International Journal of Control, Automation, and Systems (2009) 7(5):809-816 DOI 10.1007/s12555-009-0513-0
http://www.springer.com/12555
Global Asymptotic Stability of FAST TCP in the Presence of Link Dynamics Joon-Young Choi, Kyungmo Koo, Jong-Wook Kim, and Jin S. Lee Abstract: We propose a continuous-time model to describe a single-link single-source network with the FAST TCP source. The proposed model explicitly includes both the queuing delay dynamics for the link dynamics and the time-varying network feedback delay. Based on the proposed model, we establish a sufficient condition for the global asymptotic stability of the FAST TCP network. We prove the sufficient condition by constructing two sequences that represent the variations of the lower and upper bound of the source’s congestion window with respect to time, and by showing that the two sequences converge to the equilibrium point of the congestion window. The simulation results illustrate the validity of the sufficient condition for the global asymptotic stability. Keywords: FAST TCP, global asymptotic stability, internet congestion control, nonlinear time delay systems, time-varying time delay.
1. INTRODUCTION Internet congestion control is a distributed feedback control algorithm used to allocate network capacity among competing users according to a specified strategy. Internet congestion control has been commonly implemented by TCP Reno and its variants, which control their congestion windows via the well-known additive increase multiplicative decrease (AIMD) mechanism [1,2]. However, it has been shown that these algorithms are not scalable with increasing network bandwidth-delay product because additive increase is too slow and multiplicative decrease is too severe in the large bandwidth-delay product network [3-5]. In order to cope with this problem, several congestion control algorithms have been proposed for high speed networks with large bandwidth-delay products: HSTCP [6], STCP [7], FAST TCP [5], and BIC TCP [8]. Of these, FAST TCP has a feature that the queuing delay is used as a congestion measure. While the packet loss that is used as a congestion measure in TCP Reno has only binary information about the congestion, the queuing delay indicates the level of congestion, which means how __________ Manuscript received November 22, 2007; revised January 8, 2009; accepted April 13, 2009. Recommended by Editorial Board member Young Soo Suh under the direction of Editor YoungHoon Joo. This work was supported by the Grant of the Korean Ministry of Education, Science and Technology (The Regional Core Research Program/Institute of Logistics Information Technology). Joon-Young Choi is with the Department of Electronic Engineering, Pusan National University, 30 Jangjeon-dong, Geumjeong-gu, Busan 609-735, Korea (e-mail:
[email protected]). Kyungmo Koo and Jin S. Lee are with the Department of Electrical Engineering, Pohang University of Science and Technology, San 31, Hyoja-Dong, Pohang 790-784, Korea (emails: {pumpkins, jsoo}@postech.ac.kr). Jong-Wook Kim is with the Department of Electronics Engineering, Dong-A University, Hadan-dong, Saha-gu, Busan 604-714, Korea (e-mail:
[email protected]). © ICROS, KIEE and Springer 2009
far the current state is from the equilibrium point. Accordingly, the congestion control mechanism that uses the queuing delay as a congestion measure is more responsive to the network congestion and it ensures that the network state is always close to the equilibrium point when the algorithm is stable [5]. Even though extensive experiments on FAST TCP have been conducted with promising results [5], the stability property of FAST TCP has not been studied in depth. Regarding the local stability of FAST TCP, it was shown in [5] that FAST TCP in a single-link network is locally asymptotically stable when the network feedback delay is negelected. A sufficient condition for the local asymptotic stability of FAST TCP in a homogeneous multi-source network was achieved in the presence of the network feedback delay in [9]. Based on a new link model, the local stability of FAST TCP was analyzed for a single-link network in [10]. On the other hand, regarding the global stability of FAST TCP, in [11], a sufficient condition for the global asymptotic stability of FAST TCP in a single-link single-source network was established in the presence of the network feedback delay. In [12], it was shown that FAST TCP in a singlelink multi-source network is globally exponentially stable in the presence of the time-varying network feedback delay, and this was based on a different model from that used in [11]. It is noticeable that all of the published results on the global stability of FAST TCP were achieved under the neglect of the link dynamics in order to avoid the mathematical difficulty in analysis. In this paper, however, we analyze the global asymptotic stability of a single-link single-source network with the FAST TCP source in the presence of both the queuing delay dynamics for the link dynamics and the time-varying network feedback delay. We employ a dynamic model for the FAST TCP source proposed in [9] and we propose a dynamic model for the queuing delay characteristics at the link. Based on the proposed model,
