On the performance limitation of Active Queue

TuC03.3 43rd IEEE Conference on Decision and Control December 14-17, 2004 Atlantis, Paradise Island, Bahamas On the performance limitation of Active Queue Management (AQM) Khushboo Shah, Stephan Bohacek and Edmond Jonckheere Abstract— Traffic properties relevant to Active Queue Management (AQM) and the resulting pole/zero and Bode limitations are examined. ARX models of TCP are derived from data collected in a fixed "dumbbell" simulation environment. The pole dynamics at short time scales can be traced to the RTT, which serves as a confirmation of the relevance of the models. It is shown that the weighted sensitivity function is a reasonable metric for AQM performance. However, K " design shows that the sensitivity can only be slightly reduced. The conclusion of this work is that the goal of smoothing TCP’s oscillations (a principle design goal of AQM) will likely not be met in general. I. I NTRODUCTION One of the chief services provided by TCP is congestion control. The goal of TCP congestion control is to share the available bandwidth among the many flows that utilize a common link. In order to achieve this goal, TCP dictates a sending rate that is dependent on the past packet losses [1]. Since 1993 [2], there has been extensive research focused on the prospect of deliberately dropping packets according to a time-varying loss probability in order to control TCP connections’ data rate and meet various performance objectives. Such proactive dropping of packets is known as Active Queue Management (AQM). Early attempts of AQM focused strictly on the development of a mapping between the filtered version of the queue occupancy and the loss probability, without explicitly considering the dynamics of TCP. The most popular of these approaches is known as Random Early Drop (RED) [2]. The problem with this approach is that the design parameters are difficult to adjust and chaotic behavior can result [3]. More recently, there has been a number of AQM designs that take a more control theoretic approach and consider dynamic models of the TCP [4], [5], [6], [7]. The major difference between this recent work and the present paper is that we do not use a “first principles” model of TCP. Instead, a very large number of observations are made and a local linear model is identified from I/O data. By local, we mean that the loss probability varies by only a small amount. This allows the linear model to be very accurate. (See [8] for a comparison between linear and nonlinear models.) A second difference between Khushboo Shah and Edmond Jonckheere are with the Department of Electrical Engineering, University of Southern California, Los Angeles, CA, USA. khushboo, [email protected] Stephan Bohacek is with the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE, USA. [email protected] 0-7803-8682-5/04/$20.00 ©2004 IEEE much of the work on AQM and this paper is that this paper focuses on the performance of the closed-loop from a control theoretic perspective. Specifically, we examine the sensitivity function as a measurement of performance. The conclusions of this investigation bode poorly for the prospect of AQM. In short, the non-minimum phase zeros and the Bode limitations lead to design difficulties. Specifically, the incorporation of a dynamic controller to smooth the variations in packet arrivals has only a minor impact on the variation of the packet arrivals. The paper proceeds as follows. In Section II, the approach used to model TCP is discussed. Section III discusses the accuracy of the models. This model is further analyzed in Section IV where the pole zero configuration is investigated. In Section V, the motivation and rationale for the controller design is discussed. Section VI discusses the Bode limitation and how it affects the controller. Section VII investigates several controllers and shows that the control has little impact on the performance. II. TCP DYNAMICS IDENTIFICATION METHODOLOGY This section describes the identification methods used to develop a model of the dynamics of packet arrivals. This approach to modeling is quite different from the typical approach to AQM that uses first principles models. However, those models suffer from a number of drawbacks that are avoided by the approach taken here. The first principles approach models the mean packet arrival rate, whereas in practice, the actual arrival of packets does not exactly follow the mean but is some stochastic process. The approach here embraces the stochastic nature of packet arrivals. Another serious problem with the first principles approach to TCP modeling is that these models do not model TCP time-out; time-out has been shown to play a significant role in TCP [9]. The approach here will account