4 fuzzy aqm

50 The International Arab Journal of Information Technology, Vol. 4, No. 1, January 2007
Fuzzy Active Queue Management for Congestion
Control in Wireless Ad-Hoc
Essam Natsheh, Adznan B. Jantan, Sabira Khatun, and Shamala Subramaniam
Department of Computer and Communication Systems, University Putra Malaysia, Malaysia
Abstract: Mobile ad-hoc network is a network without infrastructure where every node has its own protocols and services for
powerful cooperation in the network. Every node also has the ability to handle the congestion in its queues during traffic
overflow. Traditionally, this was done through Drop-Tail policy where the node drops the incoming packets to its queues
during overflow condition. Many studies showed that early dropping of incoming packet is an effective technique to avoid
congestion and to minimize the packet latency. Such approach is known as Active Queue Management (AQM). In this paper, an
enhanced algorithm, called Fuzzy-AQM, is suggested using fuzzy logic system to achieve the benefits of AQM. Uncertainty
associated with queue congestion estimation and lack of mathematical model for estimating the time to start dropping incoming
packets makes the Fuzzy-AQM algorithm the best choice. Extensive performance analysis via simulation showed the
effectiveness of the proposed method for congestion detection and avoidance improving overall network performance.
Keywords: Active queue management, ad-hoc networks, fuzzy systems, intelligent networks, network congestion.
Received September 24, 2005; accepted March 28, 2006
1. Introduction
Mobile ad-hoc network is a network without
infrastructure where every node can work as a router.
Every node has protocols and services to request and
provide services to other nodes with the congestion
handling capability. Traditionally, the congestion
handling is done through Transmission Control
Protocol (TCP). This protocol sends congestion signal
(drop incoming packets) when the node's queue is full
(queue length is maximum). Some studies [4, 11]
showed that early dropping of incoming packet before
reaching the maximum queue length is an effective
technique to avoid congestion and to minimize the
packet latency, e. g., Active Queue Management
(AQM) drops incoming packets before the queue is full
in contrast to traditional queue management which
starts dropping only when the queue in overflowed.
Mobile ad-hoc networks suffer high network
congestion due to high Bit Error Rate (BER) in the
wireless channel, increased collisions due to the
presence of hidden terminals, interference, location
dependent connection, uni-directional links, frequent
path breaks due to mobility of nodes and the inherent
fading properties of the wireless channel [20]. This
substantiates the need for high adaptive AQM
algorithms with adapting capabilities to high variability
and uncertainty for these types of networks. The
proposed fuzzy logic based AQM, called Fuzzy-AQM,
is such types of algorithms to overcome the above
shortcoming in ad-hoc networks. The application of
fuzzy logic to the problem of congestion control allows
us to specify the relationship between queue
parameters and packets dropping probability using
“if...then...” type of linguistic rules. The fuzzy logic
algorithm would be able to translate or interpolate
these rules into a nonlinear mapping.
In this study, the focus is to investigate the impact
of the traditional and Fuzzy-AQM algorithms on the
ad-hoc network. The considered strategy is as follows:
When the congestion is detected, the node uses one of
the AQM policies to drop the incoming data packets.
Meanwhile, it allows the control packets to pass to the
queue using Drop-Tail policy. Therefore, the data
packets are dropped first when the packets drop
probability exceeds a certain threshold while the
control packets are still acceptable until the queue is
full.
Control messages are preferred to pass to the queue
during congestion time for the following reasons:
1. Control messages are used to update the changes of
the network topology. Therefore, they prevent data
packet to be transmitted through broken paths.
2. Data packets are “connection oriented”, that is,
guaranteed delivery to their destinations by TCP. In
contrast, control messages are “connectionless”; that
is, the dropped message will not be retransmitted
again.
3. Control message size is very small compared to data
packet. Normally in ad-hoc routing protocols,
control message size is 64 bytes while data packet is
512 bytes, i. e., the control message takes small
space in the queue and fast processing time in the
node.