Joon-Young Choi, Kyungmo Koo, Jong-Wook Kim, and Jin S. Lee
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we establish a sufficient condition for the global asymptotic stability. The sufficient condition is proved by constructing two sequences that represent the variations of the lower and upper bound of the source’s congestion window with respect to time, and by showing that the two sequences converge to an identical point, which is the equilibrium point of the source’s congestion window. This paper is organized as follows. Section 2 describes the single-link single-source network model of FAST TCP. Section 3 analyzes the global boundedness property of FAST TCP. Section 4 presents a sufficient condition for the global asymptotic stability of FAST TCP. Section 5 provides simulation results and discussions. Section 6 presents conclusions. 2. NETWORK MODEL In this section, we develop a network model in order to describe the characteristics of FAST TCP. We construct the network model by employing the basic structure of the fluid flow model, where the feedback loop consists of the source’s sending rate, the forward delay, the link’s congestion measure, and the backward delay [4,13-15]. In our model, however, the congestion window is chosen as the state variable at the source instead of the sending rate. The network model based on the source’s congestion window has the advantage that it properly describes the window-based operation of FAST TCP. At the link, the queuing delay is chosen as the state variable in order to indicate the congestion measure. We consider a single-link single-source network, where a pair of sender and receiver nodes is connected via a single bottleneck link as depicted in Fig. 1. The link has a finite transmission capacity c and it is assumed to have an infinite size of buffering storage. The link is associated with the queuing delay p (t ), and the source is associated with the congestion window w(t ). We assume that the source observes the queuing delay q (t ) as a feedback signal in its path: q(t ) := p (t − τ b ),
(1)
where τ b denotes the backward delay in the feedback path from link to source, and the link observes the congestion window y (t ) := w(t − τ f ),
(2)
where τ f denotes the forward delay from source to link. The round trip time (RTT) for the source T (t ) is defined as T (t ) := d + q (t ), where d is the constant round trip propagation time. We assume that T (t ) =
τ f (t ) + τ b (t ) and T (t ) is bounded by a constant T M such that T (t ) ≤ T M < ∞ for all t ≥ t0 . We adopt the following model from [9] as the continuous-time model for the FAST TCP source:
Fig. 1. Network topology. ⎛ ⎞ d w (t ) = γ ⎜ − w(t ) + w(t ) + α ⎟ , d + q(t ) ⎝ ⎠
(3)
where w(t ) is the congestion window size of the source, α > 0 and γ ∈ (0,1] are the congestion control parameters, and the time is measured in the unit of the update period during the FAST TCP operation. Here, we develop a dynamic model in order to describe the queuing delay characteristics at the link. For this purpose, we define a sending rate of the source in order to calculate the capacity consumed by the source. Based on the fact that the queuing delay is generated at the link in itself, we propose the following definition of the source’s sending rate: x(t ) :=
w(t − τ f ) , d + p (t )
(4)
which is not estimated at the source but rather is observed at the link at time t. By applying the newly defined sending rate to the queuing delay model in [9,1618], we obtain a model for the queuing delay dynamics: 1 p (t ) = ( x(t ) − c) +p , c
(5)
where h if z > 0 ⎧ (h) +z := ⎨ max(0, ) z = 0. h if ⎩
(6)
Based on (3) and (5), the whole closed-loop model is rewritten in terms of w(t) and p(t) as ⎛ ⎞ d w (t ) = γ ⎜ − w(t ) + w(t ) + α ⎟ b ⎜ ⎟ d + p (t − τ ) ⎝ ⎠
(7)
+
⎞ 1 ⎛ w(t − τ f ) p (t ) = ⎜ −c⎟ ⎜ ⎟ c ⎝ d + p (t ) ⎠p
(8)
with the initial condition that w(t0 + θ ) = η (θ ) ≥ 0, p(t0 + θ ) = ς (θ ) ≥ 0,
θ ∈ [−T M , 0], M
θ ∈ [−T , 0],
(9) (10)
where η :[−T M , 0] 6 R+ and ς :[−T M , 0] 6 ℜ+ are continuous, and R+ is the set of all nonnegative real numbers. In order to reflect the fact that the congestion
Global Asymptotic Stability of FAST TCP in the Presence of Link Dynamics
window and the queuing delay are always nonnegative in physical networks, we assume in (9) and (10) that the initial conditions are such that w(t) and p(t) are nonnegative. It is obvious from (8) and (10) that p(t ) is nonnegative for all t ≥ t0 , and the non-negativity of w(t ) for all t ≥ t0 is shown in Lemma 1 of Section 3. As shown in (7) and (8), the dynamic model for the congestion window and queuing delay is represented by highly nonlinear delay differential equations with timevarying delays τ f (t ) and τ b (t ). The corresponding equilibrium points w* and p* of (7) and (8) are uniquely computed as w* = α + cd ,
p* =
α c
(11)
.
