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IEICE TRANS. COMMUN. VOL. E85-A/B/C/D, No. 1 JANUARY 2006

Letter

Nonlinear Control of Active Queue Management for Multiple Bottleneck Network Yang Xiao† , Member and Moon Ho Lee††, Nonmember Summary Active Queue Management (AQM) based on nonlinear difference equations has been proposed to solve the end-to-end TCP network congestion problem recently. The proposed AQM scheme can guarantee the stability of the multiple bottleneck network by nonlinear control for dropping probability of the routers. Nonlinear control often relies on some heuristics and network traffic controllers that appear to be highly correlated with the multiple bottleneck network state. Based on the proposed nonlinear difference equations for TCP flows control across the network, this paper reveals the reasons of congestion of multiple bottleneck AQM, and provide three theorems for avoiding network congestion. Moreover, we give simulations to verify the results for nonlinear control of the multiple bottleneck network congestion.

Key words: Active Queue Management (AQM), nonlinear control, TCP, RED

1. Introduction AQM challenges are best solved by the signal processing community including prediction, detection, estimation, and quantization [1-5]. The goal of TCP's congestion control is to utilize all available bandwidth in a fair and efficient way, while TCP has served its purpose exceptionally well and is partly responsible for the communication explosion of the last decade, there is a fundamental conceptual flaw. Routers have queues to enable statistical multiplexing. The goal of Floyd and Jacobson's paper was to correct this apparent conflict through active queue management (AQM) [1]. The idea of AQM is as follows. When the router determines that the bandwidth is fully utilized, the received packets are dropped. Even the queue is not full, so as the router alerts TCP and thereby keeping TCP flows in check. The existing random early detection (RED) configuration is the most investigated AQM. RED is based on a single bottleneck network (SBN) assumption that may not prevent from traffic instability when congestion occurs at the same time [2-4]. As the complex case with many flows over multiple bottleneck link, the equilibrium (or the fixed point) of the fluid-based model can be much different from the true average of the stochastic counterpart, in which we have some randomness either through random Manuscript received March 23, 2006. The author is with Institute of Information Science, Beijing Jiaotong University, Beijing 100044, China.E-mail: [email protected]. †† The author is with Institute of Information & Comm., Chonbuk National University, Jeonju 561-756, Korea. E-mail: [email protected] *This project is supported by National Natural Science Foundation of China under grant: 60572093 and Brain Pool Program of Korea under grant: 051S-3-5. †

packet marking or random packet arrivals. Depending on the marking function employed at the link, we prove that the fixed point can be strictly larger or smaller than the real expected value of the rate (or throughput). This implies that the equilibrium point of the fluid model may not be the actual steady-state point of the system and the corresponding (linear) stability criterion [6-10] can also be problematic since the system would have been linearized around possibly an inaccurate equilibrium point. This paper provides a stochastic description of congestion control based on difference equations and address some possible limitations for the control of dropping probability. In section II and III, we establish nonlinear difference equation for analyzing the existing single and multiple bottleneck network of RED_AQM scheme, two theorems are proposed to avoid network congestion. In section IV, we give the simulation results according to the proposed theory. At the end of the paper, we draw the conclusion and future research in the end.

2. Single Bottleneck Network RED is the most investigated SBN_AQM (RED_AQM)[3-5]. Therefore, RED determines the packet drop probability p (t ) based on a filtered version of the queue occupancy in a single bottleneck network. The different buffer occupancy between RED_AQM and MBN_AQN is showed in Fig. 1. Therefore, a dynamic model of TCP behavior was developed using fluid-flow and stochastic differential equation analysis [5-10]. The model relates the average value of key network variables and is described by the following coupled, nonlinear differential equations as follows

1 W (t )W (t − R(t )) W (t ) = − p(t − R(t )) R(t ) 2 R(t − R(t ))

(1)

⎧ N (t ) W (t ) − C (t ), q (t ) > 0 ⎪ q (t ) = ⎨ R(t ) ⎪⎩ 0, else (2) where W(t) denotes the expected TCP sending window

Y. Xiao and M. –H. Lee: Nonlinear

Control of Active Queue Management for Multiple Bottleneck Network 2

size (packets), C (t ) is the actual capacity, p (t ) is

q (t ) denotes the expected queue length (packets) and N (t ) is number of probability

packet

marking/drop,

TCP sessions. The round-trip time is

R (t ) =

q (t ) + Tp C (t )

(3)

where T p is propagation delay (secs).