Fuzzy Active Queue Management for Congestion Control in Wireless Ad-Hoc 51
The rest of this paper is organized as follows.
Section 2 summarizes related work on the common
AQM polices issues and focuses on previous
implementations of fuzzy AQM policies. Followed by
congestion in ad-hoc networks, the fuzzy dropping
algorithm as a new AQM policy (Fuzzy-AQM),
performance analyzes of the proposed algorithm, and
finally the conclusions.
2. Related Work
The most famous AQM algorithm is Random Early
Detection (RED) [11]. The RED algorithm manages
the queue in an active manner by randomly dropping
packets with increasing probability as the average
queue size increases. It maintains two thresholds that
determine the rate of packet drops: A lower threshold
(denoted by minth) and an upper threshold (denoted by
maxth). For each packet k arrives to the queue, the drop
probability for that packet pd (k) is given by:
 













th
ththp
thth
th
thc
d
avgif
avgif
avg
qif
kp
max1
maxminmax
minmax
min
min0
(1)
Where qc is current queue size, avg is current average
queue size and maxp is maximum drop probability.
Some previous studies showed the difficulties of
choosing the RED parameters [13, 18, 19]. Other
studies showed that there is no significant benefit to
RED over Drop-Tail for the web traffic [5, 6, 13].
Those drawbacks are the main reasons to default
disable of the RED function (or some vendor-specific
variant of RED, e. g., Cisco’s Weighted RED (WRED)
[9]) in most of the available routers today. To
overcome these drawbacks, extensions of the RED
algorithm had been proposed to make it more robust
and/or adaptive, for example, Stabilized RED (SRED)
[22], Flow RED (FRED) [3], Dynamic RED (DRED)
[16] etc. The most famous dynamic configured RED is
the Adaptive RED (ARED) algorithm proposed by
Floyed et al. [12]. In ARED, the maxp is configured
dynamically to keep the average queue size avg within
a target range.
Many studies used the fuzzy logic system to
dynamically calculate the drop probability behavior of
AQM policy. Wang et al. [30] proposed Adaptive
Fuzzy-based RED (AFRED) algorithm to calculate the
drop probability using the current queue size as the
only input for the fuzzy system. Some other studies
calculate the drop probability based on Fuzzy Explicit
Rate Marking (FERM) algorithm using two queue state
inputs: The current queue size 'qc' and its rate of
change '∆qc'. The FERM was implemented in [25] for
ATM networks, while in [8, 28] it was implemented
for differentiated services (Diff-Serv) networks.
In [1, 2, 7, 10, 17, 27, 32], the authors calculate the
drop probability using Fuzzy Proportional Derivative
Controller (FPDC) with two inputs: The error 'e'
(which is the difference between the current queue size
and the desired queue length) and the change of the
error '∆e' (which is the difference between the current
error and the previous error). A conventional fuzzy
controller use (e, ∆e) as inputs to observe the
controlled system response and its parameters. These
parameters are overshoot, rise-time and settle-time.
This set of parameters is not only used to evaluate the
stability, but the performance of a system as well, and
often is given in specification. Using the same inputs
(e, ∆e) to calculate the drop probability of AQM is
meaningless and the fuzzy “if...then...” rules will not
accurately represent the queue system behavior.
Li et al. [15] have used the current average queue
size 'avg' and its variance '∆avg' as the input for the
Fuzzy Logic Adaptive RED (FLARED) algorithm to
adaptively modifying the changes of step-size of the
parameter maxp. This scheme tune only one parameter
of ARED algorithm and its drawback is the lack to
tune other ARED parameters.
In this study, we have used fuzzy logic system to
calculate the drop probability in ad-hoc networks
using: The current queue size and the number of
neighboring nodes. This scheme can be generalized to
be used in any network where the number of neighbors'
nodes represents the number of communication links,
or precisely number of TCP sessions. Table 1
compares various schemes to design fuzzy AQM
algorithms.
Table 1. The fuzzy AQM schemes.
Fuzzy AQM Scheme Congestion Metric
Optimized
Variable
AFRED [30] Current queue size 'qc'
Drop
Probability
FERM [8, 25, 28] qc and its change '∆qc'
Drop
Probability
FPDC [1, 2, 7, 10,
17, 27, 32]
Error 'e' and its change '∆e'
Drop
Probability
FLARED [15]
Average queue size 'avg'
and its change '∆avg'
∆maxp
Our scheme:
Fuzzy-AQM
qc and node neighbors
density
Drop
Probability
3. Congestion in Ad-Hoc Networks
In ad-hoc networks, congestion control is handled
through transport layer protocols. The connection-
oriented transport layer protocol used in ad-hoc
networks is Transmission Control Protocol (TCP) [14].
The objectives of this protocol include the setting up of
an end-to-end connection, end-to-end delivery of data
packets, flow control and congestion control. TCP uses
window-based flow control mechanism. The sender
maintains a variable size window whose size limits the