In the subsequent analysis, we assume that γ = 1 in (7) for brevity of expression, and note that this assumption does not affect any results of this paper. 3. BOUNDEDNESS
In the following lemmas, we show the boundedness properties of w(t ) and p(t ) described by (7) and (8) with respect to time t. Lemma 1: w(t) described by (7) is bounded below as w(t ) > 0 for all t > t0 .
Proof: Suppose that w(t0 ) > 0 and w(t ) = 0 for some value of t. Since w(t0 ) > 0, by continuity of solutions, such a value of t must be strictly greater than zero. Let t1 = inf{t : t > t0 , w(t ) = 0}, then w (t1 ) should be less than or equal to zero. Based on (7), however, it follows that w (t ) at t1 as w (t1 ) = − w(t1 ) +
d d + p(t1 − τ b )
w(t1 ) + α = α > 0, (12)
which is a positive value, and this gives us a contradiction. Therefore, no such t1 exists, and it follows that w(t ) > 0 for all t > t0 . Next, we consider the case when w(t0 ) = 0. In this case, it follows from (7) that w (t0 ) is w (t0 ) = − w(t0 ) +
d d + p(t0 − τ b )
w(t0 ) + α = α > 0, (13)
which implies that w(t ) is increasing at t = t0 , and by continuity of solutions, there must exist a time t2 > 0 such that w(t ) > 0 for t0 < t < t2 . Then, in the same manner as in the case when w(t0 ) > 0, we can prove that w(t ) > 0 for all t > t0 . Accordingly, we prove that w(t ) > 0 for all t > t0 when the initial condition is such that w(t0 ) ≥ 0.
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Note that Lemma 1 shows that w(t ) in (7) is nonnegative for all t ≥ t0 and this justifies selecting the initial condition of w(t ) in (9) as nonnegative values. The following lemma shows an ultimate lower bound of w(t ) in (7). Lemma 2: w(t ) described by (7) is ultimately bounded below as w(t) > α for all sufficiently large t. Proof: From Lemma 1 and the nonnegative definition d of p(t ) in (8), it follows that w(t ) > 0 d + p (t − τ b ) for all t > t0 . By applying this inequality to (7), we can derive the inequality w (t ) > (α − w(t )), from which we have w(t) > α for all sufficiently large t because w (t ) > 0 when w(t ) ≤ α . The following lemma shows the relationship between an ultimate lower bound of w(t) and an ultimate lower bound of p(t) in (7) and (8), respectively. Lemma 3: Let δ be a positive constant such that δ > cd . If w(t ) > δ for all sufficiently large t in (7), then p(t ) >
δ
− d for all sufficiently large t in (8). c Proof: From w(t ) > δ > cd for all sufficiently large
t and τ f (t ) ≤ T M < ∞, it follows that w(t − τ f ) > δ for all sufficiently large t , which along with (8) implies 1 δ − c for all sufficiently large t. that p > c d + p (t ) This inequality implies that p (t ) > 0 when p(t ) ≤
δ
− d . Hence, it follows that
p(t ) >
δ
− d for all c c sufficiently large t. The following lemma shows the relationship between an ultimate upper bound of w(t ) and an ultimate upper bound of p(t) in (7) and (8), respectively. Lemma 4: Let ∆ be a positive constant such that ∆ > cd . If w(t ) < ∆ for all sufficiently large t in (7),
∆ − d p(t) for all sufficiently large t in (8). c Proof: In the same manner as in the proof of Lemma 3, this lemma can be proved. The following lemmas show the relationship between ultimate upper bounds and ultimate lower bounds of w(t ) in (7). Lemma 5: Let φ be a constant such that α ≤ φ ≤ (α + cd ). If α > cd and w(t ) > φ for all sufficiently large t in (7), then it follows that then p(t )
φ > cd for all sufficiently large t , from Lemma 3 it follows that p(t ) >
φ c
− d for all
sufficiently large t , which implies that p(t − τ b ) >
φ c
− d for all sufficiently large t because τ b (t ) ≤ T M
< ∞.
