestion control for self-similar network traffic,” Perform. Eval., vol. 36–37, no. 1–4, pp. 359–386, 1999. [10] G. Gripenberg and I. Norros, “On the prediction of fractional brownian motion,” J. Appl. Prob., vol. 33, pp. 400–410, 1996. [11] S. Östring, H. Sirisena, and I. Hudson, “Dual dimensional ABR control scheme using predictive filtering of self-similar traffic,” in Proc. ICC’99, June 1999. [12] K. Åström and B. Wittenmark, Computer-Controlled Systems Theory and Design, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1997. [13] P. Brockwell and R. Davis, Time Series: Theory and Methods, 2nd ed. New York: Springer-Verlag, 1991. [14] V. Klemeˇs, “The Hurst phenomenon: A puzzle?,” Water Resources Res., vol. 10, no. 4, pp. 675–688, 1974. [15] P. Abry and D. Veitch, “Wavelet analysis of long-range-dependent traffic,” IEEE Trans. Inform. Theory, vol. 44, pp. 2–15, Jan. 1998.

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[16] L. C. Hu, W. K. Tsai, Y. Kim, and M. Iyer, “Temporal flow control for ABR traffic management in integrated networks,” in Proc. GLOBECOM’98, Nov. 1998. [17] J. Beran, Statistics for Long-Memory Processes. New York: Chapman & Hall, 1994. [18] M. Roughan, D. Veitch, and P. Abry, “Real-time estimation of the parameters of long-range dependence,” IEEE/ACM Trans. Networking, vol. 8, pp. 467–478, Aug. 2000. [19] S. Kalyanaraman, R. Jain, S. Fahmy, R. Goyal, and B. Vandalore, “The ERICA switch algorithm for ABR traffic management in ATM networks,” IEEE/ACM Trans. Networking, vol. 8, pp. 87–99, Feb. 2000.

Sven A. M. Östring (S’95) was born in Hong Kong in 1974. He received the B.E.(Hons) and Ph.D. degrees in electrical engineering from the University of Canterbury, Christchurch, New Zealand, in 1996 and 2001, respectively. He has held visiting research positions with SERC, Royal Melbourne Institute of Technology, Melbourne, Australia, in 1999, and INRIA, Sophia-Antipolis, France, in 2000. He was a Research Engineer at Industrial Research Ltd., Christchurch, New Zealand. His research interests include traffic management algorithms in integrated broad-band networks, self-similar traffic modeling, and wavelet analysis of network traffic. Dr. Östring is a student member of ACM. Harsha R. Sirisena (M’78) received the B.Sc.(Eng) degree in electrical engineering from the University of Ceylon, Sri Lanka, in 1964, and the Ph.D. degree in control engineering from the University of Cambridge, Cambridge, U.K., in 1968. From 1964 to 1965, he was an Electrical Engineer with the Government Electricity Department, Sri Lanka, and between 1968 and 1971, he was a Lecturer in Electrical Engineering at the University of Ceylon. Since 1971, he has been with the Department of Electrical and Electronic Engineering, University of Canterbury, Christchurch, New Zealand, where he is currently an Associate Professor. He has held visiting academic positions at the Universities of Lund and Minnesota, the Australian National University, and the National University of Singapore. His research interests are mainly in networking, including control-theoretic analysis and design of congestion control schemes, micromobility and TCP performance in wireless IP networks and IPv6 transition mechanisms. Dr. Sirisena is a member of IEE. Irene Hudson was born in the United Kingdom in 1953. She received the B.Sc.(Hons) degree from the University of Adelaide, Adelaide, Australia, in 1974, the M.Sc. degree from the Australian National University in 1976, the diploma in mathematics and statistics from Cambridge University in 1977, and the Ph.D. degree from La Trobe University, all in mathematics and statistics. She is currently a Senior Lecturer and, since October 2000, has been the Head of the Statistics Section in the Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand. She also holds an Adjunct Research Fellowship at the School of Forestry, University of Melbourne. She is the Director of the Research Centre for Health Care Technology NZ (CHCTnz). Previously, she was a Medical Research Council Statistician at the Department of Animal Behavior, Cambridge University, from 1977 to 1978, a Statistics Tutor, Monash University, from 1982 to 1983, the Head of the Biostatistics Unit at the Department of Paediatrics, University of Melbourne, from 1983 to 1989, a Senior Statistical Consultant at the Department of Mathematics and Statistics, University of Melbourne, from 1990 to 1994, and a Sub Program Leader of the Wood Fiber CRC in the School of Forestry, University of Melbourne, from 1994 to 1997. She has published 60 journal papers, one book, and 11 conference papers. Her research interests include spatial statistics, image analysis, forestry and biological statistics, psychometrics, transient state modeling, biostatistics and epidemiology, public health and bioengineering. Dr. Hudson is a member of the Australian and New Zealand Statistics Association, Appita, the International Biometrics Society, Australasian Epidemiology Association, and the International Association Wood Anatomists.