number of packets the sender can send. The destination
sends ACKnowledgment (ACK) for packets that are
received. When the window size is exhausted, the
sender must wait for an ACK before sending a new
packet based on a sliding window principle. This
waiting time is known as Retransmission TimeOut
(RTO) period. If the ACK does not arrive within the
RTO period, then the sender will assume the packet is
lost. The loss of packet is due to the congestion in the
network which will yield TCP to start the congestion
control mechanism.
Mobile ad-hoc networks experience dynamic
changes in the network topology due to unrestricted
mobility of nodes. The topology changes lead to
frequent changes in the connectivity of wireless links
and hence routes reestablishment may be repeated very
often. This route reestablishment process takes a
significant amount of time. The route reestablishment
time is a function of transmission range of the nodes,
distance between the source and destination, number of
intermediate nodes between the source and destination
and node's velocity. If the route reestablishment time is
greater than RTO period of the source node, then it
will not receive the ACK and assumes congestion in
the network, followed by retransmission of the lost
packets and initiation of the congestion control
mechanism [20]. A schematic illustration of congested
ad-hoc network is shown in Figure 1. The source sends
its data packets through node A, which passes those
packets to node B then to the destination. As soon as
the link between the source and node A is broken, it
starts route reestablishment process and creates a direct
link with node B. If this processing time is less than
RTO, the source will receive the ACK and send other
data packets, or it will resend the previous lost packets.
4. Fuzzy-AQM Algorithm
In this section, concepts and rules of the proposed
Fuzzy-AQM algorithm for ad-hoc networks are
introduced. In the following two subsections, we
studied the effect of some node parameters on packets
drop probability. These parameters are used in
subsection C to create the rules of the fuzzy system.
Method to design their membership functions is
presented in the later subsection. Overall system design
and its implementation complexity are presented in
subsection E and F. Compatibility of the proposed
algorithm with other conventional algorithms
discussed in the last subsection.
4.1. Effect of qc on Drop Probability
Current queue size qc is the most used indicator in
AQM policy for estimating the probability of dropping
the incoming packets. The drop probability pd can be
calculated as [26]:
 2
2
2
cp
d
qCT
N
p


Where N is a load factor, C is a transmission capacity
(in packets/seconds) and Tp is a propagation delay (in
seconds). Assuming a 10 Mbps (2500 packets/sec)
transmission capacity with a 100 msec propagation
delay, Figure 2 shows the relation between the drop
probability and the load for various queue sizes. It is
evident that the probability of a packet dropping
increases as the load increases. More packets in the
queue wait for processing as load increases. Thus, it
can be stated that when the used space of the queue is
high, the drop probability of incoming packets is also
high and vice versa. Consequently, the following rules
are proposed:
R1: If qc is low then pd ought to be low.
R2: If qc is medium then pd ought to be high.
R3: If qc is high then pd ought to be high.
Figure 1. Congestion in ad-hoc networks.
4.2. Effect of Node Neighborhood Density on
Drop Probability
In ad-hoc networks, the traffic is categorized as: Data
packets and control messages. The control messages
are used to continuously update the nodes about the
topology changes (new created or lost links). For
example, if a node has two neighbors that means it will
receive two hello messages every second from them.
Besides, receiving a route request messages, a route
breaks messages, or data packets. If that node has ten
neighbors, this means it will receive, in every second,
ten hello messages beside bulk amount of control
messages and data packets. Hence, it is clear that the
traffic pass through the nodes with few neighborhoods
is less than the others with many neighbors. In
equation 2, the load N can be written as:


n
i
iN
0

Where λi denote flow's rate from the neighbor node i
and n is the number of neighbors. The congestion will
happen at:
mc
n
i
id qqandCifp  0
1 
(2)
(4)
(3)