p(t − τ b ) >
φ
−d for all c sufficiently large t to (7) yields that w (t ) < − w(t ) + cd
φ
Then,
applying
w(t ) + α for all sufficiently large t , from which we
cd obtain that w (t ) < 0 when w(t ) ≥ α 1 + . This φ − cd cd implies that w(t ) < α 1 + for all sufficiently φ − cd
α
≥ 1 when α ≤ φ ≤ φ − cd cd (α + cd ), it follows that α 1 + ≥ α + cd . φ − cd Lemma 6: Let Φ be a constant such that Φ ≥ (α + cd ). If α > cd and w(t ) < Φ for all sufficiently large t in (7), then
large t. Furthermore, since
cd w(t ) > α 1 + for all sufficiently large t , (15) Φ − cd cd where α 1 + ≤ α + cd . Φ − cd Proof: In the same manner as in the proof of Lemma 5, this lemma can be proved.
4. GLOBAL ASYMPTOTIC STABILITY
In this section, we establish a sufficient condition for the global asymptotic stability of (7) and (8). The common approach to analyzing the stability of nonlinear time delay systems is to use the Krasovskii stability theory or the Razumikhin stability theory [19-21]. In order to apply these theories, we first need to construct or find a Lyapunov-Krasovskii functional or LyapunovRazumikhin function that is specific to each nonlinear time delay system considered. However, the nonlinear time delay system (7) and (8) has high nonlinearity and time-varying delays, which makes it extremely difficult to obtain Lyapunov-Krasovskii functionals or LyapunovRazumikhin functions for the stability analysis. In order to deal with this problem, we exploit the inherent property of (7) and (8) instead of applying the Krasovskii or Razumikhin stability theory. We construct two sequences {φk } and {Φ k } that represent the variations of the lower and upper bound of w(t), respectively, with respect to time under the condition that α > cd . We start by choosing φ1 as α ≤ φ1 ≤ (α + cd ) ,
and assume that w(t ) > φ1 for all sufficiently large t. The existence of such φ1 is guaranteed by Lemma 2, and from Lemma 5 it follows that cd w(t ) < α 1 + := Φ1 ≥ (α + cd ) φ1 − cd for all sufficiently large t.
(16)
Then, applying Φ1 to Lemma 6 yields cd w(t ) > α 1 + := φ2 ≤ (α + cd ) Φ 1 − cd for all sufficiently large t.
(17)
In the same manner, it follows that cd Φ 2 := α 1 + ≥ (α + cd ), φ2 − cd cd φ3 := α 1 + ≤ (α + cd ), and so on. Φ 2 − cd
(18)
By repeating this procedure, we can construct the two sequences {φk } and {Φ k } such that φk ≤ (α + cd ) ≤ Φ k for all k ≥ 1. Consequently,
{φk }
represents
the variation of lower bounds of w(t) with respect to time; {Φ k } represents the variation of upper bounds of w(t) with respect to time. From (16), (17), and (18), we can derive the recurrence formulae for {φk } and
{Φ k } :
φk +1 = f (φk ) = α 1 +
cd (φk − cd )
, (19) (α − cd ) (φk − cd ) + α cd
cd ( Φ k − cd ) Φ k +1 = f (Φ k ) = α 1 + , (20) (α − cd ) ( Φ − cd ) + α cd k
where the fixed points φ * and Φ* satisfying φ * = f (φ * ) and Φ* = f (Φ* ) are uniquely calculated as
φ * = Φ* = (α + cd ). Since φ * and Φ* are identical to the equilibrium point of w(t) as shown in (11), it is evident that if the sequences {φk } and {Φ k } converge to their fixed points φ * and Φ* , respectively, then w(t) and p(t) in (7) and (8) also converge to their equilibrium points, w* and p* , respectively. In the following theorem, we present a sufficient condition for the global asymptotic stability of (7) and (8). Theorem 1: The FAST TCP network described by (7) and (8) is globally asymptotically stable provided that the congestion control parameter α is chosen such that
α > cd .
