Fuzzy adaptive predictive flow control of atm ... - Semantic Scholar

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 4, AUGUST 2003

Fuzzy Adaptive Predictive Flow Control of ATM Network Traffic Bor-Sen Chen, Fellow, IEEE, Yu-Shuang Yang, Bore-Kuen Lee, Member, IEEE, and Tsern-Huei Lee, Senior Member, IEEE

Abstract—In order to exploit the nonlinear time-varying property of network traffic, the traffic flow from controlled sources is described by a fuzzy autoregressive moving-average model with auxiliary input (fuzzy ARMAX process), with the traffic flow from uncontrolled sources (i.e., cross traffic) being described as external disturbances. In order to overcome the difficulty of the transmission delay in the design of congestion control, the fuzzy traffic model is translated to an equivalent fuzzy predictive traffic model. A fuzzy adaptive flow control scheme is proposed to avoid congestion at high utilization while maintaining good quality of service. By use of fuzzy adaptive prediction technique, the difficulties in congestion control design due to nonlinearity, time-varying characteristics, and large propagation delay can be overcome by the proposed adaptive traffic control method. A comparative evaluation is also given to show the superiority of the proposed method. Index Terms—Asynchronous transfer mode (ATM) traffic control, fuzzy ARMAX process, fuzzy predictive model, stochastic fuzzy systems.

I. INTRODUCTION

F

UTURE broadband communication networks are expected to carry information traffic generated by a wide variety of services and applications. The asynchronous transfer mode (ATM)-based broadband integrated service digital network (B-ISDN) is a high speed transport network designed to support all variable services with different requirements of quality of service (QoS) (e.g., cell loss rate, delay, delay jitter) and a broad range of statistical characteristics [1], [2]. However, the additional flexibility needed to accommodate different traffic sources may cause serious congestion problems, resulting in severe buffer overflow, cell loss, and degradation of QoS. In order to guarantee the required QoS, effective congestion control schemes are needed. Due to the unpredictable fluctuation and burstiness of traffic flows within multimedia networks, congestion can occur frequently when the input traffic to a link exceeds the link capacity. In this case, the buffer, which Manuscript received March 14, 2002; revised August 9, 2002 and September 30, 2002. This work was supported by the National Science Council under Contract NSC-91-2213-E-007-098, Contract NSC-91-2213-E-007-099, Contract NSC-91-2213-E-216-021, and Contract NSC-91-2219-E-009-032. B.-S. Chen is with the Department of Electrical Engineering, National Tsing-Hua University, Hsinchu, Taiwan 300, R.O.C. (e-mail: [email protected]). Y.-S. Yang and T.-H. Lee are with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C. (e-mail: [email protected]; [email protected]). B.-K. Lee is with the Department of Electrical Engineering, Chung-Hua University, Hsinchu 300, Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2003.814860

stores the cells to be transmitted to the outgoing link, may overflow. As a result, this causes the degradation of QoS such as cell loss, excessive queueing delay, and even deadlock. In order to maintain QoS, the buffer length should be controlled around a reference level which is related to the tolerable cell-queueing delay and cell-queueing delay variation [10]. By tracking control of the queue level to a fixed threshold and minimizing of the variance of the queue length, one can drive the cell-queuing delay and its variation within their tolerable ranges. One can also reduce the probability of buffer overflow which contributes to cell loss rate. An accurate traffic model is crucial for successful congestion control of ATM networks. If the traffic models do not accurately represent the actual traffic, one may overestimate or underestimate network performance. Recently, traffic models have been described as either stationary or nonstationary [1], [2]. Stationary traffic models can generally be classified into two cases: short-range dependent models and long-range dependent models. Short-range dependent models include Markov models and regression models (i.e., AR, MA, ARMA) [1]–[4]. These traffic models have a correlation structure that is significant for relatively small lags. Long-range dependent traffic models, such as fractional autoregressive integrated moving average (F-ARIMA) and Fractional Brownian motion, have momentous correlation even for large lags [5], [6]. In most cases, actual traffic does not fulfill the stationary assumption, whereas it does exhibit nonstationary, uncertain, and even nonlinear characteristics with long transmission delay. For example, the buffer content cannot fall below zero and cannot exceed the buffer size. In order to overcome the nonlinear characteristics of network traffic, artificial neural networks have been employed to model the dynamics of the network traffic [7]. However, due to complexity of training, inaccuracy for long prediction, and difficulty for applying to congestion control, these methods can not be easily employed for practical applications. Fuzzy modeling has been introduced to very successfully represent real linear and nonlinear uncertain systems [18], [19], [30]. It has been employed for nonlinear control system designs. Recently, fuzzy modeling techniques have been employed to represent the network traffic with nonlinear characteristics [20]. However, they are too complex to be efficiently applied to fuzzy congestion control design for regulating the network traffic. In this situation, a new fuzzy model is necessary to effectively characterize the nonlinear time-delayed network traffic. In this paper, in order to exploit the nonlinear time-varying and time-delayed properties of high-speed network traffic, a fuzzy ARMAX process is introduced to model the network

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CHEN et al.: FUZZY ADAPTIVE PREDICTIVE FLOW CONTROL

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Fig. 1. Example of the leftover bandwidth for the ABR traffic.

traffic, in which the traffic flow from controllable sources (e.g., ABR) is described by the fuzzy ARMA part and the uncontrollable traffic (e.g., CBR and VBR) is described as the external disturbance. In order to overcome the difficulty of transmission delay in the design of congestion control, the fuzzy ARMAX traffic model is translated to an equivalent fuzzy predictive model. Fuzzy predictive model-based control can be expected to treat traffic congestion problem with large bandwidth-delay product by predicting the system behavior ahead of the propagation delay and taking early action rather than waiting for feedback to start taking corrective action. There are also several other predictive control schemes in the previous study [8]. However, most of these schemes, based on fixed parameter model without adaptation, are linear and so simplistic that they are not accurate enough to predict the nonlinear time-varying traffic behavior over long horizons [10]. In this paper, a fuzzy predictive ARMAX model and adaptive feedback control are combined together for traffic flow control design to overcome the large delay-bandwidth product and to take advantage of compensation of nonlinearity, disturbance, and model uncertainties of the network traffic. In this fuzzy ARMAX traffic model, the traffic data is divided into several linear clusters via fuzzy interpolation, with each cluster described by an ARMAX model [18]. In this situation, the actual network traffic is described by interpolating these ARMAX processes through the smoothing of fuzzy bases. The model is then translated to an equivalent predictive form. This fuzzy predictive ARMAX traffic model is suitable to describe the network traffic with nonlinear, time-delayed, and nonstationary properties. For the convenience of design, the fuzzy ARMAX process used to model the network traffic is translated to an equivalent fuzzy predictive model. Finally, an optimal adaptive fuzzy predictive control scheme is proposed for the nonlinear network traffic to obtain high utilization of resources while maintaining QoS by keeping the buffer length around a desired level and minimizing the variance of the buffer length. The traffic control strategy is first to derive an optimal prediction of the buffer length and then steer the traffic control rate so that the predicted buffer length approaches the desired buffer level.