Where qm is the maximum queue size. Hence, if the
neighbors' density of a node's is high, the node's queue
will be full quickly and increases the probability of
congestion and vice versa. Consequently, the following
rules are proposed:
R4: If neighbors' density is low then pd ought to be
low.
R5: If neighbors' density is medium then pd ought to be
high.
R6: If neighbors' density is high then pd ought to be
high.
Figure 2. Drop probability for the coming load.
4.3. The Rule-Base for Fuzzy Drop Probability
To fulfill the fuzzy sets theory, the previous six rules
(R1 to R6) can be combined within a 2-dimensional
rule-base to control the drop probability adaptively as
presented in Table 2. For example, according to Table
2 the first rule is:
If qc is Low and neighbors' density is Low then pd is
Low
Table 2. Fuzzy-AQM rules for drop probability.
Neighbors' Density
Low Medium High
Low Low Low Low
Medium Low High Highqc
High High High High
4.4. Membership Functions For Fuzzy
Variables
After defining the fuzzy linguistic ‘if-then’ rules, the
Membership Function (MF) corresponding to each
element in the linguistic set should be defined. For
example, if the queue size is 5 k bytes and qc equal to 2
k bytes, using conventional concept, it implies qc is
either ‘low’ or ‘medium’ but not both. In fuzzy logic,
however, the concept of MFs allows us to say the qc is
‘low’ with 80% membership degree and ‘medium’
with 20% membership degree.
The MFs we propose to use for the fuzzy inputs (qc,
neighbors' density) and the fuzzy output (pd) are
illustrated in Figure 3. These MFs are used due to their
economic value of the parametric and functional
descriptions. In these MFs, the designer needs only to
define one parameter; midpoint. These MFs mainly
contain the triangular shaped MF [23]. The remaining
MFs are as follows: Z-shaped membership to represent
the whole set of low values and S-shaped membership
to represent the whole set of high values.
(a) MFs used for the input variables.
(b) MFs used for the output variable.
Figure 3. Membership functions used for the fuzzy variables.
Maxpoint is the maximum queue size in qc−MF
(Table 2), and it is the number of the network's nodes
in the neighbors' density MF. Midpoint of qc−MF is a
threshold that indicates whether the queue is going to
be full soon. The threshold is simply set to 60% of the
queue size. The optimal value for this variable depends
in part on the maximum average delay that can be
allowed by the nodes. Tseng et al. [29] argue about the
cost-effectiveness to have large ad-hoc networks. They
proved by simulation that practical sizes of ad-hoc
networks would range within about five nodes.
Therefore, for neighbors' density MF, midpoint should
be equivalent to five nodes.
4.5. Fuzzification, Inference and
Defuzzification
The fundamental diagram of the fuzzy system is
presented in Figure 4. Fuzzification is a process where
crisp input values are transformed into membership
values of the fuzzy sets (as described in the previous
section). After the process of fuzzification, the
inference engine calculates the fuzzy output using the
fuzzy rules described in Table 2. Defuzzification is a
mathematical process used to convert the fuzzy output
to a crisp value; that is, pd value in this case.
There are various choices in the fuzzy inference
engine and the defuzzification method. Based on these
choices, several fuzzy systems can be constructed. In
this study, the most commonly used fuzzy system,

Mamdani method, is selected; for further details on this
system see [31].
Figure 4. Block diagram for the basic elements of the fuzzy-AQM.
4.6. Implementation Complexity
Using fuzzy logic system with AQM, we may achieve
comparable or better run-time computation than purely
conventional methods. This can be achieved using
lookup table. The input-output relationship of the fuzzy
reasoning engine for Fuzzy-AQM is illustrated in
Figure 5. This relationship can be stored as a lookup
table which will result in a very fast execution.
5. Performance Analysis of the Proposed
Fuzzy-AQM
5.1. Simulation Environment
Simulation of the proposed AQM design was done
using OMNeT++ version 2.3 with Ad-Hoc simulator
1.0 [21]. The OMNeT++ is a powerful object-oriented
modular with discrete event simulator tool. Each
mobile host is a compound module which encapsulates
the following simple modules: An application layer, a
routing layer, a MAC layer, a physical layer, and a
mobility layer.
 Application Layer: This module produces the data
traffic that triggers all the routing operations. In all
scenarios, 15 nodes are enabled to transmit. The
traffic is modeled by generating a packet burst of 64
packets sent to a randomly chosen destination that
stays the same for all the burst length. The rate of
each burst sending packets is 3 packets/sec. The
time elapsed between two application bursts is
normally distributed in [0.1, 3] sec. The packet size
is 512 bytes.
 Routing Layer: The routing model is the heart of the
simulator. This model depicts the Ad-hoc On-
demand Distance Vector (AODV) routing protocol,
all of its functions, parameters and their
implementation [24].
 MAC Layer: The simple implementation for this
layer has been used. The outgoing messages (from
routing layer) are let pass through to the physical
layer. The incoming one (from physical layer)
instead is delivered to the routing layer with an
MM1 queue policy with queue size 5k bytes. When
an incoming message arrives, the module checks a
flag that indicate if the routing layer is busy or not.
If so, the message will be saved in the queue using
Drop-Tail, Adaptive RED, or Fuzzy-AQM
algorithm. Note that Drop-Tail is a special case of
AQM with the following condition:


 

otherwise
qqif1
p mc
d
0
The parameters of Adaptive RED (see notation in
[12]) are set at minth = 1.5k bytes, maxth = 3k bytes,
maxp = 0.01, wq = 0.002,  = maxp/4, and  = 0.9.
When the routing layer is not busy, the MAC
module picks the first message from the queue and
sends it upward.
 Physical Layer: It deals with the on-fly creation of
links that allow the exchange of messages among
the nodes. Every time a node moves from its
position, an interdistance check on each node is
performed. If a node gets close enough (depending
on the transmission power of the moving nodes) to a
new neighbor, a link is created between the two
nodes with the following properties: Channel
bandwidth is 11 Mb/s (IEEE 802.11a) and delay is
10  s. Each node has a defined transmission range
chosen from a uniformly distributed number
between [90, 120] m.
 Mobility Layer: The random waypoint model was
adopted for the mobility layer. It is one of the most
used mobility pattern in the ad-hoc network
simulations. This is because of its simplicity and its
quite realistic mobility pattern. In this mobility
model, a node randomly selects a destination. On
reaching the destination, another random destination
is targeted after 3 seconds pause time. The speed of
movement of individual nodes range between [11,
16] m/sec. The direction and magnitude of
movement was chosen from a uniformly distributed
random number.
Three different network sizes are modeled:
700m×700m map size with 25 and 35 nodes and
800m×800m map size with 45 nodes. Each simulation
run takes 300 simulated seconds. Multiple runs were
conducted for each scenario and collected data was
averaged over those runs.
5.2. Performance Metrics
The following metrics were used for measuring
performance:
 Drop Ratio: The percentages of packets that are
dropped from the queue due to overflow
(congestion) to the total arrival in the queue.
 Invalid Route Ratio: Calculated as follows:
(11)





 n
i
n
i
routesvalidofNumber
routesinvalidofNumber
RatioRouteInvalid
1
1
Each time a route is used to forward a data packet, it
is considered as a valid route. If that route is
unknown or expired, it's considered as invalid route.
 Average End-to-End Delay: Average packet
delivery time from a source to a destination. First,
for each source-destination pair, average delay for
packet delivery is calculated. Then the whole
average delay is calculated from average delay of
each pair. End-to-end delay includes the delay in the
send buffer, the delay in the interface queue, the
bandwidth contention delay at the MAC layer, and
the propagation delay.
 Routing Overhead: Calculated as follows:




 n
1i
n
1i
ndestinatiobydatareceivedofNumber
by sourcektSentCtrlPofNumber
Overhead
Where n is number of nodes in the network and
SentCtrlPkt is control packets used by AODV and
described in Table 3. This metric can be employed
to estimate how many transmitted control packets
are used for one successful data packet delivery. We
use it to study the effect of AQM algorithms on the
efficiency and scalability of the routing protocol.
Figure 5. The input-output relationship of the fuzzy-AQM.
Table 3. Control packets used by AODV.
Message Description
RREQ A Route Request message
RREP A Route Reply message
RERR
A Route Error containing a list of the invalid
destinations
RREP_ACK A RREP acknowledgment message
6. Simulation Results and Evaluations
6.1. Drop Ratio Details
The average control messages drop ratio for the
proposed Fuzzy-AQM algorithm is less than other
conventional algorithms as shown in Figure 6-a. The
percentage of Fuzzy-AQM improvement compared to
Drop-Tail and Adaptive RED algorithms is: 93.9% and
74.5% for 25 nodes, 65.8% and 33.5% for 35 nodes,
and 75.1% and 49.7% for 45 nodes, respectively.
This improvement of the fuzzy algorithm is a result
of choosing the neighbors' density parameter to
estimate the size of incoming traffic and hence start the
early dropping policy as needed. Despite the data
packets drop ratio of Fuzzy-AQM is little bit higher
than adaptive RED, as shown in Figure 6-b, this is
enough to produce a higher enhancement in the control
messages drop ratio. This enhancement is a result of
the wide difference between the size of data packets
(512 bytes) and control messages (64 bytes).
Consequently, at congestion time, dropping one data
packet allows the queue to accept eight control
messages.
Drop-Tail algorithm doesn't have any mechanism to
distinguish between data and control packets like other
AQM algorithms. Moreover, the number of control
messages in ad-hoc network is much higher than data
packet; to provide continuous update of topology
changes. Those two reasons affect a high control
messages drop ratio for the Drop-Tail algorithm as
shown in Figure 6-a.
6.2. Invalid Route Ratio Details
The Fuzzy-AQM algorithm has less average invalid
route ratio compared to other conventional AQM as
shown in Figure 7. This decrement of the proposed
algorithm is about: 20.3% and 23.1% for 25 nodes,
31.1% and 14.6% for 35 nodes, and 22.4% and 12.9%
for 45 nodes, respectively.
Information about route breaks is broadcasted as an
RERR message. The Fuzzy-AQM algorithm allows
more control messages to pass the queue to the upper
routing layer as shown in Figure 6. This increased
number of received control messages helps the nodes
with Fuzzy-AQM to be more accurate to topology
changes and have precise updated routing tables,
hence, have less invalid routes.
6.3. Average End-to-End Delay Details
Figure 8 indicates that the proposed Fuzzy-AQM
algorithm has lower average end-to-end delay
compared to other conventional algorithms. This
decrement is approximately: 17.2% and 6.3% for 25
nodes, 24.1% and 11.6% for 35 nodes, and 33.6% and
21.6% for 45 nodes, respectively.
The nodes that have conventional AQM algorithms
have higher invalid route ratio as shown in Figure 7,
therefore they suffer longer routing delay to recover
from broken paths and discover new ones. To recover a
broken path, an RERR message must first be launched
from the intermediate nodes to tell the source node
about the broken link. The source node deletes the
corresponding entry from its routing table. The RREQ