for time-out and any other mechanism (e.g., the dynamics of fast retransmit/fast recovery, slow-start after a time-out, etc.). On the other hand, it is difficult to see how such an approach could be used in practice. However, the objective of this work is to understand the performance limitations in a fixed and known operating environment. This "best-case" situation then stands as a benchmark In order to identify models of TCP, extensive simulations were carried out with the Network Simulator (ns) [10]. TCP traffic was modeled as long lived FTP traffic, that is, for each source-destination pair a single TCP connection sent data for the entire simulation over a single bottleneck 1016 7 9 10 5 6 Monitored Link 11 -26.5 -32.5 Power Spectral Density (dB/ rad/sample) 1 Power Spectral Density (dB/ rad/sample) 0 4 SamplePeriod = 0.05 SamplePeriod = 0.01 8 3 Destinations Sources 2 -27 -33 -27.5 -33.5 -28 -34 -28.5 -34.5 -29 -35 -29.5 -35.5 -30 -30.50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized Frequency (u Srad/sample) Fig. 1. Dumbbell topology. The traffic is sent from sources to destinations. Link 0-1, the bottleneck link, is monitored. l=0 l=0 where % is the so-called residual noise and O denotes the order of the model. Standard least-squares techniques were used to estimate the coefficients dl and el . Because the best choice of the model order, O> is generally not known a priori, it is usually necessary in practice to postulate several model orders. Many criteria have been proposed as tentatively objective functions for selecting the ARX model order. One of the best known ones is Rissanen’s Minimum Description Length (MDL) criterion [12] which has the form (for Gaussian disturbances, which is the case here [8]), P GO[O] = Q ln(P VHO ) + O ln (Q ) > SamplePeriod = 0.5 -30 Power Spectral Density (dB/ rad/sample) Power Spectral Density (dB/ rad/sample) Normalized Frequency (u Srad/sample) SamplePeriod = 0.1 -31.5 -32 -30.5 -32.5 -31 -33 -31.5 -33.5 -32 -32.5 -34 -33 -34.5 -35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized Frequency (u Srad/sample) -33.50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized Frequency (u Srad/sample) Fig. 2. Power spectral density plots of % for sample periods T=0.01, 0.05, 0.1, and 0.5 seconds. SamplePeriod = 1 SamplePeriod = 10 -30 Power Spectral Density (dB/ rad/sample) -38 -38.5 Power Spectral Density (dB/ rad/sample) -30.5 -39 -31 -39.5 -31.5 -40 -32 -40.5 -32.5 -41 -33 -33.5 -41.5 0 -42 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SamplePeriod = 100 SamplePeriod = 50 -47.5 Power Spectral Density (dB/ rad/sample) -44.5 -45 -48 -45.5 -48.5 -46 -49 -46.5 -49.5 -50 -47 -50.5 -47.5 -48 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized Frequency (u S rad/sample) Normalized Frequency (u S rad/sample) Power Spectral Density (dB/ rad/sample) dumbbell topology, as shown in Fig 1. The nodes Vl ( l = 2> ===> 6) are set as the sources and the nodes Gl (l = 7> ===> 11) are set as the destinations. The monitored link, the bottleneck link, is 0 to 1. The starting time of the flows was varied slightly randomly so that each simulation was different. There were 4 FTP flows with a round-trip time (RTT) of 24 msec, 4 FTP flows with a RTT of 26 msec, and 4 FTP flows with a RTT of 28 msec. Each simulation was run for an extremely long time to ensure that all parameters were accurately estimated. In particular, most simulations used over 30,000 sample points. Hence, when the sample period was 100 sec, the simulation time was 3,000,000 sec or nearly 35 days. In the system under investigation, the queue imposes a loss probability on every arriving packet. This drop probability is uniformly distributed over [0.0295, 0.0305] and during each sample period, the drop probability is fixed. Hence, at time step n, the loss probability s (n) is set for the time period [nW> (n + 1)W )> where W is the sample period. We consider sample periods from 10 msec to nearly an hour and a half. In order to keep the scale of the packet arrivals the same for all sample periods, we define | (n + 1) to be the normalized packet arrivals over the period [nW> (n + 1)W ). The normalization is done by dividing the observed packet arrivals by the link speed and the sample period. In this queue discipline, drops could also occur if the queue fills. However, the queue size is set sufficiently large and the drop probability is set large enough, so that the queue does not fill up. Here we consider only linear ARX models [11] for TCP traffic ([8] shows that nonlinear models are not required). These linear ARX models are defined as O1 O1 X X | (n + 1) = dl | (n  l) + el s (n  l) + % (n) > -36 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized Frequency (u S rad/sample) -51 