Finally, several simulation examples are given in comparison with other traffic control schemes to confirm the performance of this proposed adaptive fuzzy traffic control method. The rest of this paper is organized as follows. In Section II, a congestion control system model for ATM network traffic is given. Then the associated fuzzy ARMAX model and the fuzzy predictive model are introduced. Based on the fuzzy predictive model, in Section III, the optimal ABR traffic control rate is derived and a fuzzy adaptive predictive control is established. This fuzzy adaptive predictive scheme is verified via simulation study and performance comparison with other methods in Section IV. Conclusions and discussions are made in Section V. II. SYSTEM DESCRIPTION AND PROBLEM FORMULATION The following system descriptions are broadly based on an ATM environment [11], but we should note that the concept is not constrained to the ATM framework. However, due to lack of space, we do not plan to involve the details in ATM specification [21]. A. Overview of ABR Congestion Control Consider an ATM switch fed by a mixture of constant bit rate (CBR), variable bit rate (VBR), and available bit rate (ABR) traffic. VBR service, with possible frequent burstiness, can be used to transmit real-time or nonreal-time video and audio data [11]–[13]. CBR service can be viewed as a special case of VBR service where its peak rate is equal to its average rate. CBR service is useful to emulate a circuit-switched network for fixedbandwidth real-time applications [13]. ABR service, instead, will try to make use of the excess bandwidth from other sources by adapting its rate to what is available. Nonreal-time delay-insensitive (but maybe loss-sensitive) data traffic is most suitable for ABR service. Since the total link capacity is fixed, ABR traffic should increase its rate to make most of the leftover bandwidth whenever the CBR and VBR traffic decreases, and vice versa [11]. Fig. 1 shows bandwidth assignments of the various services in a physical link.

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Fig. 2. Example of a source matching its rate to available bandwidth with feedback delay.

Since an ABR source must adjust its rate from time to time according to the remaining bandwidth, it should be informed of the latest network condition via a feedback channel with delay. This feedback delay together with the forward transmitting delay is called the round-trip delay [13]. Roundtrip delay can be logically lumped either into the forwarding path or the feedback path. Fig. 2 gives an usual feedback model in which the forward propagation delay is merged in the feedback path. Assume the link capacity is and the round-trip delay is , the buffer can in the worst case before the ABR incoming be filled up to rate to the switch is decreased. This is commonly called the delay-bandwidth product of the network [11]. Since is large in high speed network and so is especially in a wide area network, a large buffer is necessary to avoid overflow (and, thus, to keep cell loss rate low) but this in return may cause a large cell delay. Furthermore, more fluctuation of buffer length leads to larger cell-queuing delay variation. Also, a larger buffer means more cost. Therefore, a good control mechanism is important to deal with these problems [7]–[9], [21].


Fig. 3.

Congestion control system model.

. It is assumed that during the wired link connection period, the round-trip delay is assumed fixed. To overcome the congestion control problem due to round-trip delay , our purpose is to design a -step ahead of the buffer length based on the predictor current and past information of the regulation rate and the buffer length , but without the information . To keep the -step of uncontrollable cross traffic ahead predictive buffer length around the desired level and minimize the variance of the buffer level, we derive an to regulate the controllable source by optimal control rate minimizing a conditional mean of the square of the difference . In this way, it is possible to confine the QoS parameter, such as cell loss rate, cell delay, and the cell delay variation, within the desired range claimed in the traffic contract. This is important for real-time multimedia application. With the emergence of diverse communication services such as data, voice, and video, as well as increasing switching and multiplexing in networks [2], the packet traffic of high speed networks becomes a nonlinear process. The buffer dynamics in Fig. 3 can be described by the following Lindley equation [23]:

B. Congestion Control System Model The basic framework of our discrete-time congestion control system model with sampling time interval is depicted in Fig. 3. The cell emission rate and the queue length are updated for every seconds. If is too large, rapid change of the burst VBR traffic can not be well measured, which can result in bad bandwidth reservation, extra cell loss, or poor utilization of link capacity. On the other hand, if is too small, utilization of link bandwidth will decrease due to excess overhead for delivering the frequent feedback RM cells [16]. The input traffic arrived at the multiplexer buffer comes from two distinct groups of traffic [10]. One is the controllable traffic , which can adapt its rate to network status, such as flow , e.g., ABR. The other one is the uncontrollable traffic flow delay-sensitive traffic such as VBR video and CBR audio. The controllable traffic connection is established at the call setup phase and afterward its transmission rate should be regulated by the adaptive controller implemented in the switch. The uncontrollable sources comply the corresponding traffic contracts and transmit at their negotiated rate after connections are set up, and have higher priority than controllable ones. Therefore, the controllable sources can only share the bandwidth of the link left by the higher priority sources. This matches to the nature of ABR services [11]. Due to the round-trip delay in the feedback control loop, the controllable sources suffer from a delay

(1) is the incoming traffic from the controllable sources, where the traffic from the uncontrollable sources, the buffer a conlength, the sampling time interval, stant transferring Mb/s to cell per sampling time, the outthe buffer size. This is a highly going link capacity, and nonlinear time-delayed system. In this situation, conventional linear traffic models are not capable of capturing the traffic behavior of broadband and high-speed networks. The proposed fuzzy ARMAX model is a nonlinear mapping of controllable and uncontrollable input traffic data to the buffer length and it has excellent capability to describe these traffic characteristics. C. Fuzzy ARMAX Traffic Model The global operation of the nonlinear dynamic system in (1) can be divided into several local linear operating regions. A fuzzy dynamic model has been proposed by Takagi and Sugeno to interpolate locally linear input–output relations via membership functions for nonlinear systems [19]. This fuzzy dynamic model is described by fuzzy IF–THEN rules and will be employed here to deal with the network traffic flow control since the buffer dynamics is also a nonlinear system.