(21)
Proof: We prove this theorem by showing that the
Global Asymptotic Stability of FAST TCP in the Presence of Link Dynamics
813
condition (21) is a sufficient condition for the lower and upper bounds of w(t ) to monotonically converge to the equilibrium point w* from arbitrary initial conditions. This implies the Lyapunov stability as well as the global attraction property of w(t), which are equivalent to the global asymptotic stability of w(t). In order to show that f (φk ) is a contraction mapping [22], the derivative of f (φk ) with respect to φk is derived as follows: 2
⎞ df (φk ) ⎛ α cd =⎜ ⎟⎟ . ⎜ dφk ⎝ (α − cd ) (φk − cd ) + α cd ⎠
(22)
Then, from the condition such that α > cd it follows df (φk ) that 0 < < 1, which shows that f (φk ) is a dφk contraction mapping. In the same manner, we can show that f (Φ k ) is also a contraction mapping. Hence, the lower and upper bounds of w(t ) monotonically converge to φ * = Φ* = (α + cd ) from arbitrary initial conditions, and w(t ) monotonically converges to w* from arbitrary initial conditions, which proves the global asymptotic stability of w(t). Moreover, the global asymptotic stability of w(t ) implies the global asymptotic stability of p(t ) in view of (7) and (8). As shown in (21), a single-link single-source network with the FAST TCP source is globally asymptotically stable when the control parameter α is larger than a constant cd that is determined by the link capacity c and the propagation delay d . As the main contribution of this paper, we state that our results are an improvement over those of [11]. Unexpectedly, the sufficient condition (21) is identical to the sufficient condition found in [11] for the global asymptotic stability of FAST TCP in a single-link single-source network. However, note that in [11], the link dynamics was neglected and the time-varying network feedback delay was assumed to be a constant. Therefore, we can conclude that Theorem 1 is applicable to a much larger class of systems than the main theorem of [11]. According to Theorem 1, if the congestion control parameter α is chosen such that α > cd , FAST TCP is globally asymptotically stable, which implies that the queuing delay p(t ) is bounded for all t ≥ t0 when α > cd . The boundedness of p(t ) means that a finite buffering storage is sufficient for FAST TCP to operate in a normal mode. In addition, the boundedness of p(t ) implies that the RTT, T (t ), is bounded because T (t ) := d + p (t − τ b ), which means that the packet dropout never occurs. Therefore, the sufficient condition for the global asymptotic stability (21) mitigates the unrealistic assumption that the link has an infinite size of buffering storage, and ensures that the RTT is bounded.
Fig. 2. Network topology. Table 1. Equilibrium points. c (pkts/ms) / d (ms) Case 1
10 / 10
Case 2
50 / 20
α (pkts) 40 101 400 1001
Window size(pkts) 140 201 1400 2001
Queuing Delay(ms) 4 10.1 8 20.02
5. SIMULATION
We conduct a set of simulations to exemplify Theorem 1 by using both MATLAB and the ns-2 network simulator [23]. We numerically simulate the FAST TCP network model (7) and (8) via MATLAB, and conduct a network simulation of the FAST TCP network via the ns-2 network simulator. We adopt the code of [24] for the ns-2 simulation of the FAST TCP network, but it is slightly modified in order to disable some of the ad-hoc features such as filter dynamics, pacing, and MI, which are not considered when constructing the FAST TCP network model. The network used in the simulation consists of a single bottleneck link utilized by a source with the topology shown in Fig. 2. We simulate two cases of networks, which are denoted as Case 1 and 2. In both cases, the capacity c and round trip latency between routers d are set as c = 10 pkts/ms, d = 10 ms and c = 50 pkts/ms, d = 20 ms, respectively. In Case 1, we conduct the simulation by setting α = 101 and α = 40. The former value just satisfies the sufficient condition of Theorem 1, while the latter value violates the sufficient condition. Similarly, in Case 2, we set α as α = 1001 and α = 400, where the former value just satisfies the sufficient condition of Theorem 1, while the latter value violates the sufficient condition. Table 1 summarizes the calculated equilibrium points of the source’s congestion window and the queuing delay for each case, and Figs. 3 and 4 show the simulation results for each case. Figs. 3 and 4 illustrate that congestion windows and queuing delays are bounded and converge to their equilibrium points when α > cd . Furthermore, as shown in Figs. 3 and 4, the FAST TCP network is not asymptotically stable and oscillatory when the parameter α violates the sufficient condition such that α > cd . 6. CONCLUSIONS
We consider a single-link single source network with the FAST TCP source. We propose a dynamic model for the queuing delay generated at the link via a newly defined sending rate of the source. The whole closedloop model explicitly includes both the queuing delay dynamics for the link dynamics and the time-varying network feedback delay. Based on the proposed model, we present a sufficient condition for the global asymp-
814
Joon-Young Choi, Kyungmo Koo, Jong-Wook Kim, and Jin S. Lee
(a) MATLAB simulation results.