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized Frequency (u S rad/sample) Fig. 3. Power spectral density plots of % for sample periods T=1, 10, 50, and 100 seconds. where Q is the length of the data record and P VHO is the mean square error defined as Q 1 X P VHO := (| (n)  |b (n))2 > Q n=1 where | (n) is the observed packet arrivals while |b (n) is the packet arrivals predicted by the model from the past O samples. The order O is selected so as to minimize the MDL. Furthermore, we use balanced model order reduction techniques to reduce the order in the case of higher sampling rates for simplicity. III. R ELIABILITY OF IDENTIFIED MODELS Clearly the models are optimal if the residual error % is uncorrelated and Gaussian. The uncorrelated property was checked from the “flatness” of the power spectral density of the residual sequence % using the Pcov method (see Figs 2 and 3). Clearly, for W = 1=0> ===> 0=1> sec (Figs 2, 3), the power spectral density appears fairly flat. Not surprisingly, at higher sampling rates, W = 0=05> 0=01> sec (Fig 2), the models become less reliable 1017 S a m p le P e rio d = 0 .0 1 S a m p le P e rio d = 0 .0 1 S a m p le P e rio d = 0 .0 5 S a m p le P e rio d = 0 .0 5 2 -0 .3 -0 .2 -0 .1 0 0 .1 0 .2 D a ta S a m p le P e rio d = 0 .1 -0 .2 -0 .1 D a ta 0 0 .1 -2 -2 Imag Axis -0 .3 0 .2 -0 .2 -0 .1 0 D a ta 0 .1 0 .2 1 0 .8 0 .6 0 .4 0 .2 0 -0 .2 -0 .4 -0 .6 -0 .8 -1 -2 Imag Axis -1 .5 -1 -0 .5 0 Fig. 4. Normplots of the residual error % for sample periods T=0.01, 0.05, 0.1, and 0.5 seconds. S a m p le P e rio d = 1 Probability Probability 0 .0 50 .1 0 .1 5 -0 .0 6-0 .0 4-0 .0 2 0 Probability Probability -0 .0 4-0 .0 3-0 .0 2-0 .0 1 0 0 .0 1 0 .0 2 0 .0 3 0 .0 4 D a ta 0 .0 2 0 .0 4 0 .0 6 0 .0 8 D a ta S a m p le P e rio d = 1 0 0 D a ta S a m p le P e rio d = 5 0 . 0 .9 9 9 0 .9 9 7 0 .9 0 .9 9 8 0 .9 5 0 .9 0 0 .7 5 0 .5 0 0 .2 5 0 .1 0 0 .0 5 0 .0 .0 1 2 0 0 .0 0 3 0 .0 0 1 1 -1 .5 -1 -0 .5 0 0 .5 -4 -3 -2 -1 R e a l A x is S a m p le P e rio d = 0 .5 0 1 0 .2 0 .4 0 .6 0 .8 1 R e a l A x is Fig. 6. Pole/zero configurations of full-order models for sample periods T=0.01, 0.05, 0.1, and 0.5 seconds. S a m p le P e rio d = 1 0 0 .9 9 9 0 .9 9 7 0 0 .9 .9 9 8 0 .9 5 0 .9 0 0 .7 5 0 .5 0 0 .2 5 0 .1 0 0 .0 5 0 0 .0 .0 2 1 0 .0 0 3 0 .0 0 1 -0 .2 5-0 .2 -0 .1 5-0 .1 -0 .0 50 0 .5 R e a l A x is S a m p le P e rio d = 0 .1 R e a l A x is 0 .9 9 9 0 .9 9 7 0 .9 0 .9 9 8 0 .9 5 0 .9 0 0 .7 5 0 .5 0 0 .2 5 0 .1 0 0 .0 5 0 0 .0 .0 2 1 0 .0 0 3 0 .0 0 1 1 -1 .5 0 .0 50 .1 0 .1 5 D a ta S a m p le P e rio d = 0 .5 Probability Probability -0 .3 1 0 .8 0 .6 0 .4 0 .2 0 -0 .2 -0 .4 -0 .6 -0 .8 -1 -1 -0 .8 -0 .6 -0 .4 -0 .2 0 0 -1 0 .9 9 9 0 .9 9 7 0 .9 0 .9 9 8 0 .9 5 0 .9 0 0 .7 5 0 .5 0 0 .2 5 0 .1 0 0 .0 5 0 .0 .0 1 2 0 0 .0 0 3 0 .0 0 1 0 .9 9 9 0 .9 9 7 0 .9 0 .9 9 8 0 .9 5 0 .9 0 0 .7 5 0 .5 0 0 .2 5 0 .1 0 0 .0 5 0 .0 .0 1 2 0 0 .0 0 3 0 .0 0 1 1 .5 1 0 .5 -0 .5 -0 .3 5-0 .3 -0 .2 5-0 .2 -0 .1 5-0 .1 -0 .0 50 0 .3 1 0 .8 0 .6 0 .4 0 .2 0 -0 .2 -0 .4 -0 .6 -0 .8 -1 -5 1 .5 Imag Axis -0 .4 0 .9 9 9 0 .9 9 7 0 .9 0 .9 9 8 0 .9 5 0 .9 0 0 .7 5 0 .5 0 0 .2 5 0 .1 0 0 .0 5 0 .0 .0 1 2 0 0 .0 0 3 0 .0 0 1 Imag Axis Probability Probability 0 .9 9 9 0 .9 9 7 0 .9 0 .9 9 8 0 .9 5 0 .9 0 0 .7 5 0 .5 0 0 .2 5 0 .1 0 0 .0 5 0 .0 .0 1 2 0 0 .0 0 3 0 .0 0 1 0 .9 9 9 0 .9 9 7 0 .9 0 .9 9 8 0 .9 5 0 .9 0 0 .7 5 0 .5 0 0 .2 5 0 .1 0 0 .0 5 0 .0 .0 1 2 0 0 .0 0 3 0 .0 0 1 -0 .0 2 -05.0-0 2 .0 1 -05.0-0 1 .0 0 0 5 0 .0 005.0 1 0 .0 105.0 2 0 .0 2 5 D a ta Fig. 5. Normplots of the residual error % for sample periods T=1, 10, 50, and 100 seconds. as shown by the non-flat power spectral densities. However, even at these small sample periods, the ripple in the power spectral density is less than 6 × 104 . For large sample periods, W  10 sec (Fig 3), the power spectral density exhibits ripple of less than 105 . Regarding the Gauss property, Figs 4 and 5 show the outcome of the MATLAB “normplot” test. If the resulting plot is linear, then the residual sequence is Gaussian. For intermediate sampling rates, W = 0=05> 0=1> sec the Gauss property is obvious. For high sampling rate, W = 0=01 sec, the Gauss property deteriorates slightly in the tail. Specifically, 99=4 % of the probability mass is well-modeled by Gaussian distribution. This test concludes that the Gaussian assumption is a reasonable one. IV. POLE / ZERO CONFIGURATION In order to gain a deeper understanding of this modeling approach, we examine the pole/zero configuration of the open-loop system. Figs 6, 7 show a clear trend in the pole/zero configuration as the sampling period decreases from 100 to 0.01 sec. The pole/zero plots start with a pure delay (pole at zero and a pair of nearly canceling pole/zero) for W = 100 to a much more complicated