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The fuzzy ARMAX traffic model contains four components: rules, a fuzzifier, an inference engine, and a defuzzifier. The role of the fuzzifier is to map the crisp input data to fuzzy sets, defined by their membership functions, and depending on the degree of “possibility” of the input data. The goal of the defuzzifier is to map the output fuzzy sets to crisp output values. The fuzzy inference engine defines how the system should make inferences through the fuzzy rules contained in the rule base in order to determine the output fuzzy sets. The fuzzy ARMAX model for the network traffic in Fig. 3 can be represented by combining several ARMAX models via fuzzy rules as follows [17]–[19]:

(2) where grees

, , and , and , respectively, with

are polynomials of de-

is the number of IF–THEN rules, the number of premise varithe fuzzy set for and , and ables, the premise variables which depend on the data . The symbol denotes the delay opset . The polynomial is erator, i.e., stable in the sense that all its zeros are inside the unit circle. , we assume it satisfies the following For the driving noise usual properties:

Remark 1: We assume that the uncontrollable traffic sequence is a zero mean process. However, the mean of the uncontrollable traffic sequence is always positive in the real network. It seems to be a problem, but it does not matter in fact. Note that a switch buffer with service rate to serve with mean is equal to a switch an input sequence to serve an input sequence buffer with service rate with zero mean (see Fig. 3). In the following discussion, we will not use the information of and directly, but with and . Therefore, this problem is indeed transparent to us. Without loss of generality, is a zero mean process in the following if the we assume is changed to . In practice, an on-line estimation of can be constructed. For example, a simple estimate is given by with a window size . On the other hand, as the traffic parameters of the CBR and VBR traffic are known in advance due to call admission control, an estimated mean rate of the aggregate uncontrollable traffic can be derived by using the parameters, such as sustained cell rate (SCR) and peak cell rate (PCR) for the VBR and CBR traffic, respectively. For the convenience of adaptive fuzzy traffic control design, the th fuzzy subsystem at the th operation point in (2) will be rewritten by using transfer function representation in the following: (5) Then, the overall fuzzy network traffic system (2) is equivalent to

(6) where

(3) (4) the field spanned by the where denotes the expectation, , and a.s. denotes almost surely. In the data set , , and are fuzzy system (2), the system parameters unknown. can be some indices to The premise variables reflect the statuses of the switch and the traffic condition. They can be buffer length, change rate of buffer length, and buffer has been length variation. For example, the buffer length used to describe the load condition of the traffic or the degree of congestion for the EFCI switch [24], [11]. In the simulation and study to be described later, we choose with which three fuzzy subsystems are defined with respect to light load, nominal load, and high load (congestion) conditions. Then, the ARMAX models in (2) can be regarded as the linearized systems for the nonlinear queue dynamics (1) under the three different load conditions. Dynamics under any load condition is then represented by the smooth fuzzy interpolation. Therefore, the overall fuzzy traffic controller may automatically adjust the ABR cell rate according to the interpolated queue dynamics according to the traffic load condition.

and is the grade of membership of the following property:

in

with

In (6), local ARMAX stochastic processes are interpoto achieve a lated by nonlinear smoothing functions can be nonlinear fuzzy stochastic system. The term viewed as the overall weight (score) of the rule which decides how much contribution the rule has imposed on the output re. sult D. Fuzzy Predictive ARMAX Traffic Model The conventional representation of fuzzy stochastic system in (6) is unsuitable for controller design because the noise

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term is unavailable. Under such circumstance, its equivalent -step ahead fuzzy prediction model is more appealing for our design. For the th ARMAX subsystem in (5), define the optimal -step ahead prediction as . Then obeys the following recursive equation:

Remark 2: In the approximation of (11), the FIR sequence has been adopted to approximate the IIR sequence. Using the FIR approximation, adaptation algorithm of parameter estimation can be simplified and convergence of parameter estimates can be ensured. In most cases, if the order of FIR sequence is large enough, the error between IIR and FIR sequences may be ignored.

(7) where and satisfying

, as well as

, are the unique polynomials

(8) Meanwhile, the -step prediction system satisfies

for each sub-

(9) Derivations of (7) and (9) from (5) are given in Appendix A which follows the methodology developed in [17]. With the prediction model in (7), the equivalent -step ahead fuzzy prediction model for the fuzzy ARMAX model in (6) can be constructed as :

III. FUZZY ADAPTIVE PREDICTIVE TRAFFIC FLOW CONTROL In this section, we will deduce the optimal flow control rate of the controllable sources. First, we want to construct the of future buffer length optimal prediction based on the current and past information, i.e., the sequences and . According to the optimal prediction , an optimal traffic control rate is then solved in order to make the best effort to fulfill the QoS. Since the system parameters are assumed known in the deduction of the optimal traffic control rate, we need to estimate these parameters through an adaptive method and use these estimated values . If these parameters to calculate the adaptive control rate will surely approach were convergent, the flow control rate the optimal solution derived previously. With the skill of adaptation, the nonstationary property of network traffic can also be overcome. The above is the key idea that we will follow in this section. A. Optimal Traffic Flow Control The purpose of the optimal traffic control design purpose is in (2) or (10) so that to specify the controllable traffic rate the following -step ahead quadratic cost function is minimized [17] (12)

(10) The equivalent overall predictive system (10) is approximated as follows:

(11) is the optimal -step ahead prediction of the where fuzzy ARMAX model in (6) and the polynomials and are used to approximate the transfer functions and , respectively.

where is a weighting factor and is the reference buffer level. will be chosen as a function of The optimal control rate , , , , , , and so as to mini. Minimizing the mize the aforementioned cost function used in (12) forces the buffer term length to track the reference buffer level and to minimize the variance of the tracking error. In this way, we can confine the cell delay and minimize the cell delay variation in order to achieve the desired QoS [10]. However, if the weighting factor is not is never constrained and considered in (12), the control rate approach even it may result in a rapid response to make if a small tracking error occurs. Note that there are modeling error of the fuzzy predictive ARMAX traffic model to represent and the real traffic and parameter estimation errors of during the adaptation learning phase. Due to these uncertainties and large propagation delay, rapid, and large response of may cause oscillation of the buffer length . This in turn will lead to a severe buffer length variation and result in bad QoS. Therefore the design parameter is an important way to keep away from this situation. Our simulation results will verify this point later. Another advantage of using the weighting factor on is to avoid the design difficulty determining the control rate of the minimum variance control in the nonminimum phase case [17]. of


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By taking the operation of to both sides of (9), we get the prediction of fuzzy ARMAX traffic system (13) Using (13) and (3), the fact that

Fig. 4.