(a) MATLAB simulation results.
(b) ns-2 simulation results. Fig. 3. Congestion window and queuing delay when c = 10 pkts/ms and d = 10 ms (Case1).
(b) ns-2 simulation results. Fig. 4. Congestion window and queuing delay when c = 50 pkts/ms and d = 20 ms (Case2).
totic stability in terms of the network parameters. In order to prove the sufficient condition, we exploit the inherent property of the FAST TCP model instead of applying the Krasovskii or Razumikhin stability theory that may cause mathematical difficulty in analysis. We prove the sufficient condition by constructing two
sequences that represent the variations of the lower and upper bound of the congestion window, and by showing that the two sequences converge to an identical point, which is the equilibrium point of the congestion window. Regarding future works, we mention the following two issues. First, it needs to extend the FAST TCP model
Global Asymptotic Stability of FAST TCP in the Presence of Link Dynamics
and the sufficient condition for the global asymptotic stability to cover FAST TCP operated in a single-link multi-source network. Second, it is worthwhile to establish a sufficient condition for the global asymptotic stability of the FAST TCP network where the link has a finite buffering storage, that is, without the assumption that the link has an infinite buffering storage.
[15]
[16] [1]
[2] [3]
[4]
[5]
[6] [7]
[8] [9] [10]
[11]
[12]
[13]
[14]
REFERENCES R. Jain, K. K. Ramakrishnan, and D.-M. Chiu, “Congestion avoidance in computer networks with a connectionless network layer,” Technical Report DEC-TR-506, Digital Equipment Corporation, August 1987. V. Jacobson, “Congestion avoidance and control,” Proc. of ACM SIGCOMM’88, August 1988. C. V. Hollot, V. Misra, D. Towsley, and W. B. Gong, “Analysis and design of controllers for AQM routers supporting TCP flows,” IEEE Trans. on Automatic Control, vol. 47, no. 6, pp. 945-959, 2002. S. H. Low, F. Paganini, J. Wang, and J. C. Doyle, “Linear stability of TCP/RED and a scalable control,” Computer Networks Journal, vol. 43, no. 5, pp. 633-647, December 2003. D. X. Wei, C. Jin, S. H. Low, and S. Hegde, “FAST TCP: motivation, architecture, algorithms, performance,” IEEE/ACM Trans. on Networking, vol. 14, pp. 1246-1259, Dec. 2006. S. Floyd, “HighSpeed TCP for large congestion windows,” RFC 3649, Experimental, December 2003. http://www.icir.org/floyd/hstcp.html. T. Kelly, “Scalable TCP: improving performance in high-speed wide area networks,” ACM SIGCOMM Computer Communication Review, vol. 33, no. 2, pp. 83-91, April 2003. L. Xu, K. Harfoush, and I. Rhee, “Binary increase congestion control for fast long-distance networks,” Proc. of IEEE Infocom 2004, March 2004. J. Wang, David X. Wei, and Steven H. Low, “Modelling and stability of FAST TCP,” Proc. of IEEE Infocom 2005, April 2005. A. Tang, K. Jacobsson, L. L. H. Andrew, and S. H. Low, “An accurate link model and its application to stability analysis of FAST TCP,” Proc. of IEEE Infocom 2007, May 2007. J.-Y. Choi, K. Koo, J. S. Lee, and S. H. Low, “Global stability of FAST TCP in single-link single-source network,” Proc. 44th IEEE Conf. Decision and Control, Seville, Spain, Dec. 2005. J.-Y. Choi, K. Koo, David X. Wei, J. S. Lee, and Steven H. Low, “Global exponential stability of fast TCP,” Proc. 45th IEEE Conf. Decision and Control, San Diego, CA, USA, Dec. 2006. F. Paganini, Z. Wang, J. C. Doyle, and S. H. Low, “Congestion control for high performance, stability and fairness in general networks,” IEEE/ACM Trans. on Networking, vol. 13, no. 1, pp. 43-56, Feb. 2005. F. P. Kelly, A. Maulloo, and D. Tan, “Rate control
