- Email: [email protected]
Controlling chaos in Internet congestion control model
Chaos, Solitons and Fractals 21 (2004) 81–91 www.elsevier.com/locate/chaos
Controlling chaos in Internet congestion control model Liang Chen *, Xiaofan Wang, Zhengzhi Han School of Electronics and Information Technology and Electrical Engineering, Shanghai Jiaotong University, Shanghai 200030, PR China Accepted 6 October 2003 Communicated by Prof. B.G. Sidharth
Abstract The TCP end-to-end congestion control plus RED router queue management can be modeled as a discrete-time dynamical system, which may create complex bifurcating and chaotic behavior. Based on the basic features of the TCPRED model, we propose a time-dependent delayed feedback control algorithm to control chaos in the system by perturbing the accessible RED parameter pmax . This method is able to stabilized a router queue occupancy at a level without knowing the exact knowledge of the network. Further, we study the situation of the presence of the UDP traffic. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction The sustained explosive growth of the Internet over the past decade has made it becoming part of our life and the modern human society. The Internet is designed to operate over different underlying communication technologies including those yet to be introduced, and to support multiple and evolving applications and services. The use of the Internet protocol (IP) to communicate across different kinds of networks shields the users of the Internet from the complex reality of the network [1]. This has enormous advantages for users who need not to worry about the complexities of the networks they are using. However, these advantages are not without cost: careful design is required to guarantee the stability of the Internet and provide good service under heavy load. An uncontrolled network may suffer from severe congestion, which can cause high packet loss rates and increasing delays, and can even break the whole system by causing congestion collapse (or ÔInternet meltdown’). This is the state where any increase in the offered load leads to a decrease in the useful work done by the network. In fact, such congestion collapses already occurred in the mid 1980s when the Internet had no deployed congestion avoidance mechanism. The problem of congestion cannot be solved by introducing almost Ôinfinite’ buffer space inside the network. In fact, too much buffer space in the routers can be more harmful than too little because the packets will have to be dropped only after they have consumed valuable network resources. The stability of today’s Internet has in large part due to the congestion control and avoidance mechanisms implemented in its end-to-end transmission control protocol (TCP), developed by Jacobson during the late of 1980s [2]. TCP employs a window-based flow control mechanism. The sender keeps a congestion window (CWND) outstanding in the network. The window size limits the number of unacknowledged packets the sender can have. The destination sends acknowledgements for packets that are correctly received. When the window size is exhausted, the source must wait for an acknowledgement before sending a new packet. This is known as the Ôself-clocking’ feature of TCP. The basis of Jacobson’s congestion control algorithm lies in the additive
*
Corresponding author. Fax: +86-21-62932083-802. E-mail address: [email protected] (L. Chen).
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.09.037
82
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
increase multiplicative decrease (AIMD) mechanism, i.e., halving the congestion window for every window containing a packet loss, and increasing the congestion window by roughly one segment per round trip time (RTT) otherwise. Other components of TCP congestion control include retransmit timers, slow-start and ACK-clocking [1]. Given the importance of TCP congestion control to the health of the Internet, there have been a number of proposed modifications to its algorithms [3]. There have also been considerable efforts toward modelling and analysis of dynamics of TCP congestion control [4–6]. In particular, it has been demonstrated that TCP can produce for certain parameters simple and for others very complex chaotic behavior [7]. It has become clear that the TCP congestion control and avoidance mechanisms, while necessary and powerful, are not sufficient to provide good services in all circumstances. Basically, there is a limit to how much control can be accomplished from the edges of the network. Some mechanisms are needed in the routers to complement the endpoint congestion avoidance mechanisms. Traditionally, Internet routers have managed the queues at their links by setting a maximum length for each queue, accepting packets until the maximum length is reached and then dropping subsequent incoming packets until space becomes available in the queue. The method is known as drop-tail because packets arriving at the end of the queue (the tail) are dropped when the queue is full. Drop-tail has served the Internet well for years. However, one of the main problems with the TCP congestion control algorithm over current drop-tail Internet is that the sending sources reduce their transmission rates only after detecting packet loss due to queue overflow. This is a problem since a considerable amount of time may pass between when the packet is dropped at the router and when the source actually detects the loss. In the meantime, a large number of packets may be dropped as sources continue to transmit at a rate that the network cannot support. Thus, drop-tail buffer management forces network operators to choose between high utilization (requiring large buffers) and low delay (requiring small buffers). One solution to this problem is for the routers to drop (or mark) packets before a queue becomes full, so that end nodes can respond to congestion before buffers overflow. Such a proactive approach has been called Ôactive queue management (AQM)’ [8]. By dropping packets before buffers overflow, AQM allows routers to control when and how many packets to drop. One form of AQM recommended by the Internet engineering task force (IETF) for deployment in the network is random early detection (RED) [9]. However, it is very difficult to parameterize RED in order to give good performance under different congestion scenarios. In almost all studies the parameter settings are based on heuristics, and the proposed configuration is suitable only for the particular traffic condition studied. It is possible that the performance of a RED router approach that of a drop-tail router for a given set of configuration parameters and traffic conditions. RED try to stabilize the average queue length on a target value but the current version of RED does not succeed in this goal because the equilibrium average queue length strongly depends on the traffic loads and parameter settings. Instability in TCP-RED often leads to oscillations of average queue length [10,11]. Some researchers have even advocated against using RED because of this parameter setting difficulty. To avoid the problems of RED, several other AQM mechanisms have been proposed, which represent substantial departures from the basic RED design. These include stabilized RED (SRED) [12], random early marking (REM) [13], proportional-integral (PI) controller [14] and dynamic-RED (DRED) [15]. Recently, Ranjan and Abed showed that the TCP-RED could be approximately modelled as a first-order nonlinear map [16]. They found that the TCP-RED map exhibits a rich variety of irregular behaviors such as bifurcation and chaos. In fact, given the fact that even a very simple nonlinear map can produce complex chaotic behaviors, the appearance of bifurcation and chaos in the strongly nonlinear TCP-RED map should not be surprising. Over the past decade, there have been tremendous interests in controlling bifurcation and chaos in dynamical systems [17,18]. Although a large number of bifurcation and chaos control approaches have been proposed, many of them cannot be directly applied to the control of oscillations in TCP-RED system. This is due to the complexity of the network and physical limitation on the allowable control. In [19], a delayed feedback control (DFC) algorithm was developed, which has many attractions when applied in the physical system, such as relatively little priori system information to stabilize the equilibrium, a very low computational overhead and extremely easy implementation in hardware. In this work, ‘‘gentle’’ version of the RED [20] is considered. After describing the nonlinear dynamics of the TCPRED model, we modified the original time-delayed feedback control method and proposed a time-dependent adaptive algorithm to control chaos in the system. This technique is easy to implement and does not require the exact knowledge of the network. Further, we study the situation of the presence of the user datagram protocol (UDP) traffic, which is another transport protocol in the Internet. This paper is organized as follows. In Section 2, we introduce the TCP-RED mechanism in control system framework and model it as a discrete-time dynamical system. The properties of the TCP-RED map is also described briefly. Section 3 is devoted to describing the time-dependent DFC algorithm and giving bifurcation analysis. Numerical examples are included in Section 4 as to controlling chaos in TCP-RED map. In Section 5, the presence of UDP traffic is considered. And we conclude in Section 6.
