A Prediction Algorithm for Feedback Control Models with ... - CiteSeerX

A Prediction Algorithm for Feedback Control Models with Long Delays B. Jang, B.G. Kim, and G. Pecelli Computer Science Dept. Univ. of Mass. Lowell Lowell, MA 01854 [email protected]

Abstract It is well known that feedback control may be dicult in a network with long propagation delays (with large bandwidth-delay products). Various feedback-based congestion control algorithms considered for networks with long delays have been reviewed from control theoretical viewpoints. Most existing control algorithms are shown to provide threshold-based control technique. However, as the propagation delay becomes signi cant, the past history over the entire time lag may have to be incorporated into the control strategy. An adaptive algorithm based on the optimal least mean square error technique is presented. The algorithm attempts to predict a future buer size from weight (slope) adaptation of unknown functions which are used for feedback control. Simulation results are presented, which show the eectiveness of the algorithm.

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1 Introduction Congestion in data networks has been traditionally handled by employing a feedback control. Data sources obtain network congestion status in an implicit (from increasing response times) or an explicit manner. According to a perceived congestion status, sources increase or decrease transmission rates or window sizes. In TCP (transmission control protocol), a source's window size is set to one packet when an acknowledgement is not received in the timeout period and is incremented at each subsequent acknowledgement (up to a threshold). In the DECbit algorithm, an absence of congestion bit allows a source to linearly increase its transmission rate, but its presence forces a source's rate to drop exponentially. In a high-speed network, by the time a source receives a congestion information, many packets may have already been sent out by the source. This may result in packet losses and subsequent retransmissions. In addition, trac oscillations have been observed. When the network is congested, sources reduce their transmission rates. As the congestion is eased, those sources resume sending at higher rates forcing another congestion. In order to overcome these problems, protocols in ATM networks are designed on the basis of a connection-oriented approach. Before data is being sent, a user speci es the expected traf c characteristics and the desired quality of service. As long as the behavior of the user trac conforms to the declared characteristics at the time of the call establishment, network resources should be available all the time to meet the quality of service. However, data trac tends to be bursty. Also, data network services have been built around the connectionless services that data can be sent at any time without a prior connection establishment. ABR (Available Bit Rate) service is devised in the ATM community to help migrate data services to ATM networks. Here, the ABR bandwidth is fairly shared by ABR sources as long as ABR sources adjust their transmission rates according to the perceived level of congestion in the network. Namely, feedback congestion control algorithms are reintroduced in high-speed networks. In this article, we assume that the readers are familiar with ATM ABR services [Atmf 96]. Most studies of feedback congestion control schemes tend to focus on the thresholdbased control algorithms [Benm 93, Fend 92]. Since a long propagation delay will prevent any eective control within the round trip time window, a predictive algorithm is presented in this paper. By monitoring the past history of the system (either the input rate or the buer size), one may be able to parameterized the system states and predict future system status. For a prediction algorithm in this paper, a normalized least mean square (NLMS) algorithm is used. The model and the algorithm are described in the next section, followed by simulation results in section 3.

2 THE MODEL AND THE ALGORITHM

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2 The Model and the Algorithm 2.1 The Model The basic framework of the control model is illustrated in Fig. 1. N sources send cells to a single bottleneck switch. Cell emission rates and the bottleneck status (the queue length) at the switch are updated periodically. Without loss of generality, we assume that round trip delays between sources and the switch are in multiple units of the periodic update interval, as indicated by di in the gure. The state of the system is represented by the buer size in the node, Q(n). d . . .

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Figure 1: Feedback Control Model

2.2 The Predictor When the buer size exceed the high threshold (TH ), the switch is considered to be in a congestion and the switch computes the explicit rate (ER) at which sources have to send cells to the switch in order to avoid the congestion. If the buer size is less than the low threshold (TL) then the switch computes the ACR with exponential increasing rates. The NLMS predictor in the gures estimates the buer size in the next k steps based on a current value of the buer size and weighting factor (slope) at time n. Let Q(n) denote the buer size at time n. The k-step predictor is formulated such that the buer size at k steps in the future is estimated from the Q(n), as given by Q^ (n + k) = ak (n)Q(n)

(1)

where a(n) is an estimated weight factor at time instant n, and k = 1; 2;


; t and t is a maximum prediction interval.

2 THE MODEL AND THE ALGORITHM

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Error of the prediction at time n is e(n) = Q(n) ? Q^ (n)

(2) where Q^ (n) = a(n?1)Q(n?1). The prediction scheme uses the error to modify the weighting factor whenever the error is available at each time step. Furthermore, the weighting factor a(n) is aected in time as sources are added or removed and as the activity levels of source changes. We thus put the problem into the one of estimating the weighting factor and use the normalized least mean square error (NLMS) linear prediction algorithm. Given an initial value for a(0) = 0, the weighting factors are updated by e(n)Q(n ? 1) (3) a(n) = a(n ? 1) + j Q(n ? 1) j2 where  is a constant. If Q(n) is stationary, a(n) is known to converge in the mean squared sense to the optimal solution [Adas 96, Hayk 91]. The NLMS is known to be less sensitive to the factor . The estimated weight factor a(n) in each time-step will be used to predict the buer size ^ Q(n + k). Therefore each time step, the weighting factor indicates the direction of evolution of the functions for buer size increases/decreases in term of recent residual computed by the estimated and actual buer sizes. In the following gure, the prediction scheme is illustarated. If a  1, the predicted buer size will be inceased by Q^ (n + k) = ak (n)Q(n). Therefore k, a time to hit the TH , is predicted at time n using Q(n) and a(n) which are clearly known at time n. Q(n)

a(n) >> 1 T H

a(n) = 1

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Figure 2: The Prediction Scheme