Block diagram of the parameter adaptation scheme.

and substituting (11) into (16) leads to (14) is the optimal prediction of of shows that the stochastic fuzzy ARMAX traffic system based on the data [17]. Now, substituting (13) into (12) and using the fact set of (3), we get (15) as

(17)

After some arrangement, the optimal traffic control rate is given by

(18) where

(19)

(15) to obtain the optimal Differentiating (15) with respect to traffic control rate [note that the second term is independent of , and, hence, disappear after differentiation], we get

Following from (11), we get

, , and , , In (18), the system parameters are essential to determine the suboptimal control rate . Undoubtedly, we need a real-time adaptive algorithm to estimate these parameters with reasonably fast convergence. Remark 3: In practice, the flow control rate can never be can sometimes be negative. To negative, but the calculated avoid this situation, a saturation mechanism can be apply to directly. For example, set to zero when it is negative. Also, (the available ABR bandwidth) is set to an upper bound whenever is larger than (to avoid control rate abrupt fluctuation in traffic control rate which may lead to large buffer length variation or even overflow). In the simulation study shown later, the upper bound is given by minus a windowed for some window size averaged term . This moving averaged term can be regarded as an estimate of the bandwidth consumed by the uncontrollable traffic. B. Adaptive Tracking of Traffic System Parameters

Combining the last two equations, we have (16)

, , , and , , , Since the parameters are necessary to compute both the prediction and the control rate , it is required to make use of an adaptive algorithm to obtain these parameter estimates which are . The then used to solve the adaptive traffic control rate block diagram for parameter adaptation is depicted in Fig. 4

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[15]. According to the theory of adaptive signal processing [15], to the real buffer length we can compare the prediction and use the difference to adjust these unknown parameters. For simplicity, the famous NLMS algorithm [15] is adopted here to track the traffic system parameters as follows. of and rearLet us define an adaptive prediction range the predictor into the linear regression form as follows:

(20)

.. .

.. .

.. .

.. .

.. .

A. Traffic Source Models 1) Continuous-State Autoregressive (AR) Markov Model for Video Source [12]: We can model a VBR source as a coder with its rate being a continuous-state, discrete-time stochastic represent the bit rate of a single source during process. Let the th time slot. A first-order autoregressive Markov process is generated by the recursive relation

.. .

.. .

with initial condition as

IV. SIMULATION Before starting to simulate the proposed method presented in the previous section, we need to generate the uncontrollable sources, i.e., the cross traffic pattern, first. The uncontrollable traffic is modeled as a mixture of VBR and CBR traffic. Although existent models may not describe the real traffic adequately, the real network trace data recorded at some specific nodes may not be general enough to represent a common situation, neither. Moreover, in our model, the uncontrollable traffic is treated as a disturbance and is not used directly to solve the control rate in our approach. Therefore, it is not essential to describe the behavior of uncontrollable traffic precisely and we have chosen some well-known source models in the simulation. To perform the Monte Carlo simulation, twenty traces of the uncontrollable traffic, each with 10 000 sampling points (sampling time interval 1 ms), will be generated and used to evaluation the performance of the proposed algorithm.

where

.. .

The estimated parameters , , , and , , , of the system parameters, i.e., , , , and , , , , are used to compute the traffic control rate in (18) as if these parameters are known in advance. As the adaptive in (18) is obtained, it is sent to fuzzy predictive control law the controllable source to regulate the network traffic in Fig. 3. In the proposed adaptive fuzzy predictive traffic control scheme, , , we do not need to estimate the system parameters or , , of traffic system in (2) or (10) directly, but to , only. Therefore, the proposed estimate the parameters adaptive fuzzy traffic control is simpler and more suitable for ATM network traffic control.

. The prediction error is defined

(21) and the NLMS adaptation algorithm employed for parameter estimation [15] is given by (22) where is an arbitrary small positive number to prevent from dividing by zero and is the normalized step size. Note that the . NLMS algorithm converges when

where is a sequence of independent Gaussian random has variables and as well as are constants. Assume that and unit variance. Further, assume that ; thus, mean the process achieves steady-state with large . Moreover, since source rate cannot be negative, we simply set it to zero if goes under zero. Because the probability of being negative is very small, it will not affect the analysis of this source model much. We also truncate first few values to get a better stationary sequence. There are two video sources, each with 20 traces, are generated. The parameters used for video sources are given in Table I. 2) Discrete-State Continuous-Time Markov Process for Voice Source [13]: Consider a single voice source. It is well-represented by a two-state on-off source model as shown in Fig. 5. When in talk spurt, the source rate is fixed (assume in our case), and zero if in silent state. The composite model for such sources is simply a -state birth-death process


TABLE I PARAMETERS USED FOR VIDEO SOURCES. (UNIT: Mb/s)

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TABLE III PARAMETERS FOR FOUR VIDEOCONFERENCING SOURCES (UNIT: Mb/s)

TABLE IV PARAMETERS FOR CBR SOURCES (UNIT: Mb/s)

Fig. 5.

Markov process voice model.