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
83
2. A discrete-time model of TCP-RED congestion control 2.1. General discription We first construct a discrete-time dynamical model of TCP congestion control at the hosts coupled with RED active queue management at the routers. Upon detection of congestion, the sources should inject their packets into the network more slowly. In order for a host to be able to detect congestion, the routers must be able to provide the feedback information that the network is currently (or is about to become) overloaded. Packet drops were, and to great extent are still, the only means for a router to fight congestion. The sources become aware of the packet drops, interpret them as a feedback congestion indication, and reduce their rates. The feedback from the network and the response from the source are the foundations of the Internet congestion control and are very important because they facilitate decentralized resource allocation. Congestion control mechanisms were mainly implemented in the end hosts. However, with decisions made at the end hosts and treatment of the routers as black boxes that simply drop packets, there is clearly a limit on how much control can be achieved over the allocation of network resources. This also limits the range of services the network is capable of offering. In fact, routers know exactly how congested they are and can therefore perform more drastic resource management. Thus, the introduction of router mechanisms for congestion control that will enable the network to more actively manage its own resources seems inescapable. This motivates the avocation of active queue management (AQM) in the routers. The most prominent AQM mechanism is RED. RED controls congestion by randomly dropping packets with a probability that is a function of the average queue length in the buffer of the router. There are two distinct algorithms operating in a RED router: (i) the algorithm for computing the average queue length, which determines the degree of burstiness that will be allowed; (ii) the algorithm for calculating the packet-dropping probability, which determines how frequently packets get dropped given the current level of congestion. To derive a discrete-time, analytical description of the TCP-RED congestion control system, we consider a system of n TCP flows passing through a common link with capacity c as in Fig. 1. We assume that all access links, Ai –B and C–Di have enough capacity so that B–C is the only bottleneck link, i.e., the only link where incoming traffic rate can surpass the link capacity. We further assume that each TCP flow has the same Round Trip Time (RTT) R. The actual queue size q in the router is sampled every Dt ¼ R units of time, and the RED controller provides a new value of the drop probability p every Dt units of time. We can model the whole TCP-RED congestion control system as a discrete-time feedback dynamical system (see Fig. 2). Suppose that at time tk the actual queue length is qk . Due to the burstiness of the network traffic and other perturbations, the actual queue length is highly fluctuating so that a low-pass filter A is desirable. The filtered average queue length is given by qk ¼ Aðqk1 ; qk Þ:
ð1Þ
RED manages the queue length by randomly dropping packets with a probability pk that is an increasing function H of the average queue length, i.e., pk ¼ H ðqk Þ:
ð2Þ
Fig. 1. A system of n TCP flows passing through a common link.
84
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
Fig. 2. TCP-RED as a discrete-time feedback control system.
At time tk þ 1 ¼ tk þ Dt, the TCP hosts react to the drop probability pk and adjust their sending rates. This results in a new queue length qkþ1 that is a function of pk , i.e., qkþ1 ¼ Gðpk Þ:
ð3Þ
Eqs. (1)–(3) together imply that the TCP-RED congestion control can be described by the following discrete-time dynamical system model for average queue length: qkþ1 ¼ Aðqk ; GðH ðqk ÞÞÞ f ðqk Þ:
ð4Þ
2.2. An analytical model of TCP-RED Here we present a simplified analytical formula for the map f in Eq. (4). RED computes the average queue as following: qkþ1 ¼ Aðqk1 ; qk Þ ¼ ð1 wÞqk1 þ wqk ;
0 < w < 1;
ð5Þ
which appears as an exponential weighted moving average of the queue length and can be expressed as qkþ1 ¼ w
k X ð1 wÞi qki ;
if q0 ¼ 0:
i¼0
The RED algorithm manages the queue length by randomly dropping packets with a probability p. In the ‘‘gentle’’ version of RED, the packet drop rate increases linearly from zero when the average queue length is at the RED parameter minthresh (denoted by qmin ), to a drop rate of pmax when the average queue length reaches maxthresh (denoted by qmax ). Then the packet-dropping probability increases linearly from pmax to 1 as the average length varies from qmax to 2qmax (see Fig. 3). More precisely, at time tk , RED computes the drop probability pk as following:
Fig. 3. The drop probability of the RED algorithm.
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
8 0; > > > qk qmin > > > < q q pmax ; max min pk ¼ H ðqk Þ ¼ 1 pmax > > > ðqk qmax Þ; p þ > > max qmax > : 1;
85
06 qk < qmin ; qmin 6 qk < qmax ; ð6Þ qmax 6 qk < 2qmax ; 2qmax 6 qk 6 B;
where B is the buffer size. Now we derive an explicit expression of the map G in Eq. (3). Following the procedure suggested in [10], we can define the function G as follows: 8 p P p1 ; < 0; ð7Þ GðpÞ ¼ Mc ðTR1 p; nc R0 Þ; p2 6 p < p1 ; : B; p < p2 ; and where T is the throughput of a TCP flow (in bits/sec), M is the packet size (in bits) and R0 isthe propagation transmission time. TR1 and Tp1 are the inverse of T ðp; RÞ in R and p, respectively. p1 ¼ Tp1 nc ; R0 represents the maximum dropping probability for which the system is fully utilized, i.e., for p > p1 senders will have their rates too represents the minimum dropping probability for which no small to keep the link fully utilized. p2 ¼ Tp1 nc ; R0 þ BM c packet overflows the buffer, i.e., for p < p2 senders will have their rates too large to keep all packets in the buffer. A complex throughput function T ðp; RÞ is given in [10]. Here we adopt a simple and easy to analyze throughput function described as follows [16]: MK T ðp; RÞ ¼ pffiffiffi ; R p pffiffiffiffiffiffiffiffi where K is a constant and 1 6 K 6 8=3. The plant function G can be computed as 8 0; p P p1 ; > > < nK R c 0 ; p2 6 p < p1 ; GðpÞ ¼ pffiffiffi p M > > : B; p < p2 ;
2
2 nMK , p . where p1 nMK 2 R0 c BMþR0 c
ð8Þ
Given the map A, H and G in Eqs. (5), (6) and (8), respectively, the TCP-RED congestion control system (4) can be described as a 1 D map for the average queue length: qkþ1 ¼ f ðqk ; qÞ ¼ ð1 wÞqk þ wGðH ðqk ÞÞ;
ð9Þ
where q represents a vector of system parameters. According to the different parameter settings, the TCP-RED model can be described as three kinds of maps. Here we assume p1 < pmax , which is the interesting region from the application point of view and can guarantee good performance of RED. Then the TCP-RED map (9) can be expressed as follows: 8 > qk P b1 ; < ð1 wÞqk ; qkþ1 ¼ f ðqk ; qÞ ¼ f ðqk ; qÞ; ð10Þ b2 6 qk < b1 ; > : ð1 wÞq þ wB; q < b : k k 2 Here, 0
1
nK 0 R0 c C B f ðqk ; qÞ ð1 wÞqk þ w@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A: pmax ðqk qmin Þ M
ð11Þ
qmax qmin
Borders b1 and b2 are given by b1
p1 ðqmax qmin Þ þ qmin pmax
and
b2
p2 ðqmax qmin Þ þ qmin : pmax
Clearly, due to the extreme complexity of the network, it is impossible to derive an exact, analytical description of the TCP-RED congestion control system. Even exact knowledge of the fixed point of the system is not available.