must then be broadcasted from the source to the
destination, and an RREP consequently has to be
transmitted back to the source. Data packets are
buffered at the source node during this process and the
duration of their buffering adds more time delay to the
end-to-end delay. The nodes with Fuzzy-AQM
algorithm, on the other hand, have reliable routing
tables that minimize the need to this recovery process.
(a) Control messages drop ratio comparison.
(b) Data packets drop ratio comparison.
Figure 6. Drop ratio comparison.
Figure 7. Invalid route ratio comparison.
6.4. Routing Overhead Details
As expected, the AQM algorithms don't have major
effect on the routing protocol efficiency or scalability
as shown in Figure 9. These algorithms maximize the
number of 'received' control messages, meanwhile they
have no effect on 'sent' control messages (see equation
10). This is because the control messages used in
AODV are broadcast messages; that is, they will not be
resent if they are dropped or lost.
The Drop-tail algorithms has worst routing
overhead ratio as the number of node increase as a
result of increasing data packets drop ratio which is
clear in Figure 6-b. Meanwhile, the data packets
dropping ratio is nearly the same for adaptive AQM
algorithms (ARED and Fuzzy-AQM) that results in no
major difference in routing overhead ratio.
6.5. Drop Probability Values
In Drop-Tail algorithm, pd always take a static value of
1 to start packet dropping at overflow. In Adaptive
RED algorithm, pd increases linearly between the two
thresholds minth and maxth in dependent on the average
queue size 'avg'. Some studies [26] showed that using
linear pd function can result in forced drops when qc
exceeds maxth or link under-utilization when qc
decreases to zero. This is an evident that the original
linear drop function does not perform well within a
wide range of loads.
The pd values used by the proposed Fuzzy-AQM for
randomly chosen node in the 25 nodes simulated
network are shown in Figure 10. It is evident that the
drop function is non-linear and a high load requires a
disproportionately higher pd than a low load to keep
the queue size in the same range. Non-linearity of pd
function is also clear in the input-output relation as
shown in Figure 5.
The comparison between the average pd values used
by every node in the 25 nodes and the 35 nodes
networks is shown in Figure 11. Due to higher
neighbors' density, 35 nodes network have higher pd
values than 25 nodes network. This is a result of
increasing neighbors' density which will also increase
the number of control messages.
7. Conclusions
In this study, a novel AQM algorithm (Fuzzy-AQM)
based on fuzzy logic system was suggested. This
algorithm for early packets dropping is implemented in
wireless ad-hoc networks in order to provide effective
congestion control by achieving high queue utilization,
low packet losses and delays. The proposed scheme is
contrasted with a number of well-known AQM
schemes through a wide range of scenarios. From the
simulation results, the efficiency of the proposed fuzzy
AQM policy in terms of routing overhead, average
end-to-end delay and average packet losses are

