Int. J. Communications, Network and System Sciences, 2011, 4, 241-248 doi:10.4236/ijcns.2011.44029 Published Online April 2011 (http://www.SciRP.org/journal/ijcns)

Copyright © 2011 SciRes. IJCNS

Active Queue Management within Large-Scale Wired Networks

Ping-Min Hsu1, Chun-Liang Lin1, Ching-Han Yu2 1Department of Electrical Engineering, National Chung Hsing University, Taichung, Chinese Taipei

2Department of Electrical Engineering, National Taiwan University, Chinese Taipei E-mail: chunlin@dragon.nchu.edu.tw

Received March 3, 2011; revised March 24, 2011; accepted March 30, 2011

Abstract Signal transmission control protocol sources with the objective of managing queue utilization and delay is actually a feedback control problem in active queue management (AQM) core routers. This paper extends AQM control design for single network systems to large-scale wired network systems with time delays at each communication channel. A system model consisted of several local networks is first constructed. The stability condition guaranteeing overall stability is subsequently derived using Lyapunov stability theory. The results developed have been successfully verified on a network simulator. Keywords: Large-Scale System, Network Control, Stability, Time Delay

1. Introduction

Traffic characterization and modeling are generally rec-ognized as two important steps toward analysis and con-trol of network transmission performance. With the aid of the characterized model, control theory can be effi-ciently applied in solving the congestion control problem. There have been various delayed differential equation models developed in [1-3], in which the fluid-flow model utilized in [1] was previously proposed in [2]. Those characterized data traffic as fluid used a set of differen-tial equations to describe the Active Queue Management (AQM) policy and the router queuing process. In [4], the end-to-end congestion control mechanisms were em-ployed in transmission control protocol (TCP) flow con-trol.

While a bunch of research focusing on modeling and analysis of network control systems have been published, most approaches addressed the issue for a single sender- receiver network connection [5].

This paper is motivated by the requirement of an ap-proach for modeling and stabilization of large-scale communication networks. An extension of the fluid- based model in wireless networks was proposed in [6,7], in which their parameters for representing the probability of data transmission failure was applied in system de-velopment. A system for mixed wired and wireless net-works was concerned in [6], in which a more general

fluid-based model compared with that in [1] was pro-posed. A robust H controller design was proposed in [7], in which the stability analysis was conducted by Lyapunov theory and modern control theory was ex-tended to deal with the network congestion problem. Fur- thermore, linear matrix inequalities were proposed in control design and less oscillation under TCP was achieved. Packet dropout not only affects stability in data transmission, but also plays an important role in the net-work control systems. The authors in [8-10] have inves-tigated stability for the network control systems with packet dropout.

To the best of the authors’ knowledge, while the Lya-punov stability theory has been widely applied to the stability analysis of large-scale systems [11-13], investi-gation for stability of the large-scale network control systems (LWNCSs) are quite rare. This motivates this research. The presented work shows that an appropriate control design based on an appropriate Lyapunov func-tional ensuring asymptotic stability of the LWNCS can be constructed when there are a large number of signal flows.

The major contributions can be summarized as fol-lows.

1) A control strategy for congestion control in the large-scale wired network is constructed.

2) All local networks are interconnected according to the dropping probability. That is, how the local net-

P. M. HSU ET AL. 242

works mutually affect each other within a wired network scheme is modeled via a particular form of the dropping probability.

An example is given to demonstrate validity of the re-sult using the network simulation platform NS2 [14].

2. Modeling of Large-Scale Wired Network

The large-scale wired network under consideration is as illustrated in Figure 1, which consists of s locally wired networks within TCP under random early detec-tion (RED). In each TCP loop, signal packets are trans-mitted from a sender to a receiver, which is also the sender to next TCP loop. Furthermore, they are sequen-tially passed from 1TC to P sTCP . Some packets are dropped in the congested router under RED, in which the dropping probability is computed by the proposed con-troller. Regarding the topology illustrated in Figure 1, the current dropping probability not only determines the seriousness of transmission congestion in the present router, but also affects those in other congested routers in the large-scale wired network.