86
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
However, the map f described in Eq. (10) characterizes some important features of the TCP-RED, which can be useful to study and improve the performance of Internet congestion control. 2.3. Properties of the TCP-RED model If the average queue length is greater than b1 , the TCP sending rates would be too small to keep the link highly utilized. On the other hand, if the average queue length is less than b2 , the TCP sending rates would be too large to keep low delay. Therefore, it is desirable to keep the average queue length greater than b2 and less than b1 . In this region the system is represented as the strongly nonlinear map f . This map has a fixed point q 2 ðb2 ; b1 Þ, which is a real solution of the following equation: 2 R0 c ðnKÞ2 ¼ ðqmax qmin Þ: ðq qmin Þ q þ M pmax The eigenvalue of the system at this fixed point is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of ðq ; qÞ wnK qmax qmin a1 ¼1w < 1: 3=2 oq pmax 2ðq qmin Þ
ð12Þ
Furthermore, of ðq ; pmax Þ 1 ¼ wnKq3=2 a2 max opmax 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qmax qmin < 0: q qmin
A necessary and sufficient condition for the asymptotic stability of the fixed point q is 1 < a1 < 1. However, it has been found that the TCP-RED can exhibit complex oscillating behavior, which implies the instability of the fixed point. Here we show the bifurcation behavior of the TCP-RED model with the exponential averaging weight w as the bifurcation parameter. System parameters are chosen as follows [16]: pffiffiffiffiffiffiffiffi B ¼ 300 packets; K ¼ 8=3; R0 ¼ 0:1 s; M ¼ 0:5 kb; ð13Þ n ¼ 20; c ¼ 1500 kbps; qmax ¼ 100; qmin ¼ 50; pmax ¼ 0:1: 0:047. For w < w , the average It can be seen from Fig. 4 that the system exhibits a period-doubling bifurcation at w 0:056, two-band chaotic oscillations become
Fig. 4. Bifurcation diagram of the average queue length w.r.t. the weight w. The upper dashed line represents b1 and the lower dashed line represents b2 .
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
87
a single band chaotic oscillation. Such a border-collision-induced chaotic oscillation is Ômalignant’ and should be avoided, since it can result in significantly decreasing throughput and increasing dropping rates. In summary, the TCP-RED system has the following properties: (i) The system can exhibit complex bifurcation and chaotic behaviors. It has an unique unstable fixed point q 2 ðb2 ; b1 Þ; (ii) The eigenvalue of the system at this fixed point q ;pmax Þ satisfies a1 ¼ a1 ðq ; pmax Þ < 1; (iii) a2 of ðop < 0. max 3. Controlling chaos in TCP-RED via delayed feedback control The DFC method, originally designed to controlling chaos in continuous-time systems, is based on a feedback of the difference between the current state and the delayed state, which vanishes after stabilizing an equilibrium embedded in the original system. Later on, Socolar et al. [21] changed it into a discrete version and suggested the control be added to the accessible parameter for easy implementation. Based on the properties (i)–(iii) of the TCP-RED system, we can add a delayed feedback perturbation on the parameter pmax in the RED algorithm so as to guarantee the stability of the fixed point q . Let us consider the following controlled TCP-RED system qkþ1 ¼ f ðqk ; pmax;k Þ ¼ f ðqk ; pmax þ dpmax;k Þ;
ð14Þ
where dpmax;k is the parameter perturbation. The traditional DFC method is given by dpmax;k ¼ hðqk qk1 Þ
ð15Þ
with a constant feedback gain h. Our study shows that the controller (15) can at best stabilize the system at the periodtwo orbit. Moreover, the oscillation is sensitive to noises and may go beyond the Ôsafe’ region (qmin ; qmax ), which is not desired in the complex network. However, based on the idea of the DFC method, the process of stabilization can be looked upon as the process of minimization of the difference between the current average length and the previous one. When the difference tends to zero, the system becomes stable. A well-known example of a minimization procedure is the gradient decent method. This motivates us to introduce the following time-dependent delayed feedback control algorithm: dpmax;k ¼
of wnK qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hð hðqk qk1 Þ ¼ qk1 Þ: qk qk qmin opmax 1:5 2p max
ð16Þ
qmax qmin
As we expected, this approach can successfully stabilize the TCP-RED system at the original fixed point, expect that we have to know the concrete expression of the partial derivation, which also limit its application. To remove the disadvantage without sacrificing other benefits of the controller (16), we make some modification and assume that the parameter H can be selected such that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wnK qmax qmin of h¼ h: ð17Þ H¼ 1:5 opmax 2pmax Then the parameter perturbation given in (16) can be written as qk qk1 dpmax;k ¼ H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : qk qmin
ð18Þ
Theorem. Consider the control system described by (14) and (18). The fixed pointed q is asymptotically stable if and only if pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q qmin a1 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q ;pmax Þ . where a1 of ðq o;pq max Þ and a2 of ðop max
qk q . Linearizing system (14) about the fixed point gives Proof. Denote dqk ¼ dqkþ1 ¼ a1 dqk þ a2 dpmax;k :
ð20Þ
88
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
Substituting (18) into (14) and denoting xk ¼ dqk , yk ¼ d qk1 , we have the following second-order nonlinear difference equation ! x k yk xkþ1 a1 xk þ a2 H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð21Þ xk þ q qmin : ykþ1 xk Linearizing system (21) about (0,0) gives 0 1 a2 H a2 H xk xkþ1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p a þ 1 A q q ¼@ : q q min min ykþ1 yk 1 0
ð22Þ
The stability of (22) is governed by the characteristic equation 0 1 a2 H a2 H B C ¼ 0: z2 @a1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Az þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q qmin q qmin
ð23Þ
The fixed point is asymptotically stable provided that both solutions of (23) fall inside the unit circle, i.e., jzj < 1. The condition is satisfied if and only if (19) holds. h Now we study the effectiveness of the time-dependent DFC method (18) from the view of bifurcation analysis. We define the bifurcation point to be the exponential averaging weight w at which the eigenvalue of the system becomes )1 and the initial period-doubling bifurcation occurs [24]. After the bifurcation point the system becomes quickly unstable and the queue size oscillates widely(see Fig. 4). Hence an effective approach of improving the stability is to make the bifurcation point as larger as possible. In the following, we demonstrate that the TCP-RED system with time-dependent delayed feedback control (18) is more stable than the original TCP-RED map for a given set of configuration parameters. Note that from (12) the bifurcation point of the original TCP-RED system is given by w R ¼
2 ; 1þK
ð24Þ
where K¼
nK 2ðq
qmin Þ1:5
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 0: pmax qmax qmin
ð25Þ
On the other hand, note that from (23) the bifurcation point of the controlled system (14) and (18) is given by w D ¼
2 1 þ K þ 2KH
pffiffiffiffiffiffiffiffiffiffiffi ffi: q qmin pmax
ð26Þ
From the properties of the TCP-RED system, we know that a2 < 0 and a1 < 1 when the system is unstable. Then H must be chosen as a negative value to guarantee the stability of the fixed point q . Hence, we have w R < w D :
4. Example Considering the recommended range of pmax in [22], we have the following control form 99 8 8 >> > > < < qk qk1 == pmax;k ¼ min 0:5; max 0:01; pmax þ H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; > > qk qmin > ;> : ; :
ð27Þ
which does not influence the results of Theorem. System parameters are given by (13) with w ¼ 0:07. The control is actuated at k ¼ 100 with the constant gain H ¼ 0:01. Fig. 5 shows results of numerical simulations. One can see that
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
89
Fig. 5. Controlling chaos in the TCP-RED map using time-dependent delayed feedback perturbation actuated at time k ¼ 100.
after a transient process the system is stabilized at the originally fixed point and the perturbation tends to zero as expected.