configuration with multiple pole/zero pairs near the unit circle along with nonminimum phase zeros for W = 0=01 sec. With a sampling interval of W = 0=01 sec> the oscillatory poles/zeros near the unit circle with their arguments between l = 2 and m = 3 4 correspond to oscillations of periods equal to 2W l  0=04 A UW W and 2W m  0=03  UW W> respectively. At this very low time scale, the stable poles/zeros are interlaced and this interlaced property of the lightly damped poles/zeros produce substantial phase variation making stabilization difficult. In the case of small sampling period, this should be kept in mind, however, the poles/zeros close to the unit circle and with small phase angles could be artifacts of the fast sampling rather than intrinsically complicated features of the dynamics. As the time scale increases, we observe that there is the trend of the poles/zeros inside the unit circle going from an interlaced configuration to a configuration where the poles/zeros move on top of each other and cancel out. In this situation, balanced model reduction [13], [14] would in fact do the pole/zero cancellation in a numerically safe manner and produce a simple reduced order model. Through the same process, balanced model reduction would make the distinction between those lightly damped poles/zeros that are artifacts of the fast sampling and those that are dynamically relevant; the former would be deleted and the latter would be kept under balanced model reduction. But probably the most important feature of pole/zero configuration is the transition in the structure as revealed by a comparison between the upper left graph in Figure 8 (W = 0=01 sec) and the lower left graph in Figure 8 (W = 0=05 sec). It should be observed that, around those values, the sampling period W transits through the round trip time (UW W ), which in this case is around 0.03 sec. It appears that one should make a distinction between three different cases depending on how the sampling interval relates to the UW W : 1018 1) UW W ¿ 0=5  W  100 (Figs 6,7). In this situation, the sampling interval is much greater than the UW W , leading to an input/output transfer function with fairly well damped poles. There always seems to be a pole 0.5 0 -0.5 -1 -1.5 0.2 0.4 0.6 0.8 1 Real Axis Sam plePeriod = 100 -2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 Real Axis SamplePeriod = 0.05, DropProb = 0.03 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -5 Imag Axis Imag Axis Imag Axis Real Axis 1 0.2 0.4 0.6 0.8 1 Real Axis Fig. 7. Pole/zero configurations of full-order models for sample periods T=1, 10, 50, and 100 seconds. SamplePeriod = 0.01, DropProb = 0.003 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -2.5 -2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -2 -1.5 Imag Axis Imag Axis Real Axis Sam plePeriod = 50 2 1.5 Imag Axis 0.2 0.4 0.6 0.8 1 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1 -0.8 -0.6 -0.4 -0.2 0 Imag Axis 0.2 0.4 0.6 0.8 1 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1 -0.8 -0.6 -0.4 -0.2 0 SamplePeriod = 0.01, DropProb = 0.03 Sam plePeriod = 10 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1 -0.8 -0.6 -0.4 -0.2 0 -1.5 -1 -0.5 0 0.5 1 Real Axis SamplePeriod = 0.1, DropProb = 0.03 Imag Axis Sam plePeriod = 1 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1 -0.8 -0.6 -0.4 -0.2 0 -4 -3 -2 Real Axis -1 0 1 -1 -0.5 Real Axis 0 0.5 1 Fig. 8. Pole/zero configurations of reduced order models for sample periods T=0.01 (p = 0.03) , 0.01 (p = 0.003), 0.05 (p = 0.03), and 0.1 (p = 0.03) seconds. at the origin or near the origin, so that the system behaves more like a delay, meaning that the packet arrival does not depend as much on its past history as it depends on the drop probability. It also appears that any attempts to reduce the model using system balancing only produce models qualitatively different from the full order models. Besides, reduced models do not seem to be necessary, as the full order pole/zero diagrams are easily to interpret. 2) 0=05  W  UW W  0=1 (Figs 6 and 7). Here we are in the transition range. The strings of pole/zero pairs near the unit circle are somewhat misleading, because balanced model reduction yields models with a pole at the origin, a pole at 1, and a nonminimum phase zero on the negative real axis; the lightly damped poles/zeros have been deleted by the reduction of the model (Figure 8). The irrelevance of the pole/zero pairs was predictable from the close proximity of the poles and the zeros, which can safely be assumed to cancel. It follows that the packet arrival depends on the recent past history and the drop probability. 