Referring to [1], a simplified version of a fluid-flow model of TCP behavior involving the key network vari-ables could be modeled as

0

1,

2

0

W t R tW tW t p t R t

R t R t R t

W W

(1)

0, 0 ,

max 0, , 0

N t W t R t C q q qq t

N t W t R t C q

0

(2)

where is the average TCP window size (packets), is the average queue length (packets/sec),

Wq R t is

the round trip time (RTT), is the link capacity (packets/sec), N is the loading factor (the number of TCP sessions), and

C

[0,1]p is the probability of packet mark. The queue length 0,q q m and the window size

0, mW W where mq and mW denote the buffer ca-pacity and the maximum window size respectively. Let R t denote the RTT for and is assumed to be 0t

pR t q t C T (3)

where p is the fixed propagation delay and T q t C models the queuing delay. One is referred to [1] for the details of the mathematical modeling of the fluid-flow model of TCP.

Suppose that each dynamic model of TCP constructs a sub-network, which belongs to an interconnected bottle-neck network, then a large-scale wired network com-posed of s interconnected bottleneck networks i , S

1, 2, ,i s can be shown as in Figure 2, where , 1, ,i i i ipR t q t C T i s , with being the

propagation delay for iTCP and ipT

1 2( ) ( ) ( )q ti i i

si

q t t q( )tq . From (1) and (2), the large-scale linearized

differential equations can be written in the matrix form as

2 22 2 00 0 2

0 0

02 20 0

0

δ

δ

1δ

δ , (4) 21 δ

0

1δ

, 1, 2, ,δ

0 0

i

i

ii i

ii i i ii ioi

i i

i i

ii i

i i i i

i i

w t

q t

NR Cw tR C R C

p t RNN q tR R

Nw t R

R C R C i sq t R

with each sub-network being interconnected through

RouterSenderReceiver

, also Sender

PacketsAcknowledgements

RouterReceiver

, also Sender RouterReceiver

, also Sender

Receiver

, also Sender

Tcp 1 Tcp 2 Tcp s

PacketsAcknowledgements

PacketsAcknowledgements

dropouts dropouts dropouts

Figure 1. Topology of the large-scale wired network under consideration.

Copyright © 2011 SciRes. IJCNS

P. M. HSU ET AL.

Copyright © 2011 SciRes. IJCNS

243

1w1w

1p

1q1q

1

1

R

2w2w

2p

2q2q

2

1

R

TCP 1

TCP 2

1

2

swsw

sp

sqsq

TCP s

routed packetsdata packets

Congested Router 1

Congested Router 2

Congested Router s

11q

1sq

12q

2sq

1sq

ssq

AQMcontrol law

AQMcontrol law

1

2

1

2

1

1

R

2

1

R

1R se

2R se

1

1

R

1C

2C

1N

2N2

1

R

sN

sC1

sR

AQMcontrol law

sR se 1

sR

1

sR

21q

22q

2sq

Figure 2. Block diagram of the large-scale wired network.

0

0

0

01

0

0 1

03 3 30

02 3 30

δ , d δ

δ d

2 δ d

2 δ d , , 1, 2

i

i

i

s

i io pi ij j ijj

s

i i pi ij ijR

i i i iR

i i i iR

p t R K q q t h

p p t v v K d tv

N w t v v R Cv

N q t v v R C i jv

u

, s

(5)

where ij is the transmission time between the bottleneck networks i and

hS jS ; , 0δ i i iw w w δ iq 0iq q ,

0j jq q q , and 0; 0 0 0i denoting our operating point; i and

δp pi i ip iw q , , pq jq

iN

denote data packets belonging to the ith and jth sub-networks respectively;

i denotes the ith expected TCP sending window; i denotes the link capacity; denotes the number of the ith TCP sessions;

w C

0 0i i ipR q C

maxp

T p; i is the probabil-ity of packet mark; RED consists of a proportional con-troller and packet-marking profile, shown as in Figure 3, where , and are configurable parame-ters and pi denotes the slope of the active queue management control law which computes i as the func- tion of the measured queue length

miq n

K

maxq

p

iq by the AQM pol-icy; ij is the flow distribution ratio from the bottle-neck network

d

jS to i . If S iq t is bounded by and , the packet-marking probability is given by

minqmaxq

0

0

0

01

0

1

30

3 30

20

3 30

δ

δ d d

2 δ d

2 δ d

i

i

i

s

i pi ij j ijj

s

i pi ij iRj

iiR

i i

iiR

i i

p K d q q t h

p t v v K tv

Nw t v v

vR C

Nq t v v

vR C

u

minq maxq

maxp

ip

1

pK

iq

ip

i tu

-+

iq

packet-marking profile

Figure 3. RED drop function.