2.3 ER Algorithm We use NLMS algorithm to predict the buer size at k steps ahead of the time. Based on the predicted buer size in the future, Q^ (n + k) from the time n, the explicit rate (ER)

3 SIMULATION RESULTS

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is computed at the switch according to the algorithm in [Ritt 96]. Brie y, ER is computed as follows. 1. When no congestion is detected (the Q^ (n + k) is not greater than the high threshold, TH ) and Q^ (n + k) > TL , the value of ER does not change. The ACR (Allowed cell rate) for the sources is computed at the switch as speci ed in the source behavior in the ATM Forum standard (by the linear increase algorithm in [Atmf 96]). 2. When no congestion is detected and Q^ (n + k) < TL then the ACR for the sources is computed at the switch using the exponential increase speci ed in [Atmf 96]. 3. If the congestion is detected in terms of the predicted queue length, Q^ (n+k) from the time n, ER is computed by ER = Fair Share  Explicit Reduction Factor, where Fair Share = Link speed at switch / Number of sources.

3 Simulation Results The link speed of the switch is set to 50Mpbs, and the link speed for each source to the switch is set to 150Mpbs=N for N sources. The control algorithm is experimented for a single bottleneck switch with a buer of high and low thresholds of 5000 and 1000 cells, respectively. Ten active sources located at various distances are used for simulation. In order to examine the transitional behaviors, two abrupt changes of active sources are made. Initially, sources located at f5,6,8,9,10,13,17,19,20,23,32,39,40,43,50g are active. At time 1937, sources at f2,10,13,17,19,20,23,24,24,32,39,40,42,43,50g are active, which are further changed to those at f9,15,24,24,32,32,34,39,39,40,42,42,43,43,50g at time 2694. In Fig. 3, the switch buer sizes are plotted when the NLMS prediction algorithm is not used. In the gure, high and low threshold values of 5000 and 1000 are shown as well as transition times 1937 and 2694. It is seen that the buer size can exceed the high threshold. When the NLMS predition algorithm is used, the resulting changes in the buer sizes are shown in Fig. 3. Clearly, the gure illustrates that the buer sizes remain under the high threshold most of the time. Also, the width of uctuation is reduced as well. Details of the changes in buer sizes near the transition time of 1937 are shown in Fig. 4 without and with the NLMS prediction algorithm. Several other con gurations are simulated and their results are similar to those presented here. Although the buer sizes uctuate even with the NLMS algorithm, the buer sizes can be bounded within the high threshold and low threshold. The algorithm appears to be promising in delivering the desirable QoS (Quality of Service) demand, in particular the loss

4 CONCLUSION

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ratio of ABR connections. Further studies are needed to investigate a theoretical treatment of the prediction algorithm and to examine other prediction algorithms.

4 Conclusion In this paper, the rate control based on predicted buer sizes is studied. In particular, the normalized least mean square (NLMS) error technique is used to predict the buer size in k steps. The rate control based on the NLMS prediction shows a tighter control of buer sizes compared to those without the NLMS prediction. The prediction method was applied to a particular ER-based control algorithm which was recommended by the ATM Forum. More study is however necessary to establish whether the prediction-based control can be equally eective in other ER control algorithms.

References [Adas 96] A. Adas, \Supporting Real Time VBR Video Using Dynamic Reservation Based on

Linear Prediction," Infocom 96. [Atmf 96] ATM Forum, Trac Managment v.4.0, Aug. 1996. [Benm 93] L. Benmohamed and S.M. Meerkov, \Feedback Control of Congestion in Packet Switching Networks: The Case of a Single Congested Node," IEEE/ACM Trans. Networking, Vol. 1, No. 6, Dec. 1993, 693-708. [Fend 92] K.W. Fendick, M.A. Rodrigues and A. Weiss, \Analysis of a Rate-based Feedback Control Strategy for Long Haul Data Transport," Perf. Eval., 16, 1992, 67-84. [Haye 96] M.H. Hayes, \Statistical Signal Processing and Modeling," John Wiley & Sons, 1996. [Hayk 91] S. Haykin, \Adaptive Filter Theory," Prentice Hall, 1991. [Kesh 91] S. Keshav, \A Control-Theoretic Approach to Flow Control," Sigcomm 91, 1991, 3-15. [Masc 96] S. Mascolo,, D. Cavendish, and M. Gerla, \ATM Rate Based Congestion Control Using a Smith Predictor: an EPRCA Implementation," Infocom 96. [Ritt 96] M. Ritter, \Network Buer Requirements of the Rate-based Control Mechanism for ABR Services," Infocom 96.

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Figure 5: Comparison Result of The Feedback Control Models