3) W  0=01 ? UW W (Fig 6). Here we are sampling at a rate faster than the RTT. The raw pole/zero diagram produced by the model using the MDL is a little difficult to interpret. For this reason, we first used balancing to reduce its order. The best compromise between retaining the key features of the original model and having a model of sufficient simplicity to lend itself to analysis is shown in Figure 8. The reduced model retains some of the early features, that is, a pole near the origin (at 0=2), a pole near 1, along with a pair of nearly canceling pole/zero near 1, and a nonminimum phase zero on the negative real axis. Since the sample rate is less than the RTT, the dynamics of TCP can be seen by examining the poles. As discussed in Section II, the simulations included flows with three different round-trip times: 24, 26, 1019 and 28 msec. The reduced model has poles at these locations as well as one more set with an oscillation period of 33 msec and two real poles. The poles corresponding to the round-trip times and one of the real poles have slowest decay. The table below shows the periods of oscillation along with the times that the modes take to decay to 10% of their initial values. oscillation period decay times 26.5 msec 1.2 sec 28.5 msec 0.58 sec oscillation period decay times 24.4 msec 1.1 sec real 0.62 sec real 0.016 sec 33 msec 0.4 sec It is likely that the burstiness of the packet arrivals is the cause of the poles to appear with oscillation periods equal to the round-trip times of the flows. To see this, consider the example of a TCP flow in slowstart with congestion window equal to one. Hence, one packet is transmitted. Upon receiving the ACK for this packet, the congestion window is increased to two and two packets are sent back-to-back. These packets are then received in a back-to-back fashion by the destination host and the ACKs for these are returned, again back-to-back. Upon receiving these two ACKs, the congestion window is increased to 4 and a burst of 4 packets is transmitted. One round-trip time later, a burst of 8 packets is sent. This periodic bursting continues until there is a drop. When TCP is in the congestion avoidance state, packets are still sent in bursts every round-trip time; however, the size of the burst only increases by one packet every round-trip time. In either case, the presence of a connection with a particular round-trip time leads to periodic arrival of packets with the same period as the round-trip time. While the periods of oscillation for some of the poles have a clear physical interpretation, the damping is less clear. However, it appears that a lower average TCP Flows AQ M K(z) - drop probability p sea of poles and zeros, the relationship between round-trip times and pole oscillation periods is not apparent. QoS modeling error w + + dynamics packet G(z) arrival y number of flows, round-trip times Fig. 9. Block diagram of the overall system consisting of a feedback loop for the packet arrival rate and a feedforward structure for the Quality of Service (QoS). V. AQM DESIGN PRINCIPLES After identifying the ARX model, the transmittance J(}) from s to | can be computed. More specifically, PO1 l %(}) l=0 el } |(}) = s(}) + PO1 P O1 }  l=0 dl } l }  l=0 dl } l | {z } | {z } J(}) loss probability leads to a slower decay. This was verified by performing many simulations where the sample rate was held constant, but the average loss probability was reduced by a factor of ten to 0.003. In this case, the reduced order model is similar. The pole/zero diagram is shown in Fig 8. The frequencies of the poles are nearly identical. However, the damping is significantly different. The table below shows the oscillation periods along with the times to decay to 10% of the initial values. oscillation periods decay times 26.0 msec 12 sec 28.0 msec 21 sec oscillation periods decay times 24.1 msec 1.4 sec real 7.7 sec real 0.7 sec 35 msec 4.3 sec Clearly, the modes that correspond to the round-trip times of 26 and 28 msec decay at drastically reduced rates. This trend of slower rates of decay is consistent with other simulations. However, this trend is not exact. For example, in the two tables above, it is noticed that the mode with a period of 24 msec only has a small change in decay rate. Thus, while a direct relationship between the rate of decay of a particular mode and the packet loss probability experienced by the flow that gives rise to the mode cannot be determined, there is a strong relationship between the loss probability of all of the flows and the rate of decay of the entire system. To understand the relationship between the loss probability and the decay rate, note that a flow experiencing ten times smaller loss probability will typically send ten times more packet between packet losses. This leads to a greater average sending rate, but more importantly, a longer time between losses. Intuitively, a reasonable estimate of the future value of the congestion window is that it will increase until roughly 1/S (packet loss) packets are sent. Clearly, if the loss probability is smaller, the effect of the specific value of the congestion window will be noticeable for a longer time than if the loss probability if larger. 