5. Presence of UDP traffic Besides TCP, the Internet makes another transport protocol available to its multiple applications and services: User Datagram Protocol (UDP). UDP offers a direct way to send and receive datagrams. It is transaction oriented, and delivery and duplicate protection are not guaranteed. UDP is used with many real-time applications such as Internet telephone, real-time video conferencing, and streaming of stored audio and video. All these applications can tolerate a small fraction of packet loss, so that reliable data transfer is not absolutely critical for the success of the application. The nonadaptive UDP-based applications, to some degree, impact the stability of the Internet. In this section we consider the presence of the UDP traffic. From the perspective of TCP connections, UDP traffic takes away some of the available bandwidth. Hence, given the UDP load cu , the available capacity for the TCP connections becomes c cu ð1 pÞ [23] and the plant function G in Eq. (7) is changed into 8 0; p P p3 ; > > < c c ð1 pÞ c cu ð1 pÞ u 1 GðpÞ ¼ TR p; R 0 ; p4 6 p < p3 ; > M n > : B; p < p4 ; c cu ð1 pÞ c cu ð1 pÞ BM . ; R0 and p4 Tp1 ; R0 þ where p3 Tp1 n n c cu ð1 pÞ Then the TCP-UDP-RED map can be expressed as follows: 8 qk P b3 ; < ð1 wÞqk ; qkþ1 gðqk ; qÞ ¼ gðqk ; qÞ; b4 6 qk < b3 ; ð28Þ : ð1 wÞqk þ wB; qk < b4 with 0
1
cnK R0 c C B gðqk ; qÞ ð1 wÞqk þ w@ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A: qk qmin ÞÞÞ M gðqk qmin Þðc cu ð1 gð
p3 and p4 are the positive, real solutions of the following two equations, respectively. Here, g ¼ qmaxpmax qmin cu p33=2 þ ðc cu Þp31=2
nMK ¼ 0; R0
90
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
Fig. 6. Bifurcation diagram of the average queue length w.r.t. w. The upper dashed line represents b3 and the lower dashed line represents b4 .
Fig. 7. Control of chaos in the TCP-UDP-RED map using time-dependent delayed feedback perturbation actuated at time k ¼ 100.
nMK cu p43=2 þ ðc cu Þp41=2 BM ¼ 0: þ R0 c Borders b3 and b4 in Eq.(28) are given by b3
p3 ðqmax qmin Þ þ qmin pmax
and
b4
p4 ðqmax qmin Þ þ qmin : pmax
This TCP-UDP-RED model also exhibits complex oscillating behavior, which is shown in Fig. 6 as an example. Here, the UDP traffic cu ¼ 100 kbps and other system parameters are chosen as (13). Fig. 7 shows the numerical results when the time-dependent DFC method (18) is applied to control chaos in the TCP-UDP-RED map. One can see that the system is also stabilized at the originally fixed point.
6. Conclusions Congestion control in the Internet is an extremely important and challenging problem, which has been the main subject of intensive studies over the last decade. In this work, controlling bifurcation and chaos in an Internet con-
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
91
gestion control system model––the TCP-RED map is investigated. We have described in this paper a time-dependent DFC method, which can control chaos in the system by perturbing the accessible parameter pmax . We have shown through bifurcation analysis and simulations that the controlled system is much stabler than the original one and the router queue occupancy can be stabilized at a level. The benefits of a stabilized queue in a network are high resources utilization, predictable maximum delays, ease in buffer provisioning, and traffic intensity and number of connections [15]. Moreover, it was also shown that using the proposed method for the presence of the UDP traffic gives similar performance. We will further investigate the effectiveness of the proposed approach to improving the performance of the RED router queue management mechanism.
Acknowledgements This work was supported by the National Natural Science Foundation of China with the grant number 60174005 and 69874025.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
Peterson L, Davie B. Computer networks: a system approach. 2nd ed. Morgan Kaufmann; 1999. Jacobson V. Congestion avoidance and control. Proc ACM SIGCOMM 1988;18(4):314–29. Floyd S. A report on recent development in TCP congestion control. IEEE Commun Mag 2001;39(4):84–90. Mathis M, Semke J, et al. The macroscopic behavior of the TCP congestion avoidance algorithm. Comput Commun Rev 1997;27(3):633–43. Padhye J, Firoiu V, et al. Modeling TCP throughput: a simple model and its empirical validation. ACM Comput Commun Rev 1998;28(4):303–14. Hespanha JP, Bohacek S, et al. Hybrid modeling of TCP congestion control. Lect Notes Comput Sci 2001;2034:291–304. Veres A, Boda M. The chaotic nature of TCP congestion control. In: Proceedings of IEEE Infocom. 2000. Available from: http:// www.ieee-infocom.org/2000. Braden B et al. Recommendations on queue management and congestion avoidance in the Internet. IETF Request for Comments (RFC) 1998, p. 2309. Floyd S, Jacobson V. Random early detection gateways for congestion avoidance. IEEE Trans Network 1997;1:397–413. Firoiu V, Borden M. A study of active queue management for congestion control. In: Proceedings of IEEE Infocom. 2000. Available from: http://www.ieee-infocom.org/2000. Hollot C, Misra VA. Control theoretic analysis of RED. In: Proceedings of IEEE Infocom. 2001. Available from: http://www.ieeeinfocom.org/2001. Ott T, Lakshman T, et al. SRED: stabilized RED. In: Proceedings of IEEE Infocom. 1999. Available from: http://www.ieeeinfocom.org/1999. Athuraliya S, Li VH, et al. REM: active queue management. IEEE Network 2001;15:48–53. Hollot C, Misra V et al. On designing improved controllers for AQM routers supporting TCP flows. In: Proceedings of IEEE Infocom. 2001. Available from: http://www.ieee-infocom.org/2001. Aweya J, Ouellette M, et al. A control theoretic approach to active queue management. Comput Networks 2001;36:203–35. Ranjan P, Abed EH. Nonlinear instabilities in TCP-RED. In: Proceedings of IEEE Infocom. 2002. Available from: http:// www.ieee-infocom.org/2002. Chen G, Dong X. From chaos to order: perspectives, methodologies, and applications. Singapore: World Scientific; 1998. Chen G, editor. Controlling chaos and bifurcations in engineering systems. Boca Raton, FL: CRC Press; 1999. Pyragas K. Continuous control of chaos by self-controlling feedback. Phys Lett A 1992;170:421–8. Floyd S. Recommendation on using the ‘‘gentle’’ variant of RED. March 2000. Available from: http://www.aciri.org/floyd/red/ gentle.html. Socolar JES, Sukow DW, et al. Stabilizing unstable periodic orbits in fast dynamic systems. Phys Rev E 1994;50(4):3245–8. Floyd S, Gummadi R, Shenker S. Adaptive RED: an algorithm for increasing the robustness of RED’s active queue management, 2001. Available from: http://www.aciri.org/floyd/papers/adaptiveRed.ps. La RJ, Ranjan P, et al. Nonlinear dynamics of mixed TCP and UDP traffic under RED. MED. 2002. Available from: http:// www.ece.umd.edu/~hyongla/papers/med2002.pdf. La RJ, Ranjan P, et al. Analysis of adaptive random early detection, 2003. Available from: http://www.glue.umd/edu/~priya/ paper272.pdf.