The form of (5) is determined based on the following considerations.

1) While each sub-network is interconnected through (5), the feedback signal applied for the controller is treated as the combination of all the queue lengths in the large-scale network. Thus, the term

01δ

s

ij j ijjd q q t h

in (5) was claimed. It

was further multiplied with piK , which denotes the slope of the active queue management control law.

2) The term 0

0δ d

iiR

p t v vv

denotes the differ-

ence between the current and delayed values of the dropping probability adopted in the ith TCP. Con-sidering the fluid-flow model, the current queue length variation is affected by the delayed dropping probability while the current one is applied in real world. Considering this fact, while constructing (5),

the term 0

0δ d

iiR

p t v vv

is added.

3) 0ip is added to cancel 0ip while computing where δ

ip

0i i ip p p . 4) Additional terms in the AQM controller law are

added to obtain the transformed system (6) with the controller (7).

5) Now, substituting 0δ ,i ip t R , defined by (5), into (4) gives a new system. Based on (4), a large-scale wired network system consisting of s local net-works can be expressed in the state-space repre-sentation as follows

, 1, 2, , i ii i i ix t t t i A x B u s (6)

where δ δT

i i it w t q t x ,

20

0 0

2 iN0

1i i

iii

i i

R C

N

R R

,

20

22

0

i i

i i

R C

N

B , and

A

1

( ) ( ) ( ( )), , 1, 2, ,s

i ij j ij i ij

t F x t h t i j

u u s (7)

with 0ij pi ijK d F and i i it K t u x being the AQM control law for the ith subsystem, and

P. M. HSU ET AL. 244

0

0

0

0 01

0

30

3 30

20

3 30

δ d

2 δ d

2δ d

i

i

i

s

i i pi ij i ij

iR

iiR

i i

iiR

i i

t K d q t p

p t v vv

Nw t v v

vR C

Nq t v v

vR C

u u

Equation (6) represents the linearized system of the large-scale wired network while considering coupling effects induced by the locally wired networks. The term

implies the dropping probability while concerning the interconnection of local networks. The term

i tu i tu

is to be determined in the stability analysis. Equation (7) is treated as the control input such that 0i ip t t p u i is applied as the dropping probability.

i tu can be derived in the following steps: Step 1) Consider (5), which models how local net-

works may mutually affect each other in the large-scale wired networking environment.

Step 2)Substitute 0δ ,i ip t R into (4) and use i x to obtain system of (6) with the controller

(7). 0i it Rx

From (6) it is easily seen that is controllable. The action of the AQM control law is to mark packets as a function of the measured queue length q. The plant dynamics given by (6) relates reveals the packet-marking probability dynamically affects the queue length.

,ii iA B

Now, let the set of feasible operating conditions 0 0, mq q , 0 0,iw W m , and 0 0,1ip . Assume

that , and are continuously differentiable δ iw δ i δ ipq

on 0 , 0iR , i.e. iW

wb

v

,

δ iq

qb

v

,

δ

i

p

pb

v , 0 , 0iv R .

Therefore

0

0

0δ di

i WRw t v v b R

v

i ,

0

0

0δ di

i qRq t v v b R

v

i ,

0

0

0δ di

i pRp t v v b R

v

i .

Furthermore

20

1 1

0

s s

i i i pi ij i pi ij ij j

i i i

t t K d t K q d

p g t t

u u u u

u

t (8)

where

0

0

0

30

3 30

20

3 30

0

2δ d

2δ d

+ δ d

i

i

i

ii iR

i i

iiR

i i

iR

Ng t w t

vR C

Nq t v v

vR C

p t v vv

v v

It is seen that im i img g t g with 3

2 30

2 iim W

i i

Ng b

R C

2

02 30

2 iq i pb

i i

Nb R

R C . The gain piK is chosen as

0

01

0

01

1, when 0,

1,2, ,1

, otherwise,

i im is

ijj

pi

i ims

ijj

p gq d

K i s

p gq d

u

(9)

This ensures 0,i i i it t u u u t . With regard to the above system one can ensure that i , 1,2, ,i s satisfy

2 , , 1, 2, ,i i i i i it t t t R i s u u u u (10)

From (8) and (9), the gain reduction tolerance is given by

1

s

i pi j ijK d

where are chosen , 1, ,ijd j s 1such that

1

s

i pi ijjK d

.