TABLE II PARAMETERS FOR THE VOICE TRACE (UNIT: Mb/s) TABLE V CHARACTERISTICS OF THE UNCONTROLLABLE TRAFFIC (UNIT: Mb/s)

depicted in Fig. 5. We assume the initial state is 0 (no one in talk spurt), and hence the minimum rate is zero. Since we have treated time in a discrete fashion, there is a gap between the continuous-time and the discrete-time frameworks. To get rid of this problem, we sample the continuous rate at every unit-time and use this model to represent the voice sources in our simulation. The parameters for the total 20 voice source traces are given in Table II. 3) GBAR Model for Videoconferencing Source [14]: The GBAR model is an autoregressive process which tries to represent the H.263 videoconferencing traffic. Two inherent features of this process are that the marginal distribution is gamma and the autocorrelation function is geometric. Let denote a random variable with a gamma distribution with denote shape parameter and scale parameter . Let a random variable with a beta distribution with parameters and . The GBAR model is based on the following well-known reand random sults: the sum of independent random variable and the product of variables is a and random variables is independent random variable. Thus, if is , a is , is , and these three random variables are mutually independent, then

defines a stationary stochastic process with a marginal distribution. The process previously defined is called a GBAR process, which can be used to model data from videois as conferencing sources, and its autocorrelation function follows:

The parameters and can be estimated from the mean variance of the data as

and

Four videoconferencing sources, each with twenty traces, will be generated. The related parameters of the four videoconferencing sources used in our simulation study are listed in Table III. The first few initial values are also truncated in the generation of the sequences. 4) CBR Sources: Two CBR sources are considered and the related parameters of the CBR traffic are given in Table IV. 5) Aggregation of Uncontrollable Traffic: We sum up all the data of the previous sources to generate the overall uncontrollable traffic, including two video sources, one Markov process voice source, four videoconferencing sources, and two CBR sources. The characteristics of the accumulated data trace by counting all the 20 traces are listed in Table V and a typical trace is illustrated in Fig. 6. We can see the large and rapid fluctuations on both long and short time scales which provide a challenging input for the rate controller. 6) ABR Sources: In the real networks, there are many applications which can transmit data using the ABR service, such as e-mail, FTP, and WWW as long as they are not real-time applications. In our simulation, greedy ABR sources have been adopted to evaluate the proposed method. That is, these sources will use up all the bandwidth which is available to them. This should be reasonable since the ABR service is most likely to carry delay-insensitive data traffic which can be saved and then transmitted at any time and at any rate when allowed. Also, the greedy source is a popular model among most ABR congestion control literatures [13].

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Fig. 6. Typical trace of the uncontrollable traffic v(k ) used in the simulation.

Fig. 8. Comparisons of buffer length distributions by using the proposed control method under unit round-trip delay and different values (0, 0.01, 0.1) of the weighting factor . TABLE VI SIMULATION RESULTS WITH d = 1 (UNIT: CELLS)

of ). In the following results, the cell loss rate (CLR) is defined as [21]

Fig. 7.

Membership functions for normalized queue length y (k )=B

.

B. Simulation Results is 1) Traffic System Parameters: The buffer length chosen as the only one premise variable [i.e., and in (2)]. The buffer length can be regarded as the degree of congestion. The membership functions for are depicted in Fig. 7 which shows only three rules are used to make up for the fuzzy ARMAX traffic model. This is reasonable since nonlinearity lies in overflow and underflow which occur at the edges of the buffer. Therefore, three rules should be adequate in our situation and the three ARMAX processes are the linearized models of the nonlinear queue dynamics under light load, medium load, and heavy load (congestion), respectively. In the prediction equation (11), we choose the orders as and . In the Lindley equation (1), is chosen as 1 ms, as , as 155 Mb/s, and as 100 cell places in our simulation. The adaptation step size in the NLMS algorithm is set to 30 cell places (30% is set to 0.2. The desired level

Lost Cells Total Transmitted Cells which is measured over the total simulation time interval. Link utilization is defined as Total Input Traffic Utilization Link Capacity which is measured in each time slot and then average over the total simulation time. All the simulation results given in the following subsections are obtained from the Monte Carlo test with 20 runs. 2) Effect of the Weighting Factor Upon System Performance: In the cost function (12), the weighting factor has been introduced as a design parameter. In this simulation, we would like to explore the effect of the weighting factor upon system performance by setting delay as 1 and as 0, 0.01, and 0.1, respectively. Buffer distributions corresponding these different choices of are depicted in Fig. 8 and are summarized in Table VI. Under the Monte Carlo test, it is seen from Table VI that the results show a good performance in both high link utilization and low CLR (cell loss rate). In (12), the overall cost is a funcand tion of both the predictive tracking error the control rate . If the penalty factor is set to 0, the cost function only accounts for tracking error and, thus, leads to the minimum variance control [17]. Since the round-trip delay


is small, it is easy to predict the buffer length and thus can be determined such that the link a precise ABR rate bandwidth is highly utilized and the buffer is far from overflow. , 0.01, and 0.1, there is no buffer For all the three cases, overflow detected during the transmission of a total amount of cells or bits. We note about that a bit error probability of the order 10 is expected in an optical fiber transmission system. The proposed adaptive fuzzy system, though capable of predicting the traffic when delay is small, results in some degree of modeling error due to fuzzy approximation error and the parameter adaptation error. This modeling error leads to a steady-state buffer tracking error under the weighted minimum variance condrives a rapid response trol scheme. Recall that the choice . In the presence of modeling for adjusting the control rate error, more rapid control response usually cause larger oscilla. tions at the control system output, i.e., the buffer length On the other hand, the weighting factor in the cost function (12) is used to make a tradeoff between queue tracking conespecially trol and the response of the traffic control rate when in the presence of modeling errors. By increasing the value of , a smoother and slower control response of can be can be achieved and the fluctuation of the buffer length reduced. Comparing the buffer distributions shown in Fig. 8, we can see that the higher is, the tighter the buffer length distribution is and the smaller the buffer length standard deviation (SD) is. However, if the significance of the queue tracking , then the differerror is decreased in the cost function ence of the mean buffer length and the reference level and the variation of the difference will be increased. This phenomenon can be observed in Fig. 8. As defined in the ATM traffic management specification [21], the mean buffer length, the standard deviation, and the buffer distribution are directly related to the maximum cell delay and delay variation QoS parameters. To guarantee the required QoS of the ABR connection, usually a repeated trial-by-error routine must be performed in order to find suitable parameters, such as the buffer reference level , the weighting factor , and the orders of the fuzzy ARMAX model, to attain satisfactory mean buffer length and buffer length deviation. 3) Results of Different Delays: To study the effect of propagation delay upon the performance, we first set to 0.01 and to 1, 4, and 7, respectively. The results are illustrated in Fig. 9 and are summarized in Table VII. We can see that the variation of the buffer length becomes larger as delay increases. In the minimum variance control design, even if there is no modeling error, a higher prediction results from a larger propagaerror of the buffer length tion delay [17]. With the underlying predictive control strategy, higher prediction error leads to larger buffer length fluctuation. On the other hand, a larger delay in a feedback control loop contributes more phase lag and decreases the phase margin. The reduction of phase margin can cause oscillation at the buffer length and amplify the effect of the uncontrollable sources upon the buffer length variation. This conclusion can be verified from Table VII by comparing the buffer standard deviations under different propagation delays. However, the link utilization remains fairly good even at large delay. We note that, if the delay is further increased, then the overall traffic feedback system may become unstable due to modeling error.