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for communication networks: shadow prices, proportional fairness and stability,” Jour. Oper. Res. Society, vol. 49, no. 3, pp. 237-252, March 1998. S. H. Low and D. E. Lapsley, “Optimization flow control, I: basic algorithm and convergence,” IEEE/ACM Trans. on Networking, vol. 7, no. 6, pp. 861-874, Dec. 1999. S. H. Low, Larry L. Peterson, and L.Wang, “Understanding Vegas: a duality model,” Journal of the ACM, vol. 49, no. 2, pp. 207-235, 2002. S. H. Low, F. Paganini, and J. C. Doyle, “Internet congestion control,” IEEE Control Systems Magazine, vol. 22, pp. 28-43, 2002. H. Choe and S. H. Low, “Stabilized Vegas,” Proc. of IEEE Infocom, April 2003. S.-I. Niculescu, Delay Effects on Stability: A Robust Control Approach, Springer, London, 2001. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, 3rd ed., Prentice Hall, 2002. M. Jankovic, “Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems,” IEEE Trans. Automatic Control, vol. 46, no. 7, pp. 1048-1060, July 2001. H. K. Khalil, Nonlinear Systems, 3rd ed., Prentice Hall, 2002. The network simulator - ns-2. [Online] Available at http://www.isi.edu/nsnam/ns. FAST TCP simulator module for ns-2 version 1.1. [Online] Available at http://www.cubinlab.ee.mu.oz. au/ns2fasttcp.
Joon-Young Choi received the B.S., M.S. and Ph.D. degrees in Electronic and Electrical Engineering from Pohang University of Science and Technology (POSTECH), Pohang, Korea in 1994, 1996 and 2002, respectively. From 2002 to 2004, he worked as a Senior Engineer at Electronics and Telecommunication Research Institute (ETRI), Daejeon, Korea. From 2004 to 2005, he worked as a Visiting Associate in the Departments of Computer Science and Electrical Engineering at California Institute of Technology (CALTECH), Pasadena, CA. Since 2005, he has been an Assistant Professor in the School of Electrical Engineering at Pusan National University, Busan, Korea. His research interests include nonlinear control, internet congestion control, embedded systems and automation. Kyungmo Koo received the B.S. degree in Electrical Engineering from Pohang University of Science and Technology (POSTECH), Korea in 2001. He is now a Ph.D. candidate in Electrical Engineering and Computer Science at POSTECH, Korea since 2001. During his studies, he spent one year at the California Institute of Technology as a visiting researcher in 2004. His research interests are in the control and optimization of nonlinear systems.
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Joon-Young Choi, Kyungmo Koo, Jong-Wook Kim, and Jin S. Lee
Jong Wook Kim received the B.S., M.S., and Ph.D. degrees in the Electronic and Electrical Engineering from Pohang University of Science and Technology (POSTECH), Pohang, Korea, in 1998, 2000, and 2004, respectively. Currently, he is an Assistant Professor in the Department of Electronics Engineering at Dong-A University, Busan, Korea. His current research interests are numerical optimization methods, robot control, intelligent control, diagnosis of electrical systems, and system identification.
Jin S. Lee received the B.S. degree in Electronics Engineering from Seoul National University, Seoul, Korea, in 1975, the M.S. degree in Electrical Engineering and Computer Science from the University of California, Berkeley, in 1980, and the Ph.D. degree in System Science from the University of California, Los Angeles, in 1984. From 1984 to 1985, he worked as a Member of Technical Staff at AT&T Bell laboratories, Holmdel, NJ, and from 1985 to 1089, as a Senior Member of Engineering Staff at GE Advanced Technology Laboratories, MT. Laurel, NJ. Since 1989, he has been a Professor at Pohang University of Science and Technology (POSTECH), Pohang, Korea. From 1998 to 2000, he served as the Head of the Electrical Engineering Department, POSTECH and from 2000 to 2003, as the Dean of Research Affairs at POSTECH. From 2003 to 2004, he was with the Computer Science Department, the California Institute of Technology, Pasadena, for his sabbatical leave. His research interests include nonlinear systems and control, robotics, and automation.