Controlling chaos in Internet congestion control model Liang Chen *, Xiaofan Wang, Zhengzhi Han School of Electronics and Information Technology and Electrical Engineering, Shanghai Jiaotong University, Shanghai 200030, PR China Accepted 6 October 2003 Communicated by Prof. B.G. Sidharth
Abstract The TCP end-to-end congestion control plus RED router queue management can be modeled as a discrete-time dynamical system, which may create complex bifurcating and chaotic behavior. Based on the basic features of the TCPRED model, we propose a time-dependent delayed feedback control algorithm to control chaos in the system by perturbing the accessible RED parameter pmax . This method is able to stabilized a router queue occupancy at a level without knowing the exact knowledge of the network. Further, we study the situation of the presence of the UDP traffic. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction The sustained explosive growth of the Internet over the past decade has made it becoming part of our life and the modern human society. The Internet is designed to operate over different underlying communication technologies including those yet to be introduced, and to support multiple and evolving applications and services. The use of the Internet protocol (IP) to communicate across different kinds of networks shields the users of the Internet from the complex reality of the network [1]. This has enormous advantages for users who need not to worry about the complexities of the networks they are using. However, these advantages are not without cost: careful design is required to guarantee the stability of the Internet and provide good service under heavy load. An uncontrolled network may suffer from severe congestion, which can cause high packet loss rates and increasing delays, and can even break the whole system by causing congestion collapse (or ÔInternet meltdown’). This is the state where any increase in the offered load leads to a decrease in the useful work done by the network. In fact, such congestion collapses already occurred in the mid 1980s when the Internet had no deployed congestion avoidance mechanism. The problem of congestion cannot be solved by introducing almost Ôinfinite’ buffer space inside the network. In fact, too much buffer space in the routers can be more harmful than too little because the packets will have to be dropped only after they have consumed valuable network resources. The stability of today’s Internet has in large part due to the congestion control and avoidance mechanisms implemented in its end-to-end transmission control protocol (TCP), developed by Jacobson during the late of 1980s [2]. TCP employs a window-based flow control mechanism. The sender keeps a congestion window (CWND) outstanding in the network. The window size limits the number of unacknowledged packets the sender can have. The destination sends acknowledgements for packets that are correctly received. When the window size is exhausted, the source must wait for an acknowledgement before sending a new packet. This is known as the Ôself-clocking’ feature of TCP. The basis of Jacobson’s congestion control algorithm lies in the additive
*
Corresponding author. Fax: +86-21-62932083-802. E-mail address: [email protected] (L. Chen).
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.09.037
82
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
increase multiplicative decrease (AIMD) mechanism, i.e., halving the congestion window for every window containing a packet loss, and increasing the congestion window by roughly one segment per round trip time (RTT) otherwise. Other components of TCP congestion control include retransmit timers, slow-start and ACK-clocking [1]. Given the importance of TCP congestion control to the health of the Internet, there have been a number of proposed modifications to its algorithms [3]. There have also been considerable efforts toward modelling and analysis of dynamics of TCP congestion control [4–6]. In particular, it has been demonstrated that TCP can produce for certain parameters simple and for others very complex chaotic behavior [7]. It has become clear that the TCP congestion control and avoidance mechanisms, while necessary and powerful, are not sufficient to provide good services in all circumstances. Basically, there is a limit to how much control can be accomplished from the edges of the network. Some mechanisms are needed in the routers to complement the endpoint congestion avoidance mechanisms. Traditionally, Internet routers have managed the queues at their links by setting a maximum length for each queue, accepting packets until the maximum length is reached and then dropping subsequent incoming packets until space becomes available in the queue. The method is known as drop-tail because packets arriving at the end of the queue (the tail) are dropped when the queue is full. Drop-tail has served the Internet well for years. However, one of the main problems with the TCP congestion control algorithm over current drop-tail Internet is that the sending sources reduce their transmission rates only after detecting packet loss due to queue overflow. This is a problem since a considerable amount of time may pass between when the packet is dropped at the router and when the source actually detects the loss. In the meantime, a large number of packets may be dropped as sources continue to transmit at a rate that the network cannot support. Thus, drop-tail buffer management forces network operators to choose between high utilization (requiring large buffers) and low delay (requiring small buffers). One solution to this problem is for the routers to drop (or mark) packets before a queue becomes full, so that end nodes can respond to congestion before buffers overflow. Such a proactive approach has been called Ôactive queue management (AQM)’ [8]. By dropping packets before buffers overflow, AQM allows routers to control when and how many packets to drop. One form of AQM recommended by the Internet engineering task force (IETF) for deployment in the network is random early detection (RED) [9]. However, it is very difficult to parameterize RED in order to give good performance under different congestion scenarios. In almost all studies the parameter settings are based on heuristics, and the proposed configuration is suitable only for the particular traffic condition studied. It is possible that the performance of a RED router approach that of a drop-tail router for a given set of configuration parameters and traffic conditions. RED try to stabilize the average queue length on a target value but the current version of RED does not succeed in this goal because the equilibrium average queue length strongly depends on the traffic loads and parameter settings. Instability in TCP-RED often leads to oscillations of average queue length [10,11]. Some researchers have even advocated against using RED because of this parameter setting difficulty. To avoid the problems of RED, several other AQM mechanisms have been proposed, which represent substantial departures from the basic RED design. These include stabilized RED (SRED) [12], random early marking (REM) [13], proportional-integral (PI) controller [14] and dynamic-RED (DRED) [15]. Recently, Ranjan and Abed showed that the TCP-RED could be approximately modelled as a first-order nonlinear map [16]. They found that the TCP-RED map exhibits a rich variety of irregular behaviors such as bifurcation and chaos. In fact, given the fact that even a very simple nonlinear map can produce complex chaotic behaviors, the appearance of bifurcation and chaos in the strongly nonlinear TCP-RED map should not be surprising. Over the past decade, there have been tremendous interests in controlling bifurcation and chaos in dynamical systems [17,18]. Although a large number of bifurcation and chaos control approaches have been proposed, many of them cannot be directly applied to the control of oscillations in TCP-RED system. This is due to the complexity of the network and physical limitation on the allowable control. In [19], a delayed feedback control (DFC) algorithm was developed, which has many attractions when applied in the physical system, such as relatively little priori system information to stabilize the equilibrium, a very low computational overhead and extremely easy implementation in hardware. In this work, ‘‘gentle’’ version of the RED [20] is considered. After describing the nonlinear dynamics of the TCPRED model, we modified the original time-delayed feedback control method and proposed a time-dependent adaptive algorithm to control chaos in the system. This technique is easy to implement and does not require the exact knowledge of the network. Further, we study the situation of the presence of the user datagram protocol (UDP) traffic, which is another transport protocol in the Internet. This paper is organized as follows. In Section 2, we introduce the TCP-RED mechanism in control system framework and model it as a discrete-time dynamical system. The properties of the TCP-RED map is also described briefly. Section 3 is devoted to describing the time-dependent DFC algorithm and giving bifurcation analysis. Numerical examples are included in Section 4 as to controlling chaos in TCP-RED map. In Section 5, the presence of UDP traffic is considered. And we conclude in Section 6.