4) W ¿ UW W . While no figures are included for this case, it should be remarked that for very small sample periods, the model order is very large even after balanced model order reduction. Furthermore, in the z(}) Recall, that s is the deviation of the loss probability from the nominal loss probability and | is the deviation of the number of packet arrivals during a sample period from the nominal number of packet arrivals during a sample period. This model accurately depicts how, given a fixed number of flows, | will vary over time. This variation can lead to the queue occasionally emptying and hence a decrease in utilization1 . One possible approach to avoid empty queues is to make the queue capacity large. For example, in the single flow case, it is possible to show that if the capacity of the queue is the same as delay-bandwidth product, then the queue will never empty. The drawback is that increasing the queue size also increases the delay. Since the time it takes to transfer a small file is strongly dependent on the delay, increasing the queue size can decrease the performance of the network. Therefore, a principle objective of AQM is for the queue to remain as unoccupied as possible, yet never empty. This goal could be achieved if the number of packet arrivals in a sample period is nearly constant. The more variation in the arrival rate, the larger the variation in the queue occupancy and hence the larger the average queue occupancy required to avoid an empty queue. Considering Figure 9, we see that from the control theory perspective, AQM should minimize the impact of a disturbance z on the output. | There are two ways one can approach this performance objective; either the sup-norm of the transmittance from the disturbance to the output is minimized, or the supnorm of the transmittance from the disturbance to the integral of the output is minimized. There two approaches are nearly the same; in both cases the objective is to minimize the weighted sensitivity function of the closedloop system. The only difference is the weighting function. Specifically, if V is the transmittance from the disturbance to the output (i.e., V = (1 + JN)1 ), and Z is the transfer function of the weighting function, we seek to minimize kZ Vk4 . However, it is a good practice to also minimize W , the complementary sensitivity, where W := (1 + JN)1 we seek to find a controller N that ° ° JN. Thus ° ZV ° ° minimizes ° ° W ° . As it turns out, the inclusion of the 4 complementary sensitivity does not influence the controller. 1 If the queue never empties, then the link is always sending packet, hence full utilization is achieved. On the other hand, if the queue empties, the link will be idle, less than full utilization. 1020 1 4 0 3 2 -1 1 -2 Magnitude (dB) Next we discuss the rationale behind the selection of different weighting functions. First, we consider the weighting function an integrator. Thus, we seek to ° 1that includes ° ° Z (}) V (}) ° } ° , where we have decomposed minimize ° ° ° W 4 the weighting function into the integrator and Z= The rationale behind including an integrator is that the integrator mimics the queue and therefore this control strategy will penalize large queueing delay. Further weighting with Z is also appropriate. To see this, compare the effect of low frequency oscillations in the queue occupancy to high frequency oscillations. High frequency oscillations will appear as short-term, intermittent congestion. However, low frequency oscillations will appear as persistent congestion that will include persistent queueing delay and/or multiple drops. Such persistent congestion has adverse effects on different applications. For example, in voice-over-IP this persistent congestion can have a significant impact on quality, while short-lived congestion has little impact on quality. TCP connections can suffer time-outs if congestion is persistent. Thus, if an integrator is incorporated into the weighting function, a low pass filter should also be included. There is some motivation to not include the integrator, and simply focus on minimizing the variation in the number of packet arrivals in a sample period. To see this, consider the case where the nominal queue occupancy is very small. In this case, if the packet arrival rate falls below the link speed, then the queue will empty and remain empty until the number of arrivals in a sample period surpasses the link-capacity. On the other