⎧ 0,0 ≤ q (t ) < q min ⎪⎪ q(t ) − q min p (t ) = ⎨ , q min ≤ q(t ) < q max ∈ [0,1] (4) − q q min ⎪ max ⎪⎩ 1, q max ≤ q(t ) < C (t ) where q (t ) is the exponential weighted moving average of queue size, and q min , q max are configurable RED parameters.

The queue length q (t ) and window size W (t ) satisfy

q(t ) ∈ [0, q max ] , and W (t ) ∈ [0,Wmax ] where q max and Wmax denote buffer capacity and maximum window size respectively. The dropping probability p satisfies

p(t ) ∈ [0,1] .

We illustrate these differential equations in the block diagram of Fig.1, which highlights TCP window-control and queue dynamics. Fig. 2. TCP’s congestion-avoidance principle for AQM of one router The Eq.(1)-Eq.(4) are a nonlinear continuous system’s equations, if adopting the approaches of [6-10], the AQM system may be linearized around incorrect equilibrium points. To avoid the problem, we analyze the system to express it into a nonlinear discrete system

1 R(k ) W (k )W (k − R(k )) − Ts p (k − R(k )) 2 R(k − R(k ))

W (k + 1) = W (k ) + Ts

Fig. 1. Block-diagram of TCP’s congestion-avoidance flow-control mode We can know the principle of AQM for single link of TCP senders from Fig. 2. As packets arrive, they are placed in the real queue and a token is placed in the virtual queue of the router. The real packets leave the real queue according to the link speed, while the tokens leave the virtual queue at the virtual link speed. That is, a token is served only when all tokens that arrived before it have be served. The service time of a token is the size of the packet that the token represents divided by the virtual link speed. If a token finds the virtual queue full, then the real packet and token are dropped. The RED control law based Fig. 2 can be expressed as follows:

(5)

N (k ) ⎧ W (k ) − Ts C (k ), q (k ) > 0 ⎪q (k ) + Ts q (k + 1) = ⎨ R(k ) ⎪⎩ q (k ), else (6) where

R (k ) =

q(k ) + Tp C (k )

(7)

where Ts denotes the sampling period. In our AQM model, the router needs to sampling the queue length in its packet buffer with the sampling period Ts . Eq. (6) reflects variation of queue lengths of the router and Eq. (5) reflects variation of TSP window sizes. Compared with the linearized AQM systems of [6-10], our model have no parameters involving equilibrium points of the continuous nonlinear model in Fig.1, which will bring us a more efficient control for the AQM

IEICE TRANS. FUNDAMENTALS/COMMUN./ELECTRON./INF. & SYST., VOL. E85-A/B/C/D, No. 1 JANUARY 2002

3 network, since the window size W ( k ) , the average round-trip time R (k ) and the dropping probability p ( k ) are automatically adjusting the varying link capacity C (k ) and the number of TCP sessions N (k ) . Ref. [6]-10] only consider the simple case of the link capacity and the number of TCP sessions being constants. Thus, it is possible to reach our objective of control system to achieve full utilization of the bandwidth in the presence of these short lived flows by the dropping probability pi ( k ) ∈ [0,1] in Eq. (5).

and only if N (k )

Wi (k )

∑ R (k ) [1 − p (k )] ≤ C i =1

i

(11)

i

where pi (k ) is given by Eq. (8). The proof is trivial. In practical problems, it is some difficult to apply Theorem 2 since it needs to know the Wi (k ) , Ri (k ) and

pi (k ) of each TCP sender, while Theorem 1 only needs the average parameters.