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
83
2. A discrete-time model of TCP-RED congestion control 2.1. General discription We first construct a discrete-time dynamical model of TCP congestion control at the hosts coupled with RED active queue management at the routers. Upon detection of congestion, the sources should inject their packets into the network more slowly. In order for a host to be able to detect congestion, the routers must be able to provide the feedback information that the network is currently (or is about to become) overloaded. Packet drops were, and to great extent are still, the only means for a router to fight congestion. The sources become aware of the packet drops, interpret them as a feedback congestion indication, and reduce their rates. The feedback from the network and the response from the source are the foundations of the Internet congestion control and are very important because they facilitate decentralized resource allocation. Congestion control mechanisms were mainly implemented in the end hosts. However, with decisions made at the end hosts and treatment of the routers as black boxes that simply drop packets, there is clearly a limit on how much control can be achieved over the allocation of network resources. This also limits the range of services the network is capable of offering. In fact, routers know exactly how congested they are and can therefore perform more drastic resource management. Thus, the introduction of router mechanisms for congestion control that will enable the network to more actively manage its own resources seems inescapable. This motivates the avocation of active queue management (AQM) in the routers. The most prominent AQM mechanism is RED. RED controls congestion by randomly dropping packets with a probability that is a function of the average queue length in the buffer of the router. There are two distinct algorithms operating in a RED router: (i) the algorithm for computing the average queue length, which determines the degree of burstiness that will be allowed; (ii) the algorithm for calculating the packet-dropping probability, which determines how frequently packets get dropped given the current level of congestion. To derive a discrete-time, analytical description of the TCP-RED congestion control system, we consider a system of n TCP flows passing through a common link with capacity c as in Fig. 1. We assume that all access links, Ai –B and C–Di have enough capacity so that B–C is the only bottleneck link, i.e., the only link where incoming traffic rate can surpass the link capacity. We further assume that each TCP flow has the same Round Trip Time (RTT) R. The actual queue size q in the router is sampled every Dt ¼ R units of time, and the RED controller provides a new value of the drop probability p every Dt units of time. We can model the whole TCP-RED congestion control system as a discrete-time feedback dynamical system (see Fig. 2). Suppose that at time tk the actual queue length is qk . Due to the burstiness of the network traffic and other perturbations, the actual queue length is highly fluctuating so that a low-pass filter A is desirable. The filtered average queue length is given by qk ¼ Aðqk1 ; qk Þ:
ð1Þ
RED manages the queue length by randomly dropping packets with a probability pk that is an increasing function H of the average queue length, i.e., pk ¼ H ðqk Þ:
ð2Þ
Fig. 1. A system of n TCP flows passing through a common link.
84
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
Fig. 2. TCP-RED as a discrete-time feedback control system.
At time tk þ 1 ¼ tk þ Dt, the TCP hosts react to the drop probability pk and adjust their sending rates. This results in a new queue length qkþ1 that is a function of pk , i.e., qkþ1 ¼ Gðpk Þ:
ð3Þ
Eqs. (1)–(3) together imply that the TCP-RED congestion control can be described by the following discrete-time dynamical system model for average queue length: qkþ1 ¼ Aðqk ; GðH ðqk ÞÞÞ f ðqk Þ:
ð4Þ
2.2. An analytical model of TCP-RED Here we present a simplified analytical formula for the map f in Eq. (4). RED computes the average queue as following: qkþ1 ¼ Aðqk1 ; qk Þ ¼ ð1 wÞqk1 þ wqk ;
0 < w < 1;
ð5Þ
which appears as an exponential weighted moving average of the queue length and can be expressed as qkþ1 ¼ w
k X ð1 wÞi qki ;
if q0 ¼ 0:
i¼0
The RED algorithm manages the queue length by randomly dropping packets with a probability p. In the ‘‘gentle’’ version of RED, the packet drop rate increases linearly from zero when the average queue length is at the RED parameter minthresh (denoted by qmin ), to a drop rate of pmax when the average queue length reaches maxthresh (denoted by qmax ). Then the packet-dropping probability increases linearly from pmax to 1 as the average length varies from qmax to 2qmax (see Fig. 3). More precisely, at time tk , RED computes the drop probability pk as following:
Fig. 3. The drop probability of the RED algorithm.
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
8 0; > > > qk qmin > > > < q q pmax ; max min pk ¼ H ðqk Þ ¼ 1 pmax > > > ðqk qmax Þ; p þ > > max qmax > : 1;
85
06 qk < qmin ; qmin 6 qk < qmax ; ð6Þ qmax 6 qk < 2qmax ; 2qmax 6 qk 6 B;
where B is the buffer size. Now we derive an explicit expression of the map G in Eq. (3). Following the procedure suggested in [10], we can define the function G as follows: 8 p P p1 ; < 0; ð7Þ GðpÞ ¼ Mc ðTR1 p; nc R0 Þ; p2 6 p < p1 ; : B; p < p2 ; and where T is the throughput of a TCP flow (in bits/sec), M is the packet size (in bits) and R0 isthe propagation transmission time. TR1 and Tp1 are the inverse of T ðp; RÞ in R and p, respectively. p1 ¼ Tp1 nc ; R0 represents the maximum dropping probability for which the system is fully utilized, i.e., for p > p1 senders will have their rates too represents the minimum dropping probability for which no small to keep the link fully utilized. p2 ¼ Tp1 nc ; R0 þ BM c packet overflows the buffer, i.e., for p < p2 senders will have their rates too large to keep all packets in the buffer. A complex throughput function T ðp; RÞ is given in [10]. Here we adopt a simple and easy to analyze throughput function described as follows [16]: MK T ðp; RÞ ¼ pffiffiffi ; R p pffiffiffiffiffiffiffiffi where K is a constant and 1 6 K 6 8=3. The plant function G can be computed as 8 0; p P p1 ; > > < nK R c 0 ; p2 6 p < p1 ; GðpÞ ¼ pffiffiffi p M > > : B; p < p2 ;
2
2 nMK , p . where p1 nMK 2 R0 c BMþR0 c
ð8Þ
Given the map A, H and G in Eqs. (5), (6) and (8), respectively, the TCP-RED congestion control system (4) can be described as a 1 D map for the average queue length: qkþ1 ¼ f ðqk ; qÞ ¼ ð1 wÞqk þ wGðH ðqk ÞÞ;
ð9Þ
where q represents a vector of system parameters. According to the different parameter settings, the TCP-RED model can be described as three kinds of maps. Here we assume p1 < pmax , which is the interesting region from the application point of view and can guarantee good performance of RED. Then the TCP-RED map (9) can be expressed as follows: 8 > qk P b1 ; < ð1 wÞqk ; qkþ1 ¼ f ðqk ; qÞ ¼ f ðqk ; qÞ; ð10Þ b2 6 qk < b1 ; > : ð1 wÞq þ wB; q < b : k k 2 Here, 0
1
nK 0 R0 c C B f ðqk ; qÞ ð1 wÞqk þ w@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A: pmax ðqk qmin Þ M
ð11Þ
qmax qmin
Borders b1 and b2 are given by b1
p1 ðqmax qmin Þ þ qmin pmax
and
b2
p2 ðqmax qmin Þ þ qmin : pmax
Clearly, due to the extreme complexity of the network, it is impossible to derive an exact, analytical description of the TCP-RED congestion control system. Even exact knowledge of the fixed point of the system is not available.