hand, the integral of | will tend towards 4. In order to force the integrator to zero, the controller will eventually cause the number of packet arrivals to increase beyond the link capacity. However, since the queue was fixed at zero, this increase in data rate will cause the queue to fill. Similarly, [4] discussed the idea of the setting the desired packet arrival rate to be below the link speed. In this case, since small positive deviations of the number of packet arrivals does not affect the queue, incorporating an integrator into the weighting function makes even less sense. While it would be best if the sensitivity could be reduced over all frequencies, the Bode limitation discussed in the next section shows that this is not possible. Thus, the frequencies over which the sensitivity should be reduced must be determined. Consider low frequency variation in |. Such variations will cause the packet arrival rate to increase beyond the link speed for extended periods of time. This will cause large queueing delay and perhaps fill the queue and cause uncontrolled drops. On the other hand, high frequency variations in the number of packet arrivals will not cause the queue to fill significantly. Thus, a low-pass weighting function should be used to put emphasis on the sensitivity function at low frequencies. Next, the cut-off frequency of the weighting functions is determined. To make this selection, we consider the fundamental frequency of TCP. Consider a TCP flow with 0 -3 -4 0 10 -3 x 10 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 0 10 -1 1 10 2 10 3 -2 0 10 10 -3 x 10 1.5 1 10 2 10 3 10 1 0.5 0 -0.5 -1 1 10 2 10 3-1.5 0 1 10 10 10 Frequency (rad/sec) 2 10 3 10 Fig. 10. Design of the controllers in the upper frames did not have an integrator. Rather, the weighting function is a simple low-pass filter. The cut-off frequency of the weighting function for the sensitivity on the left was 6 rad/sec and was 50 rad/sec on the right. Design of the controllers in the lower frames had an integrator. The weighting function used in the left-hand plot was an integrator while the weighting function used in the right-hand plot was an integrator and a low-pass filter with cut-off at 50 rad/sec. nominal loss probability of 3% and RTT of 25 msec. It is well known that the average p size s of the congestion window of this flow is given by 3@2@ 0=03 = 7. If the TCP flow assumes its saw-tooth pattern, then the maximum value of the window is around 9 and the minimum is 4. The variation between 4 and 9 takes 5 round-trip times or 125 msec. Thus we see that the fundamental frequency of this flow is 50 udg@ sec. The damping of these deviations of | at these frequencies could be a design goal. However, considering how this frequency is intrinsic to TCP, it is not clear if such frequencies could be damped. Indeed, we will see that they cannot be. Since a weighting function with an integrator and one without an integrator are both justified, we examine both situations. In both situations, the conclusion is the same, the controller has a fairly minor impact on the performance of the system. Before embarking on an attempt to reduce sensitivity, we are forewarned that the sensitivity can never be arbitrarily reduced. The Bode limitation imposes strict bounds on what is possible. VI. B ODE LIMITATION From the robust control point of view, the effect of the modeling noise z on the feedback is given by the sensitivity function V via |(}) = V(})z(})> where V(}) = (1 + J(})N(}))1 and N (}) is the transmittance of the AQM. Hence, if the packet arrivals are to be controlled, then the impact of this noise must be reduced by 1021 minimizing the sensitivity function. In terms of the power spectral densities || > zz of |> z, respectively, the above becomes l l )V(hl ) zz (hl )= || (h ) = V(h that the sensitivity is only slightly below -1dB. Again, it is difficult to justify the inclusion of such a controller in a network. The drop probability is imposed in the same time interval over which the packet arrival is observed. Therefore, in general, there is a direct transmission and J(4) 6= 0= However, the controller can only decide on a drop probability over [nW> (n  1)W ] after it has observed the packet arrival on [(n 2)W> (n 1)W ]. Therefore, the loop O(}) = N(})J(}) inevitably have some delay, so that O(4) = 0. The latter, along with the open-loop stability of the system, yields the Bode limitation [15]: Z + ¡ ¢ log V(hl )V(hl ) g = 0= This work, along with [8], makes the point that one should not hope for much performance gain at small time-scales by using AQM schemes. Specifically, it seems unlikely that a controller, in a fixed environment, could actively adjust the loss probability to control the short-term variations in the queue