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Fig. 9. Comparisons of buffer length distributions by using the proposed control method under different round-trip delays (d = 1; 4; 7) and the weighting factor = 0:01. TABLE VII SIMULATION RESULTS OF DIFFERENT DELAYS (UNIT: CELLS)

A way to reduce the effects of the propagation delay upon the fluctuation of buffer length and cell loss rate is to decrease the control bandwidth of the feedback traffic control system and . To attain this slow down the response of the source rate goal, we can increase the weighting factor in the adaptive fuzzy control design. It is also shown in Table VII that by set, we can reduce the mean level and ting to 0.1 and the variance of the buffer length and result in better cell loss rate at the cost of reducing the link bandwidth utilization. If , then the weighting factor is further increased, say there will be no buffer overflow but with less link utilization ratio 74.5786%. Comparisons of the buffer distributions under , 0.1, and 0.15 with are shown in the settings Fig. 10. It is shown in Table VII that when the weighting factor is increased from 0.01 toward 0.15, the buffer usage is dramatically decreased while the CLR is greatly improved. In the extreme , the mean buffer length is close to zero case with and the link utilization is as low as 74.5786%, which imply that the ABR source is over-decreased. By increasing the weighting factor , the traffic control system becomes more conservative in that the incoming ABR source rate, the buffer usage, and the link utilization will be decreased in order to attain a satisfactory CLR. In practice, a compromise should be made between CLR and link utilization especially in the presence of long round-trip delay and large variation of the uncontrollable traffic.

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Fig. 11.

Control signal c(k ) of proportional congestion control.

Fig. 10. Comparisons of buffer length distributions by using the proposed control method under the round-trip delay d = 7 and different values (0.01, 0.1, 0.15) of the weighting factor .

Our simulation results provide a guideline to attain a satisfactory compromise. A suitable weighting factor can be obtained by a finer search procedure to comply with the compromised QoS specification. Remark 4: The sampling time interval is chosen as 1 ms which is a reasonable tradeoff between computation complexity and system dynamic performance [28]. On the other hand, under the assumptions that the signal propagation speed is approxiKm/s and that the queueing delays are nemated by glected, then a distance of 100 Km from an ABR user to an ATM node corresponds to a round-trip delay of 1 ms [28], [29]. Therefore the round-trip delay ranged from 1 to 7 represents a distance range up to 700 Km which is sufficient to cover the cases such as local area networks (LANs) and metropolitan area networks (MANs). 4) Comparisons With Other Methods: To demonstrate the advantage of our proposed algorithm, comparisons with the other four methods described in the following will be made and . under the conditions a) Binary Feedback [11]: The buffer length is monitored by setting two thresholds, and . When the buffer length exceeds , congestion is detected and the source rate is reduced with a factor of 0.98. When the buffer length falls below , there is no more congestion and the source rate is increased by one level. Otherwise, the source rate is not changed. In the simulation, the buffer desired level is set to 75 cells. and are set to 0.95 and 1.05 , The two thresholds respectively. Increasing level is set to one percent of the link capacity . Initial source rate is set to link capacity minus the mean of the uncontrollable traffic. A saturation with the lower bound 0 and the upper bound is also applied to the control rate. b) Binary Feedback With Adaptive Linear AR Prediction [22]: This is basically the same as the above method except that it adopts a linear adaptive AR prediction algorithm to predict the future buffer length. The congestion condition is then determined based on the predicted buffer length. In the simulation, the AR model for prediction is with an order of five. c) Proportional Congestion Control [9]: Consider a multiplexing buffer and denote a control reference of buffer length

Fig. 12. Comparisons of buffer length distributions by using various control methods.

as , the control signal is generated by the following congestion control algorithm (see Fig. 11): if if if where

can be viewed as the lower threshold and as the upper threshold. In general, the choice of thresholds will influence the performance of control algorithm and should be chosen carefully. The control signal is inversely proportional to the occupancy . If is of the buffer when greater than , a control signal is sent back to the source to reduce its rate, and vice versa. Let us denote the current source . Then the next source rate is given by rate as . d) Predictive Self-Tuning Fuzzy (PSTF) Control [16]: An adaptive Sugeno-type fuzzy regulator, based on predictions of buffer length, changing rate of buffer length, and cell loss rate,


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TABLE VIII SIMULATION RESULTS OF DIFFERENT CONTROL ALGORITHMS WITH y = 0:75B

(UNIT: CELLS)

TABLE IX SIMULATION RESULTS OF DIFFERENT CONTROL ALGORITHMS WITH y = 0:3B

(UNIT: CELLS)

is proposed in [16] to regulate the source rates of greedy or nongreedy ABR users. With a single greedy ABR user with source , the -step ahead prediction of the buffer rate is given length

where is the average bandwidth consumed by the ABR ser, three variables are generated by vice. With

where is the number of lost cells within the th sampling interval. The normalized buffer length prediction , the , and the normalized estimated buffer change rate are then used as inputs to the fuzzy cell loss prediction , , and system. From the measured values of , three sets of linguistic values, Small, Big, Negative, Positive, and Zero, Nonzero are defined, respectively. With the possible eight combinations of the linguistic values, eight fuzzy be the memrules are constructed in the rule base. Let bership value evaluated at the IF part of the th rule using the , , and . Then, the output of inputs the fuzzy system, which is used to determine the ABR source rate in the next time interval, is given by

where is a positive number with . The time, , are tuned by an adapvarying parameters, tive scheme based on the gradient descent method so that the at time is minimized. cost value