The dropping probability can be designed in various forms

⎧ 0, 0 ≤ q(k ) < qmin ⎪⎪ q (k ) − q min , q min ≤ q(k ) < q max p(k ) = ⎨ − q q max min ⎪ ⎪⎩ 1, qmax ≤ q(k )

(8)

We should notice that the dropping probability in (8) timely depends on the queue lengths of the router queue lengths of the router, instead of being independent like the linearization of [6-10]. Thus, [6-10] gave many strict limitations for the parameters of the router algorithm to keep the AQM network to be stable. From the dropping probability p ( k ) , the TCP window size, the round-trip time R (k ) and N ( k ) the number of TCP sessions, we can calculate throughput of the router as following

q out (k ) =

N (k ) W (k )[1 − p(k )] R(k )

(9)

Now, we can consider the simple link of two routers, shown in Fig. 3, and we can establish two theorems for the network without congestion. Theorem 1: The network of Fig. 3 has no congestion if and only if

N (k ) W (k )[1 − p(k )] ≤ C R(k ) where p (k ) is given by Eq. (8).

(10)

The proof is trivial. In Theorem 1, we assume that all TCP senders have same average dropping probability p (k ) , the average TCP window size W (k ) , the average round-trip time

R (k ) , however, the parameters may different from TCP senders. For the more complicated case, we have further theorem for the network congestion. Theorem 2: The network of Fig. 3 has no congestion if

Fig. 3 The simple link of two routers Remarks: (1) Theorem 1 and Theorem 2 tell us that the dropping probability p (k ) plays an important role in the avoidance of network congestion, the AQM program needs to adjust the p (k ) according to the arriving packets rate

N (k ) W (k ) . If we design a good control rule of R(k ) dropping probability p (k ) , then the network can be free from congestion. Following simulation will verify our results. (2) Different from many existing results, our theorems are based on nonlinear difference equations (1–9), which makes us be capable of using digital technique (software) to implement the control of the network. (3) If adopting existing linearization approaches, the nonlinear system could have been linearized around an incorrect equilibrium points. However, our nonlinear difference equations (5)–(11) have not the linearization processing at equilibrium points for the round-trip time R (k ) queue size q (k ) and TCP window size W (k ) , then our results for network will much more reach reality of working network. Following simulation will also verify the result. (4) Eq. (8) provides a design of the control of the dropping probability p (k ) , the key is how to employ the information of the buffer queue, since the router has no other way to get more information of the status of the network from Fig.2. Thus, an efficient AQM algorithm depends on the design of the control of the dropping probability p (k ) .

Y. Xiao and M. –H. Lee: Nonlinear

Control of Active Queue Management for Multiple Bottleneck Network 4

3. Muliple Bottleneck Network Now we consider multiple bottleneck network, shown in Fig. 4. The network of multiple links can be described as follows.

S mn , n = 1,..., N , m = 1,..., M slave routers Rm , m = 1,..., M , and

(1) The TCP senders

are

linked to the

the

slave

routers

are

C m , m = 1,..., M

is

with

the

link

capacity

;

Rm , m = 1,..., M are connected to mater router Ra , and Ra is with the link capacity of C a ; The mater router Ra is connected to another mater router Rb , and Rb is with the link capacity of Cb ;

(4) The

destination

receivers

Dmn , n = 1,..., N , m = 1,..., M

are receiving the TCP

data

from

packets

S mn , n = 1,..., N , m = 1,..., M

The proof is trivial.

4. Simulations

of

(2) the slave routers

(3)

pm (k ) m = 1,..., M have the same form of Eq. (8), but they are different for different routers Rm .

where

sources

.