3. Stability of LWNCS

The following theorem states the main result which cha-racterizes the stability condition for the LWNCS.

Lemma: For any scalar 0 and any real vectors X and Y with appropriate dimension, then

T T T 1

TX Y Y X X X Y Y (11)

Theorem: The large-scale wired network system de-scribed by (6), which satisfies (10), would be asymptoti-cally stable, if the state feedback control law of each sub-network is given by

1, 1, 2, ,

2T

i i i i it t i u B P x s (12)

where 1i i i with i satisfying

2 2 1 2 2 1i i i i i (13)

and 0Ti i P P satisfies the following Riccati matrix

inequality:

T T1

+1 0

ii i i ii i n i i i i i

i n

s

s

A P P A P I B B P

I (14)

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P. M. HSU ET AL.

Copyright © 2011 SciRes. IJCNS

245

in which 1 is a positive constant, 2( )i i i i 4 and : max , 1,2, ,i ji j s with ji satisfying

Proof: The stability analysis is derived around the ori-gin δ δ 0i iq w of (4), i.e. the equilibrium point 0 0 0, ,i iq p w of the fluid-flow model. Regarding the problem, we define an appropriate Lyapunov functional candidate as 1

1, ,T T

ij n ij i i ij i j

I F B B F (15)

T T 2

0 01 1

d dij

s s t t tTi i i i i ij j j i i i it h

i j

t t 2 d

V x x P x x x x x u (16)

for . The set 1,2, ,i s 0 0 0, ,i iq p w chosen as the equilibrium point of the fluid-flow model corresponding

to is determined by iTCP

20 0

0 0 0 00 0 0

10, 0, , 1

2i i

i i i i ip ii i i i

w N qp w C R T p

R R R C

It is obtained that 0 0i V while 0i u and

for . Taking differentiation with re-spect to time t and using (6) while ignoring

and 0i ix V 0i x T1

0 0s

i ii x x 2

10

s

ii u gives

T T

1

2 2 T

1 1

T

1

T T

1 1

+2

+2

s

i i i ii i i ii n ii

s s

i i i i i ij j iji i

s

i i i i ii

s s

ij j j j ij j iji j

V t I t

t t t h

t t

t t t h t h

x x A P P A x

u x P B F x

x P B u

x x x x

It is easily obtained from (11) that

T T T T

1 1

T T T

1 1 1

T1

T T

1 1 1

T1

1

1

1

s s

j ij ij i i i i i i ij j iji j

s s

j ij ij i i ij j iji j

i i i i

s sTj ij ij i i ij j ij

i j

s

i i i ii

t h t t t h

t h t h

t t

t h t h

s t t

x F B P x x P B F x

x F B B F x

x P P x

x F B B F x

x P P x

T

1

T T

1

2s

i i i i ii

s

i i i i i i i ii

t t

t P

x P B u

x P B B x t

(17)

Furthermore

T T

1 1 1

s s s

ij j j i i ii j i

t t s t t

x x x x

Therefore, it can be obtained that

T T 21

1

2

T T T

1 1 1

1

1 +

4

1 +

s

i i i ii i i ii i n ii

Ti i i i i i i i i

s s

j ij ij i i ij ij n j iji j

t s s

t

t h t h

V x x A P P A I P

P B B P x

x F B B F I x

where 1 0 . After substituting (12) into (10), one gets

(18) T T T2i i i i i i i i i i i i it x t t x P B B P x P B u t

From (12)-(15), it can be concluded that i.e.

TT T T T

1 1 1 11 1

0

01 10 , ,

is

i i i ii i i i i i n is i i is is n

J

t tdiag

V xF B B F I F B B F I

(19)

P. M. HSU ET AL.

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246

where , TT T T

1 1i i i s ist t t h t h x x x

T T1 +1 0i ii i i ii i n i i i i i i ns s J A P P A P I B B P I

and T T1 , ,ij n ij i i ij i j I F B B F and i satisfies

(13). This completes the proof.