All the settings of the predictive adaptive fuzzy control algorithm made in [16] remain in the following simulation study. Comparisons of the simulation results of the above algorithms with that of our proposed method under the setting are illustrated in Fig. 12. All the results are summarized and compared in Table VIII. As we can see that, , all the algorithms deliver good link in the case of utilization. The proposed method is, however, has the best cell loss rate among all the compared algorithms. Moreover, the proposed one is superior to all the other methods while comparing the variance of buffer length distribution. Therefore, we may say that the proposed fuzzy adaptive predictive method delivers a good performance in CLR, utilization, and buffer length variation (and hence cell delay variation). A further comparison between the proposed method and the predictive . self-tuning fuzzy control is made for the case The results are compared in Table IX. In addition to the close link utilization rates, our proposed method still outperforms the predictive self-tuning fuzzy control in CLR, buffer mean, and buffer length standard deviation. V. CONCLUSION AND DISCUSSIONS The feature of this paper is that, first, a Takagi–Sugeno stochastic fuzzy linear model is proposed to interpolate the nonlinear buffer dynamics at different traffic operating conditions through the corresponding membership functions. This fuzzy ARMAX model is translated to an equivalent fuzzy predictive model which is suitable to derive the optimal flow control rate for an ABR connection with propagation delay. Since the parameters in fuzzy predictive model are actually unknown, a NLMS algorithm is adopted to estimate these parameters in real time. The adaptive flow control rate is then constructed based on this recursive parameter estimates. As shown in the simulation study, the adaptive fuzzy predictive model is able to approximate the queue dynamics under different traffic load conditions and thus the adaptive minimum

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variance control can lead to a good traffic control system. Due to the prediction mechanism, as shown in the simulation study, the proposed method can be used to deal with the problem of large delay-bandwidth product. Compared with the other methods, our algorithm delivers much better performance such as CLR, utilization, and buffer length variation even in the case of large delay. Usually, a tradeoff should be made between CLR and link utilization in the presence of large round-trip delay and high variation of the uncontrollable CBR and VBR traffic. Our algorithm provide a systematic approach to make this compromise by adjusting the weighting factor defined in the cost function in the minimum variance control. It is expected that, instead , which of being fixed, an online tuning weighting factor adapts to some premise variables such as the queue length and the estimated fluctuation of the uncontrollable traffic to reflect different operating conditions of the queue, will be beneficial to make a better QoS. The stability and performance issues of the tuning scheme will be a good topic for future study. The fairness issue of link bandwidth allocation is not addressed in this paper. Also, the role of the scheduler in a switch, whose function is essential for traffic management in an ATM network, is not discussed. As the communication technology rapidly advances, ATM chipsets and switches of newer generation now have the capability of providing flow isolation and fair bandwidth allocation by utilizing the scheduling mechanism [25]. For example, equal fairness and weighted fairness can be achieved by round-robin and weighted round-robin scheduling strategies, respectively [26]. Therefore, the ABR traffic control can focus on the congestion control problem without the need to consider the fairness issue [27]. APPENDIX The derivations of (7) and (9) are described as follows. First, to get multiply each subsystem in (2) by

Substituting (8) in the previous equation, we then have

which can be rearranged into the following form:

Now, multiplying tion, one can get

to both sides of the aforementioned equa-

(23) or

Since

is a stable polynomial, both

can be implemented as causal filters and, thus, we have

which implies that

Therefore, (9) is obtained. Substituting (9) into (23), it is easy to see that

which is exactly (7). ACKNOWLEDGMENT The authors wouldlike to thank the anonymous referees whose constructive and helpful comments led to significant improvements in this paper. REFERENCES [1] V. Frost and B. Melamed, “Traffic modeling for telecommunication networks,” IEEE Commun. Mag., vol. 32, pp. 70–81, Mar. 1994. [2] A. Adas, “Traffic models in broadband networks,” IEEE Commun. Mag., pp. 82–89, July 1997. [3] H. Heffes and D. L. Lucantoni, “A Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance,” IEEE J. Select. Areas Commun., vol. 4, pp. 856–868, Sept. 1986. [4] C. Shim, I. Ryoo, J. Lee, and S.-B. Lee, “Modeling and call admission control algorithm of variable bit rate video in ATM networks,” IEEE J. Select. Areas Commun., vol. 12, pp. 332–344, Feb. 1994. [5] W. E. Lehand, M. S. Taqqu, W. Willinger, and D. V. Wilson, “On the self-similar nature of Ethernet traffic (extended version),” IEEE/ACM Trans. Networking, vol. 2, pp. 1–15, Feb. 1994. [6] J. Beran, R. Sherman, M. S. Taqqu, and W. Willinger, “Long-range dependence in variable-bit-rate video traffic,” IEEE Trans. Commun., vol. 43, pp. 1566–1597, July 1995. [7] Z. Fan and P. Mars, “Access flow control scheme for ATM network using neural network-based traffic prediction,” Proc. IEE Commu., vol. 144, no. 5, pp. 295–300, Oct. 1997. [8] L. Benrnohamed and S. M. Meerkov, “Feedback control of congestion in packet switch networks: The case of single congested node,” IEEE/ACM Trans. Networking, vol. 1, pp. 693–708, Dec. 1993. [9] I. W. Habib and T. N. Saadawi, “Access flow control algorithms in broadband networks,” Comput. Commun., vol. 15, no. 5, pp. 326–332, 1992. [10] A. Pitsillides and J. Lembert, “Adaptive congestion control in ATM based network: quality of service and high utilization,” Comput. Commun., vol. 20, pp. 1239–1258, 1997. [11] T. M. Chen, S. S. Liu, and V. K. Samalam, “The available bit rate service for data in ATM networks,” IEEE Commun. Mag., vol. 34, pp. 56–71, May 1996. [12] B. Maglaris, D. Anastassiou, P. Sen, G. Karlsson, and J. D. Robbins, “Performance models of statistical multiplexing in packet video communications,” IEEE Trans. Commun., vol. 36, pp. 834–844, July 1988. [13] M. Schawartz, Broadband Integrated Networks. Upper Saddle River, NJ: Prentice-Hall, 1996. [14] D. P. Heyman, “The GBAR source model for VBR videoconferences,” IEEE/ACM Trans. Networking, vol. 5, pp. 554–560, Aug. 1997. [15] S. Haykin, Adaptive Filter Theory, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1996. [16] R. Q. Hu and D. W. Petr, “A predictive self-tuning fuzzy-logic feedback rate controller,” IEEE/ACM Trans. Networking, vol. 8, pp. 697–709, Dec. 2000.