We have following theorem for the network in Fig. 9 without congestion

We verify our discrete AQM system via simulations using the Eq. (5) –Eq. (11). We look at a single bottlenecked router running RED. In addition to infinite duration, greedy flows such as the one we model, we introduce short lived, http flows into the router, to generate a more realistic traffic scenario. The TCP flows were simulated using the TCP module provided by Eq.(5). The effect of flows which are very short lived is essentially that of introducing noise to the queue. In all our plots we depict the time evolution of TCP window size, the round-trip time (RTT) of the network, the instantaneous queue length and the dropping probability with the unit of the time axis being milliseconds. In our experiment, we look at the queues of the slave routers Rm , m = 1,..., M with TCP flows

N m (k ) ∈ [60,90], m = 1,..., M , the link capacities of the slave routers are C m (k ) ∈ [2400,3000],1,..., M packets/ms, and satisfy the condition (15) and (16) of Theorem 3, Nm

∑C m =1

m

(k ) ≤ C a = 90 × 3000 packets/ms

C a ≤ C b = 90 × 3000 The propagation delays T p for the flows range uniformly Fig. 4 The multiple links of many routers Theorem

3:

Assume

that

the

slave

routers

Rm , m = 1,..., M in the network of Fig. 4 have same the link capacity of C m , m = 1,..., M , and mater router Ra and Rb have the link capacity of C a and Cb , the network of Fig. 10 has no congestion if and only if 1)

N m (k ) Wm (k )[1 − pm (k )] ≤ C m Rm (k )

(12)

2) N

∑C m =1

m

(k ) ≤ Ca

(13)

3)

C a ≤ Cb

(14)

between 2 ms and 4 ms. We attempt to control the queue to provide a queuing delay Ts of around 1 ms, and hence set the q min and

q max of the queue as 200 packets and

250 packets respectively, with average packet size being 100 Bytes. The buffer has a maximum capacity of B=800 packets. The TCP window size, the round-trip time (RTT) of the network, the instantaneous queue length, the dropping probability, the throughputs of the router and the state orbit (W ( k ), q ( k )) of the router are shown in Figure 5Fig.10. From the state orbit (W ( k ), q ( k )) of the router queue in Fig. 10, we know that the nonlinear AQM network is stable for different dropping probability p (k ) in Eq.(8), the nonlinear AQM network with control p (k ) has a

IEICE TRANS. FUNDAMENTALS/COMMUN./ELECTRON./INF. & SYST., VOL. E85-A/B/C/D, No. 1 JANUARY 2002

5 small solution range, which makes the dropping probability to be smaller that 0.2 after 3ms, the throughput of network has a stable dynamic range from 1200 to 2400 packets/ms.

Fig.5 TCP window sizes controlled by p (k )

Fig. 6 The round-trip times (RTT) of the network

problems in a network by two theorems based on nonlinear difference equations. For congestion control, the proposed difference equations’ fluid model can be a tool to derive control of AQM algorithms, though the actual behavior in a network is always stochastic, as there always exists inevitable multiple bottleneck links due to random packet arrivals as well as random marking/dropping at routers. This paper reveals that there may exist some limitation on the control of the dropping probability of routers, the equilibrium points can be quite different for different controls of the dropping probability. The proposed theorems consider that under multiple bottleneck network, the control of dropping probability of routers ensures the convergence of the average rate to its equilibrium point, may impose excessive restriction of AQM parameters. Simulations verify the proposed control design can avoid of the congestion problems [6-10]. References

Fig.7 The router’s probability instantaneous queue lengths

Fig.8 The dropping

Fig. 9 The throughputs of Fig. 10 The state orbits the router of the router Fig. 8 shows that the AQM algorithm of control p(k ) in Eq.(8) can guarantee the throughput stability of TCP sender in the multiple bottleneck network in Fig. 4 for a time varying link capacity C m (k ) ∈ [2400,3000] packets/ms, after the AQM algorithm enter stable status after 3ms, the result also can be seen from Fig. 10, the state orbit (W ( k ), q ( k )) of the router convergences to a small range.

5. Conclusions This paper proves that the fluid-based modeling approach to be extremely powerful and versatile for many

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