occupancy. The conclusion of this work is not that AQM is not a realistic goal, but rather to understand what objectives AQM could achieve. Since it appears fruitless to attempt to control TCP dynamics by varying the loss probability around some a priori imposed static probability, the real problem would be to find, for a given number of flows and their RTTs, the static probability that would secure the desired mean output |. However, changes in the environment (e.g., changes in the number of flows and round-trip times) would require adjusting the static loss probability. Thus, future work will focus on schemes that quickly find the appropriate static loss probability in a way that leads to a stable closed-loop system. Note this stands in contrast to the original idea of AQM where it was hoped that overflow could be minimized by predicting the future value of the queue occupancy.  This result implies that it is not possible to construct a controller that makes the sensitivity uniformly small; if V is decreased at some frequencies, it must increase at others. As a result, we see that one cannot hope to smooth the variations in the packet arrivals at all frequencies. Next we examine several examples to determine to what degree the sensitivity can be reduced at any frequency. We will see that it is difficult to get the sensitivity below -1dB. Thus, the controller has little impact on damping disturbance and smoothing the variations in |. VII. K 4 CONTROLLER We examine two examples where no integrator is incorporated into the weighting function and two examples with an integrator. We employ K 4 control so that we are assured that the controller is optimal with respect to minimizing the weighted sensitivity function. The upper frames of Figure 10 show the closed-loop sensitivity function where no integrator is incorporated into the weighting function. The right-hand plot in the upper frames shows the case where the cut-off for the weighting function is slightly above 50 udg@ sec which is the fundamental frequency of the TCP flows considered here. In this case, the sensitivity in the low frequencies is above -1dB, indicating rather poor performance. By decreasing the cut-off frequency of the weighting function, the sensitivity can be decreased at low frequencies. However, the left-hand plot in the upper frames of Figure 10 shows that even if the cut-off is at very low frequencies, the sensitivity is only slightly below -3dB. Thus, such a controller would decrease the disturbances with periods of one second or greater (recall that the TCP flows considered here have a fundamental period of oscillation of around 125 msec). The controller acts to cut the impact of the disturbance by 30 %. While a reduction by 30% is useful, it is not particularly large. This controller also has no impact on the disturbances with period below 1 sec. We have found significant disturbances at these moderate frequencies. The lower frames of Figure 10 show the sensitivity when an integrator is included in the weighting function. We see VIII. C ONCLUSION R EFERENCES [1] W. Stevens, TCP/IP Illustrated, Vol, Addison Wesley, New York, 1994. [2] Sally Floyd and Van Jacobson, “Random early detection gateways for congestion avoidance”, IEEE/ACM Transactions on Networking, vol. 1, pp. 397–413, 1993. [3] Eyad Abed and Priya Ranjan, “Nonlinear instabilities in TCP-RED”, in INFOCOM, 2002. [4] Srisankar Kunniyur and R. Srikant, “Analysis and design of an adaptive virtual queue (AVQ) algorithm for active queue management”, in Proceedings of ACM SIGCOMM, 2001. [5] S. Low, F. Paganini, and J. Doyle, “Internet congestion control: An analytical perspective”, IEEE Control Systems Magazine, February 2002. [6] Sanjeeva Athuraliya, Steven H. Low, and David E. Lapsley, “Random early marking”, in QofIS, 2000. [7] G. Vinnicombe, “On the stability of end-to-end congestion control for the internet”, in CUED/F-INFENG/TR.398, 2000. [8] K. Shah, S. Bohacek, and E. Jonckheere, “The predictability of data network traffic”, in American Control Conference, 2003. [9] Stephan Bohacek and Khushboo Shah, “TCP throughput and timeout – steady state and time-varying dynamics”, Globecom, 2004. [10] “Http://www.isi.edu/nsnam”. [11] Lennart Ljung, System Identification; Theory for the User, 1987. [12] J. Rissanen, “Modeling by shortest data description”, Automatica, vol. 14, pp. 465–471, 1978. [13] U. M. Al-Saggaf and G. F. Franklin, “An error bound for a discrete reduced order model of a linear multivariable system”, IEEE transactions on Automatic Control, vol. 32, pp. 815–819, 1987. [14] E. Jonckheere, “Principal component analysis of flexible systemsopen-loop case”, IEEE trans. on AUTOMATIC CONTROL, 1984. [15] Bing-Fei Wu and E. 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