[17] G. C. Goodwin and K. S. Sin, Adaptive Filtering Prediction and Control. Upper Saddle River, NJ: Prentice-Hall, 1984. [18] I. S. R. Jang, C. T. Sun, and E. Mizutani, Neuro-Fuzzy and Soft Computing. Upper Saddle River, NJ: Prentice-Hall, 1997. [19] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. SMC-15, pp. 116–132, Jan./Feb. 1985. [20] R. G. Cheng and C. J. Chang, “Design of a fuzzy traffic controller for ATM network,” IEEE/ACM Trans. Networking, vol. 4, pp. 460–469, June 1996. [21] The ATM forum traffic management specification, ATM Forum, V 4.1, AF-TM-0121.000, Mar. 1999. [22] B. Qiu, “The effect of traffic prediction on ABR buffer requirement,” in Proc. APCC/OECC’99, vol. 1, 1999, pp. 151–154. [23] D. V. Lindley, “The theory of queues with a single server,” in Proc. Camb. Phil. Soc., vol. 48, 1952, pp. 277–289. [24] Y.-C. Lai and Y.-D. Lin, “Choice of high and low thresholds in the ratebased flow control,” IEEE Proc. Commun., vol. 146, pp. 95–101, Apr. 1999. [25] U. Briem, E. Wallmeier, C. Beck, and F. Mathhiesen, “Traffic management for an ATM switch with per-VC queuing—Concept and implementation,” IEEE Commun. Mag., pp. 88–93, January 1998. [26] Y. H. Long, T. K. Ho, and A. B. Rad, “Fair flow control of ABR service by per-VC virtual queuing,” Comput. Commun., vol. 23, pp. 71–83, 2000. [27] L. Benmohamed and Y. T. Wang, “A control-theoretic explicit rate algorithm for ATM switches with per-VC queuing,” in IEEE Proc. INFOCOM ’98 17th Annu. Joint Conf. IEEE Computer Communications Societies, vol. 1, 1998, pp. 183–191. [28] S. Chong, S. Lee, and S. Kang, “A simple, scalable, and stable explicit rate allocation algorithm for max-min flow control with minimum rate guarantee,” IEEE/ACM Trans. Networking, vol. 9, pp. 322–335, June 2001. [29] A. Kolarov and G. Ramamurthy, “A control-theoretic approach to the design of an explicit rate controller for ABR service,” IEEE/ACM Trans. Networking, vol. 7, pp. 741–753, Oct. 1999. [30] B. S. Chen, S. C. Peng, and K. C. Wang, “Traffic modeling, prediction, and congestion control for high-speed networks: A fuzzy AR approach,” IEEE Trans. Fuzzy Syst., vol. 8, pp. 491–508, Oct. 2000.

Bor-Sen Chen (M’82–SM’89–F’01) received the B.S. degree from Tatung Institute of Technology, Taiwan, R.O.C., the M.S. degree from National Central University, Taiwan, R.O.C., and the Ph.D degree from the University of Southern California, Los Angeles, in 1970, 1973, and 1982, respectively. From 1973 to 1987, he was a Lecturer, Associate Professor, and Professor at Tatung Institute of Technology. Currently, he is a Professor with National Tsing Hua University, Hsin-Chu, Taiwan, R.O.C. His current research interests include control signal processing and bioinformatics. Dr. Chen has received the Distinguished Research Award from the National Science Council of Taiwan four times. He is a Research Fellow of the National Science Council and the Chair of the Outstanding Scholarship Foundation. He was the Chairman of the Taipei Chapter of IEEE Control Society in 1991. He is now an Associate Editor of the IEEE TRANSACTIONS ON FUZZY SYSTEMS and the Editor of the Journal of Asian Control.

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Yu-Shuang Yang was born in Tainan, Taiwan, R.O.C., in 1963. She received the B.S. and M.S. degrees in communication engineering from National Chiao Tung University, Taiwan, R.O.C., in 1984 and 1986, respectively. She is currently working toward the Ph.D. degree at the same university. Her current research interests are in ATM QoS, communication system architecture, network traffic management, and wireless communications.

Bore-Kuen Lee (S’87–M’91) was born in Taichung, Taiwan, R.O.C., in 1961. He received the M.S. and Ph.D. degrees in automatic control engineering from National Chiao Tung University, Taiwan, R.O.C., in 1986 and 1991, respectively. From 1986 to 1987, he worked in the robotic control area as an Associate Researcher at the Mechanical Industry Research Laboratories of Industrial Technology Research Institute, Hsinchu, Taiwan. Since 1991, he has been with the Department of Electrical Engineering at Chung Hua University, Taiwan, where he is currently an Associate Professor. His research interests include variable structure control theory, estimation theory, fuzzy control, and traffic control of networks. Dr. Lee is a member of Phi Tau Phi.

Tsern-Huei Lee (S’86–M’87–SM’98) received the B.S. degree from National Taiwan University, Taipei, Taiwan, R.O.C., the M.S. degree from the University of California, Santa Barbara, and the Ph.D. degree from the University of Southern California, Los Angeles, in 1981, 1984, and 1987, respectively, all in electrical engineering. Since 1987, he has been a Member of the Faculty of National Chiao-Tung University, Hsinchu, Taiwan, where he is a Professor in the Department of Communication Engineering and a Member of the Center for Telecommunications Research. His current research interests are in communication protocols, broad-band switching systems, network traffic management, and wireless communications. Dr. Lee received an Outstanding Paper Award from the Institute of Chinese Engineers in 1991.