pronounced than other AQM polices, with capabilities
of adapting to high variability and uncertainty in the
mobile ad-hoc networks.
(a) 25 nodes.
(b) 35 nodes.
(c) 45 nodes.
Figure 8. Average end-to-end delay comparison.
Figure 9. Routing overhead comparison.
Figure 10. Drop probability values used by a node.
Figure 11. Average pd values used by 25 and 35 nodes networks.
References
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Essam Natsheh obtained his MSc in
computer engineering from the Arab
Academy for Science and
Technology in Egypt, in 1999. He
subsequently worked as a lecturer at
Al-Alamiah Institute for Computer
and Technology, SA, in 2002. He
worked as a lecturer also at the Information Systems
Department of the King Faisal University, SA, from
2002 to 2003. Since 2003, Natsheh has been a member
of a research group headed by Dr. Jantan A. at the
University Putra Malaysia, which investigates issues
related to the design and analysis of ad-hoc wireless
networks.
Adznan B. Jantan obtained his
MSc in digital systems from
Cranfield Institute of Technology,
UK, in 1982 and his PhD in speech
recognition systems from the
University College of Swansea, UK,
in 1988. Since 2002, he is an
associate professor at the Department of Computer and
Communication System at the University Putra
Malaysia, where he has been conducting research in
computer networking, pattern recognition, and digital
systems design.
Sabira Khatun received her BSc
(Hons.), MSc in applied mathematics
and PhD on hydromagnetic stability
from the University of Rajshahi,
Bangladesh in 1988, 1990, and 1994,
respectively. She received her
second PhD in communications and
networking from University Putra Malaysia in 2003.
She became a lecturer at the Department of Computer
Science and Engineering, University Khulna,
Bangladesh in 1991, and promoted to assistant
professor in 1994. She joined the Department of
Computer & Communication Systems Engineering,
University Putra Malaysia in 1998. She is an active
researcher of Teman project and MyREN Research
Community. She is a member of IEEE. Her research
interest spans broadband and wireless communications,
and network management, including software defined
radio and IPv6.
Shamala Subramaniam completed
her PhD from University Putra
Malaysia in 2002. Currently, she is a
lecturer at the Department of
Communication Technology and
Networks, Faculty of Computer
Science and Information
Technology, University Putra Malaysia. Her research
interest includes scheduling algorithms, congestion
control, real-time systems, modeling, and simulation.

International Journal of Fuzzy Systems, Vol. 15, No. 2, June 2013
© 2013 TFSA
203
Robust Fuzzy Congestion Control of TCP/AQM Router via Perturbed
Takagi-Sugeno Fuzzy Models
Wen-Jer Chang, Po-Hsun Chen, and Cheng-Ting Yang
Abstract1
Because the number of transmission control pro-
tocol sessions usually varies from time to time, a per-
turbed Takagi-Sugeno fuzzy model is used in this pa-
per to represent the dynamic transmission control
protocol network systems. According to the proposed
perturbed Takagi-Sugeno fuzzy model, a robust fuzzy
controller design approach is investigated to achieve
the congestion avoidance for the transmission control
protocol network systems. In order to accomplish the
above mission, some sufficient conditions are derived
based on the Lyapunov stability theory. By solving
these sufficient stability conditions, an active queue
management router can be obtained via the proposed
robust fuzzy congestion control technique.
Keywords: Perturbed Takagi-Sugeno fuzzy model, TCP
network system, congestion control and robust fuzzy
control.
1. Introduction
Fuzzy systems based on the Takagi-Sugeno (T-S)
fuzzy model [1] have attracted great interests from sci-
entific and engineering communities. The T-S fuzzy
model provides an efficient method to represent complex
nonlinear systems via fuzzy sets and fuzzy reasoning. By
extending the linear control and constructing Parallel
Distributed Compensator (PDC), the T-S fuzzy model
based control has been applied to cope with complex
nonlinear systems. The analysis and synthesis problems
of T-S fuzzy systems have been extensively studied
[2-12]. The T-S fuzzy systems can be separated into the
homogeneous fuzzy systems [2-6] and the affine fuzzy
systems [7-12]. The homogeneous fuzzy system is re-
ferred to the T-S fuzzy system of which consequent part
is linear and which does not have a constant bias term. In
contrast, the affine fuzzy system means the T-S fuzzy
system of which consequent part is affined with a con-
Corresponding Author: Wen-Jer Chang is with the Department of
Marine Engineering, National Taiwan Ocean University, Keelung 202,
Taiwan, R. O. C.
E-mail: wjchang@mail.ntou.edu.tw
Manuscript received 10 Aug. 2012; revised 25 March 2013; accepted
21 May 2013.
stant bias term.
In this paper, an Active Queue Management (AQM)
router design problem for the Transmission Control
Protocol (TCP) network system is discussed. The dy-
namic model of TCP network system has been proposed
such that three key parameters, number of TCP sessions,
link capacity and round-trip time are related to a feed-
back control problem. In order to understand the behav-
ior of a dynamic TCP network, modeling and analysis of
such network are very important. In [13], the authors
used jump process driven stochastic differential equa-
tions [14] to model the behavior of a TCP network. In
[15], the authors exploited fluid modeling to present a
general methodology for the analysis of a network of
routers supporting AQM with TCP flows. They modeled
the data traffic as a fluid and specifically used Poisson
counter driven stochastic differential equations to model
sample path description of TCP traffic. By employing
the small signal linearization about an operating point,
the authors of [16] approximated the nonlinear
fluid-based TCP dynamic model in [15] as a linear con-
stant system. It can be found that the TCP network sys-
tem is a nonlinear perturbed affine time-delay system
because it has time delay, constant link capacity and
varying sessions. Thus, the linear constant system will
sacrifice the original nonlinear properties for applying
the linearized technique. Therefore, the motivation of
this paper is to use the fuzzy modeling technique to re-
place the linear modeling method such that the nonlinear
TCP network system can be represented by a perturbed
time-delay T-S fuzzy model.
The AQM design problems are important in TCP net-
work system because AQM is embedded in the router
and has much information about circumstances of cur-
rent networks. Some kinds of AQM schemes have been
proposed, such as Random Early Detection (RED)
[16-17], Random Exponential Marking (REM) [18],
Adaptive Virtual Queue (AVQ) [19], Proportional Inte-
gral (PI) controller [20-22], predictive controller [23]
and fuzzy controller [24-25]. By applying the linearized
model, references [18, 20-22] applied control theory to
address congestion avoidance issues. However, the con-
trol approaches developed in [18, 20-22] sacrifice the
original nonlinear properties because they applied the
linear modeling technique to implement the TCP net-
works. To overcome the problem, the authors of [25]