4. Design Process

Control design procedure is given bellows to summarize the previous analysis.

Step 1) Set parameters of the LWNS including i ,

i , 0i , ip , w , , and C

N R T b qb pb . The system model of (6) is then constructed.

Step 2) Consider the model defined by (6) and (7) with the positive constant 1 , the parameter ij is chosen to satisfy (15).

Step 3) Find i , ima g , i , i , i and to solve for the Riccati matrix inequality (14). The solution

can be calculated by transforming (14) into T 0i i P P

linear matrix inequalities and solved via the available computational software.

Step 4) Obtain the control gain T 2i i i iK B P . The

AQM control law i tu is determined by (7) with

i i it t u K x .

5. Illustrative Example

Consider a large-scale wired network described by (4) with 0 175q packets, , , 1 60N 2 50N 3 20N ,

4N 5 , , 20 , ,

40 , 10R

0.1 s C0.2 s

1 3,7500.1 s RR 30 0.1 s

R packets/s, pack-ets/s,

2 4,C 000

3 4,0C 00 packets/s, 4 packets/s, and 1

4,000C0.001 . The large-scale wired network consists

of four sub-networks described, respectively, by

4

1 1 1 1 1 11

4

2 2 2 2 21

4

3 3 3 3 3 31

4

8 0 390,

300 5 0

2.5 0 320,

500 100 0

1 0 889,

200 100 0

0.

j j jj

j j jj

j j jj

t t A t h t

t t A t h

t t A t h

t

2 t

t

x x x u

x x x

x x x u

x

u

4

4 4 4 41

25 0 32000

50 10 0 j j jj

t A t h

4 t

x x u

Select 0.04ij

W p qb b b for each and as-

sume so that , 1, 2, 3, 4i

1 40mg

200 2 3m mg g . Referring to [1], 4mg 20 0 0i iw R C i Ni and

. From 20 0i iw p 2 0 0i i im qp g then ,

2 3 4

1 0.20.1 , and 1 2 3 4 0.4 are

chosen to meet (13). Solving for the Riccati matrix ine-quality (14) gives

1 2

3 4

0.321 0.017 0.028 0.006,

0.017 0.002 0.006 0.003

0.0009 0.0007 0.00004 0.00004,

0.0007 0.0018 0.00004 0.0042

P P

P P

,

and the corresponding control gain matrices are obtained as 1 27.07 1.45 K , 2 9.22 2.01 K ,

3 , and 2.77 2.05 K 4 2.16 1.92 K re-spectively. The size of the packet is assumed to be 500 bytes.

The numerical experiments are conducted on a net-work simulator-NS2 [14]. The simulation results with the

chosen parameters show satisfactory performance of the queuing response in the presence of random delays, see Figure 4. It is found that larger TCP flows cause higher link utilizations, but with larger queuing delays.

The results for the case of 1 , 2100N 50N ,

3 20N , 4 5N are displayed in Figure 5. As it can be seen from Figure 5(a), large queue oscillations may cause considerable variations in the RTT of packets for the corresponding sub-network. However, with the pro-posed controllers, the network still remains to be stable when it holds a large number of data flows.

6. Conclusions

This paper has modeled and analyzed stability of large- scale wired networks under control where the AQM stra- tegy uses RED to fulfill the queue management. The problem of feedback control has been solved for the LWNS with delayed perturbations in the interconnections and a new condition ensuring the overall loop stability is

P. M. HSU ET AL. 247

(a)

(b)

(c)

(d)

Figure 4. Transient response of queueing with 1 60N ,

2 , 350N 20N and 4 . (a) Queue size for TCP1; (b) Queue size for TCP2; (c) Queue size for TCP3; (d) Queue size for TCP4.

5N

(a)

(b)

(c)

(d)

Figure 5. Transient response of queueing with 1 100N ,

2 50N , 3 20N and 4 5N . (a) Queue size for TCP1, (b) Queue size for TCP2; (c) Queue size for TCP3; (d) Queue size for TCP4.

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P. M. HSU ET AL.

Copyright © 2011 SciRes. IJCNS

248

presented. The simulation study conducted on the NS2 has been verified successfully.

7. Acknowledgements

This research was sponsored by National Science Council, Taiwan under the grant NSC No. 95-2221-E-005-017.

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