IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 1, JANUARY 2005
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On Control Gain Selection in Dynamic-RED Liansheng Tan, Yan Yang, Wei Zhang, and Moshe Zukerman, Senior Member, IEEE Abstract— Based on control theory, this letter provides guidelines for the selection of the control gain for dynamic-RED to stabilize a congested queue at a target and hence to improve network performance. Simulations demonstrate that indeed satisfactory performance can be achieved if the control gain is selected based on the guidelines. Index Terms— Active queue management, random early detection (RED), congestion control, stability.
I. I NTRODUCTION ANDOM early detection (RED) [2] is a widely studied active queue management (AQM) scheme implemented in routers for Internet congestion control. RED is based on early detection and probabilistic signaling of congestion. As other congestion control and avoidance schemes [9], the goal of RED is to stabilize the queue length at a given target [1, 3, 5, 8] to reduce data drop and improve system throughput. Unfortunately it is difficult to provide any systematic and robust law to configure RED’s parameters [6]. Although certain stability conditions for various RED controllers are discussed in [3, 4], no efficient method for dynamically and optimally adapting control gains has been proposed so far. Ref. [1] proposes the so-called dynamic-RED (DRED) scheme that uses a proportional control function to enhance the performance of RED, but its control gain selection is based on empirical investigation and simulation analysis. In this paper, we investigate the issue of stabilizing the queue length at a given target from a model-based and control theoretic standpoint. We use a model of TCP and RED dynamics [3] and the DRED controller of [1]. We evaluate the stability range of the gain, and provide guidelines for control gain selection to stabilize the router queue and thus to improve the performance of DRED. Simulations provide further insight into performance implications.
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II. S TABILITY A NALYSIS AND THE P ROPOSED A LGORITHM We use the TCP model of [3] as described by the following equations: 2 ˙ (t) = − 2N W (t) − RC p(t − R) W R2 C 2N 2 N 1 q(t) ˙ = W (t) − q(t) R R
(1) (2)
Manuscript received January 31, 2004. The associate editor coordinating the review of this letter and approving it for publication was Prof. Nasir Ghani. This research has been partially supported by the Projects of Development Plan of the State Key Fundamental Research in China under Grant No. 2003CB314804. Liansheng Tan, Yan Yang and Wei Zhang are with the Computer Science Department of Central China Normal University, Wuhan 430079, PR. China (e-mail:
[email protected]). M. Zukerman is with the Australian Research Council Special Research Centre for Ultra-Broadband Information Networks (CUBIN) an affiliated program of National ICT Australia, EEE Dept., The University of Melbourne, Victoria 3010, Australia (e-mail:
[email protected]). Digital Object Identifier 10.1109/LCOMM.2005.01026.
where the variables are defined as follows: W : Expected ˙ (t): Time-derivative of W ; q: TCP window size (packets); W Current queue length (packets); C: Capacity of link (packets per second); N : Load factor (number of TCP connections); p: Probability of packet drop; p(t): ˙ Time-derivative of p; R: Round-trip time, i.e., the sum of q/C (the processing time of buffer queue) and Tp (propagation time of packet in the link). The function of the drop probability is defined [1] as p(t) ˙ =
α(q(t) − qref ) B
(3)
where α is the proportional control gain, qref is the target (packets) and B is the (allocated) router buffer size serving only as a normalization parameter. A crucial issue in DRED is the setting of the control gain α. To obtain a theoretic guideline for setting the control gain, we take Laplace transform of equations (1), (2) and (3). This gives 2N RC 2 −Rs W (s) − e p(s) R2 C 2N 2 N W (s) Q(s) sQ(s) − q(0) = − R R qref α ). sp(s) − p(0) = (Q(s) − B s
sW (s) − W (0) = −
(4) (5) (6)
By the above equations, we obtain µ ¶ 2N q(0) N W (0) 2 3 ∆(s)Q(s) = q(0)s + + s R2 C R µ ¶ C2 R 2 s2 − 1 − sR + (p(0)s − αqref )(7) 2N 2 2 2
where we assumed e−Rs ≈ 1 − sR + R 2s and denoted µ ¶ 1 2N R2 C 2 3 ∆(s) := s[s + + − α s2 R R2 C 4N B µ ¶ 2N RC 2 C2 + − α s + α]. (8) R3 C 2N B 2N B The component ∆(s) is approximately the characteristic equation (CE) of the closed-loop system (1), (2) and (3). Therefore, we only consider the following polynomial ¶ µ 2N R2 C 2 1 + 2 − α s2 ∆1 (s) := s3 + R R C 4N B µ ¶ 2 2N RC C2 + − α s+ α. (9) 3 R C 2N B 2N B The system described by (1), (2) and (3) is stable if and only if all the zeros of ∆1 (s) are in the open left-half plane. We now construct the Routh table [5] in Table 1, and further compute the following coefficient γ31 =
RC 2 C2 1 2N R2 C 2 2N − α − α/( + − α). R3 C 2N B 2N B R R2 C 4N B
c 2005 IEEE 1089-7798/04$20.00 °
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IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 1, JANUARY 2005
Routh-Hurwitz stability test [5] states that the closed-loop system is stable if and only if the values of the second column are all greater than zero, i.e., 1/R + 2N/(R2 C) − R2 C 2 α/(4N B) > 0, C 2 α/(2N B) > 0, and γ31 > 0. Therefore, the TCP dynamic system (1) and (2) is considered to be stable in terms of the queue level under the DRED scheme (3) if and only if the control gain α in the DRED scheme satisfies ¡C ¢ q¡ C ¢ 1 3 2 − N + 2R + − NCR N + 2R 0