86
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
However, the map f described in Eq. (10) characterizes some important features of the TCP-RED, which can be useful to study and improve the performance of Internet congestion control. 2.3. Properties of the TCP-RED model If the average queue length is greater than b1 , the TCP sending rates would be too small to keep the link highly utilized. On the other hand, if the average queue length is less than b2 , the TCP sending rates would be too large to keep low delay. Therefore, it is desirable to keep the average queue length greater than b2 and less than b1 . In this region the system is represented as the strongly nonlinear map f . This map has a fixed point q 2 ðb2 ; b1 Þ, which is a real solution of the following equation: 2 R0 c ðnKÞ2 ¼ ðqmax qmin Þ: ðq qmin Þ q þ M pmax The eigenvalue of the system at this fixed point is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of ðq ; qÞ wnK qmax qmin a1 ¼1w < 1: 3=2 oq pmax 2ðq qmin Þ
ð12Þ
Furthermore, of ðq ; pmax Þ 1 ¼ wnKq3=2 a2 max opmax 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qmax qmin < 0: q qmin
A necessary and sufficient condition for the asymptotic stability of the fixed point q is 1 < a1 < 1. However, it has been found that the TCP-RED can exhibit complex oscillating behavior, which implies the instability of the fixed point. Here we show the bifurcation behavior of the TCP-RED model with the exponential averaging weight w as the bifurcation parameter. System parameters are chosen as follows [16]: pffiffiffiffiffiffiffiffi B ¼ 300 packets; K ¼ 8=3; R0 ¼ 0:1 s; M ¼ 0:5 kb; ð13Þ n ¼ 20; c ¼ 1500 kbps; qmax ¼ 100; qmin ¼ 50; pmax ¼ 0:1: 0:047. For w < w , the average It can be seen from Fig. 4 that the system exhibits a period-doubling bifurcation at w
Fig. 4. Bifurcation diagram of the average queue length w.r.t. the weight w. The upper dashed line represents b1 and the lower dashed line represents b2 .
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
87
a single band chaotic oscillation. Such a border-collision-induced chaotic oscillation is Ômalignant’ and should be avoided, since it can result in significantly decreasing throughput and increasing dropping rates. In summary, the TCP-RED system has the following properties: (i) The system can exhibit complex bifurcation and chaotic behaviors. It has an unique unstable fixed point q 2 ðb2 ; b1 Þ; (ii) The eigenvalue of the system at this fixed point q ;pmax Þ satisfies a1 ¼ a1 ðq ; pmax Þ < 1; (iii) a2 of ðop < 0. max 3. Controlling chaos in TCP-RED via delayed feedback control The DFC method, originally designed to controlling chaos in continuous-time systems, is based on a feedback of the difference between the current state and the delayed state, which vanishes after stabilizing an equilibrium embedded in the original system. Later on, Socolar et al. [21] changed it into a discrete version and suggested the control be added to the accessible parameter for easy implementation. Based on the properties (i)–(iii) of the TCP-RED system, we can add a delayed feedback perturbation on the parameter pmax in the RED algorithm so as to guarantee the stability of the fixed point q . Let us consider the following controlled TCP-RED system qkþ1 ¼ f ðqk ; pmax;k Þ ¼ f ðqk ; pmax þ dpmax;k Þ;
ð14Þ
where dpmax;k is the parameter perturbation. The traditional DFC method is given by dpmax;k ¼ hðqk qk1 Þ
ð15Þ
with a constant feedback gain h. Our study shows that the controller (15) can at best stabilize the system at the periodtwo orbit. Moreover, the oscillation is sensitive to noises and may go beyond the Ôsafe’ region (qmin ; qmax ), which is not desired in the complex network. However, based on the idea of the DFC method, the process of stabilization can be looked upon as the process of minimization of the difference between the current average length and the previous one. When the difference tends to zero, the system becomes stable. A well-known example of a minimization procedure is the gradient decent method. This motivates us to introduce the following time-dependent delayed feedback control algorithm: dpmax;k ¼
of wnK qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hð hðqk qk1 Þ ¼ qk1 Þ: qk qk qmin opmax 1:5 2p max
ð16Þ
qmax qmin
As we expected, this approach can successfully stabilize the TCP-RED system at the original fixed point, expect that we have to know the concrete expression of the partial derivation, which also limit its application. To remove the disadvantage without sacrificing other benefits of the controller (16), we make some modification and assume that the parameter H can be selected such that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wnK qmax qmin of h¼ h: ð17Þ H¼ 1:5 opmax 2pmax Then the parameter perturbation given in (16) can be written as qk qk1 dpmax;k ¼ H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : qk qmin
ð18Þ
Theorem. Consider the control system described by (14) and (18). The fixed pointed q is asymptotically stable if and only if pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q qmin a1 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q ;pmax Þ . where a1 of ðq o;pq max Þ and a2 of ðop max
qk q . Linearizing system (14) about the fixed point gives Proof. Denote dqk ¼ dqkþ1 ¼ a1 dqk þ a2 dpmax;k :
ð20Þ
88
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
Substituting (18) into (14) and denoting xk ¼ dqk , yk ¼ d qk1 , we have the following second-order nonlinear difference equation ! x k yk xkþ1 a1 xk þ a2 H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð21Þ xk þ q qmin : ykþ1 xk Linearizing system (21) about (0,0) gives 0 1 a2 H a2 H xk xkþ1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p a þ 1 A q q ¼@ : q q min min ykþ1 yk 1 0
ð22Þ
The stability of (22) is governed by the characteristic equation 0 1 a2 H a2 H B C ¼ 0: z2 @a1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Az þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q qmin q qmin
ð23Þ
The fixed point is asymptotically stable provided that both solutions of (23) fall inside the unit circle, i.e., jzj < 1. The condition is satisfied if and only if (19) holds. h Now we study the effectiveness of the time-dependent DFC method (18) from the view of bifurcation analysis. We define the bifurcation point to be the exponential averaging weight w at which the eigenvalue of the system becomes )1 and the initial period-doubling bifurcation occurs [24]. After the bifurcation point the system becomes quickly unstable and the queue size oscillates widely(see Fig. 4). Hence an effective approach of improving the stability is to make the bifurcation point as larger as possible. In the following, we demonstrate that the TCP-RED system with time-dependent delayed feedback control (18) is more stable than the original TCP-RED map for a given set of configuration parameters. Note that from (12) the bifurcation point of the original TCP-RED system is given by w R ¼
2 ; 1þK
ð24Þ
where K¼
nK 2ðq
qmin Þ1:5
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 0: pmax qmax qmin
ð25Þ
On the other hand, note that from (23) the bifurcation point of the controlled system (14) and (18) is given by w D ¼
2 1 þ K þ 2KH
pffiffiffiffiffiffiffiffiffiffiffi ffi: q qmin pmax
ð26Þ
From the properties of the TCP-RED system, we know that a2 < 0 and a1 < 1 when the system is unstable. Then H must be chosen as a negative value to guarantee the stability of the fixed point q . Hence, we have w R < w D :
4. Example Considering the recommended range of pmax in [22], we have the following control form 99 8 8 >> > > < < qk qk1 == pmax;k ¼ min 0:5; max 0:01; pmax þ H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; > > qk qmin > ;> : ; :
ð27Þ
which does not influence the results of Theorem. System parameters are given by (13) with w ¼ 0:07. The control is actuated at k ¼ 100 with the constant gain H ¼ 0:01. Fig. 5 shows results of numerical simulations. One can see that
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
89
Fig. 5. Controlling chaos in the TCP-RED map using time-dependent delayed feedback perturbation actuated at time k ¼ 100.
after a transient process the system is stabilized at the originally fixed point and the perturbation tends to zero as expected.
5. Presence of UDP traffic Besides TCP, the Internet makes another transport protocol available to its multiple applications and services: User Datagram Protocol (UDP). UDP offers a direct way to send and receive datagrams. It is transaction oriented, and delivery and duplicate protection are not guaranteed. UDP is used with many real-time applications such as Internet telephone, real-time video conferencing, and streaming of stored audio and video. All these applications can tolerate a small fraction of packet loss, so that reliable data transfer is not absolutely critical for the success of the application. The nonadaptive UDP-based applications, to some degree, impact the stability of the Internet. In this section we consider the presence of the UDP traffic. From the perspective of TCP connections, UDP traffic takes away some of the available bandwidth. Hence, given the UDP load cu , the available capacity for the TCP connections becomes c cu ð1 pÞ [23] and the plant function G in Eq. (7) is changed into 8 0; p P p3 ; > > < c c ð1 pÞ c cu ð1 pÞ u 1 GðpÞ ¼ TR p; R 0 ; p4 6 p < p3 ; > M n > : B; p < p4 ; c cu ð1 pÞ c cu ð1 pÞ BM . ; R0 and p4 Tp1 ; R0 þ where p3 Tp1 n n c cu ð1 pÞ Then the TCP-UDP-RED map can be expressed as follows: 8 qk P b3 ; < ð1 wÞqk ; qkþ1 gðqk ; qÞ ¼ gðqk ; qÞ; b4 6 qk < b3 ; ð28Þ : ð1 wÞqk þ wB; qk < b4 with 0
1
cnK R0 c C B gðqk ; qÞ ð1 wÞqk þ w@ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A: qk qmin ÞÞÞ M gðqk qmin Þðc cu ð1 gð
p3 and p4 are the positive, real solutions of the following two equations, respectively. Here, g ¼ qmaxpmax qmin cu p33=2 þ ðc cu Þp31=2
nMK ¼ 0; R0
90
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
Fig. 6. Bifurcation diagram of the average queue length w.r.t. w. The upper dashed line represents b3 and the lower dashed line represents b4 .
Fig. 7. Control of chaos in the TCP-UDP-RED map using time-dependent delayed feedback perturbation actuated at time k ¼ 100.
nMK cu p43=2 þ ðc cu Þp41=2 BM ¼ 0: þ R0 c Borders b3 and b4 in Eq.(28) are given by b3
p3 ðqmax qmin Þ þ qmin pmax
and
b4
p4 ðqmax qmin Þ þ qmin : pmax
This TCP-UDP-RED model also exhibits complex oscillating behavior, which is shown in Fig. 6 as an example. Here, the UDP traffic cu ¼ 100 kbps and other system parameters are chosen as (13). Fig. 7 shows the numerical results when the time-dependent DFC method (18) is applied to control chaos in the TCP-UDP-RED map. One can see that the system is also stabilized at the originally fixed point.
6. Conclusions Congestion control in the Internet is an extremely important and challenging problem, which has been the main subject of intensive studies over the last decade. In this work, controlling bifurcation and chaos in an Internet con-
L. Chen et al. / Chaos, Solitons and Fractals 21 (2004) 81–91
91
gestion control system model––the TCP-RED map is investigated. We have described in this paper a time-dependent DFC method, which can control chaos in the system by perturbing the accessible parameter pmax . We have shown through bifurcation analysis and simulations that the controlled system is much stabler than the original one and the router queue occupancy can be stabilized at a level. The benefits of a stabilized queue in a network are high resources utilization, predictable maximum delays, ease in buffer provisioning, and traffic intensity and number of connections [15]. Moreover, it was also shown that using the proposed method for the presence of the UDP traffic gives similar performance. We will further investigate the effectiveness of the proposed approach to improving the performance of the RED router queue management mechanism.
Acknowledgements This work was supported by the National Natural Science Foundation of China with the grant number 60174005 and 69874025.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
Peterson L, Davie B. Computer networks: a system approach. 2nd ed. Morgan Kaufmann; 1999. Jacobson V. Congestion avoidance and control. Proc ACM SIGCOMM 1988;18(4):314–29. Floyd S. A report on recent development in TCP congestion control. IEEE Commun Mag 2001;39(4):84–90. Mathis M, Semke J, et al. The macroscopic behavior of the TCP congestion avoidance algorithm. Comput Commun Rev 1997;27(3):633–43. Padhye J, Firoiu V, et al. Modeling TCP throughput: a simple model and its empirical validation. ACM Comput Commun Rev 1998;28(4):303–14. Hespanha JP, Bohacek S, et al. Hybrid modeling of TCP congestion control. Lect Notes Comput Sci 2001;2034:291–304. Veres A, Boda M. The chaotic nature of TCP congestion control. In: Proceedings of IEEE Infocom. 2000. Available from: http:// www.ieee-infocom.org/2000. Braden B et al. Recommendations on queue management and congestion avoidance in the Internet. IETF Request for Comments (RFC) 1998, p. 2309. Floyd S, Jacobson V. Random early detection gateways for congestion avoidance. IEEE Trans Network 1997;1:397–413. Firoiu V, Borden M. A study of active queue management for congestion control. In: Proceedings of IEEE Infocom. 2000. Available from: http://www.ieee-infocom.org/2000. Hollot C, Misra VA. Control theoretic analysis of RED. In: Proceedings of IEEE Infocom. 2001. Available from: http://www.ieeeinfocom.org/2001. Ott T, Lakshman T, et al. SRED: stabilized RED. In: Proceedings of IEEE Infocom. 1999. Available from: http://www.ieeeinfocom.org/1999. Athuraliya S, Li VH, et al. REM: active queue management. IEEE Network 2001;15:48–53. Hollot C, Misra V et al. On designing improved controllers for AQM routers supporting TCP flows. In: Proceedings of IEEE Infocom. 2001. Available from: http://www.ieee-infocom.org/2001. Aweya J, Ouellette M, et al. A control theoretic approach to active queue management. Comput Networks 2001;36:203–35. Ranjan P, Abed EH. Nonlinear instabilities in TCP-RED. In: Proceedings of IEEE Infocom. 2002. Available from: http:// www.ieee-infocom.org/2002. Chen G, Dong X. From chaos to order: perspectives, methodologies, and applications. Singapore: World Scientific; 1998. Chen G, editor. Controlling chaos and bifurcations in engineering systems. Boca Raton, FL: CRC Press; 1999. Pyragas K. Continuous control of chaos by self-controlling feedback. Phys Lett A 1992;170:421–8. Floyd S. Recommendation on using the ‘‘gentle’’ variant of RED. March 2000. Available from: http://www.aciri.org/floyd/red/ gentle.html. Socolar JES, Sukow DW, et al. Stabilizing unstable periodic orbits in fast dynamic systems. Phys Rev E 1994;50(4):3245–8. Floyd S, Gummadi R, Shenker S. Adaptive RED: an algorithm for increasing the robustness of RED’s active queue management, 2001. Available from: http://www.aciri.org/floyd/papers/adaptiveRed.ps. La RJ, Ranjan P, et al. Nonlinear dynamics of mixed TCP and UDP traffic under RED. MED. 2002. Available from: http:// www.ece.umd.edu/~hyongla/papers/med2002.pdf. La RJ, Ranjan P, et al. Analysis of adaptive random early detection, 2003. Available from: http://www.glue.umd/edu/~priya/ paper272.pdf.