used a time-delay affine T-S fuzzy model to reconstruct
the TCP networks. However, the number of TCP ses-
sions was assumed to be constant in [25]. In fact, the
number of TCP sessions usually varies with time. In or-
der to obtain a better approximation to the original
nonlinear TCP network system, a perturbed time-delay
affine T-S fuzzy model is constructed in this paper. Ac-
cording to this perturbed T-S fuzzy model, the contribu-
tion of this paper is to investigate the robust fuzzy con-
gestion control for the TCP/AQM router design.
Employing the Linear Matrix Inequalities (LMI) tech-
nique [26], one can find a suitable common positive
definite matrix for the stability conditions, and then to
obtain a stable fuzzy controller for the closed-loop ho-
mogeneous T-S fuzzy models. However, the fuzzy con-
troller design of the perturbed time-delay affine T-S
fuzzy models is a challenging problem for the designers
because the closed-loop stability conditions are not LMI
formulations but Bilinear Matrix Inequalities (BMI) ones.
The BMI conditions cannot be easily solved via a con-
vex optimization algorithm [26]. In order to solve the
BMI problem, an Iterative Linear Matrix Inequality
(ILMI) algorithm has been presented in [7-10]. Extend-
ing the control approach of [7-10], a modified ILMI al-
gorithm is developed in this paper to find feasible solu-
tions for the synthesis problem of fuzzy controller design
for perturbed time-delay affine T-S fuzzy models.
Therefore, a novel congestion control approach for the
AQM router can be accomplished in this paper based on
the robust fuzzy controller design for the perturbed
time-delay affine T-S fuzzy models.
The organization of this paper is presented as follows.
In Section 2, the time-delay affine T-S fuzzy model of
TCP network with AQM routers is introduced. Section 3
provides the stability conditions for the existence of
fuzzy control based AQM routers. In order to find the
feasible solutions of these stability conditions, an ILMI
algorithm is developed in Section 3. The effectiveness of
proposed fuzzy control on managing queue utilization is
illustrated in Section 4 by a numerical simulation. Fi-
nally, a conclusion is given in Section 5.
2. Perturbed Time-Delay Affine T-S Fuzzy Model
of TCP/AQM Routers
According to [15], the TCP network system can be
formulated as the following stochastic differential equa-
tions:
 
 
    
  
  
W t R tW t1
W t p t R t
R t 2 R t R t
-
-
-
    (1a)
 
   
    
    
m
m
r t -C if 0<q t <q
q t = max 0, r t -C if q t =0
min 0, r t -C if q t =q






 (1b)
where    mW t 0, W is the TCP window size; mW is
the maximum window size;  R t is the round-trip times;
   p t 0, 1 is the probability of packet mark/drop;  r t
is the aggregate incoming rate; C is link capacity; mq
is the finite buffer size. The queue length,  q t , changes
depending on the queue occupancy rate, which is the
accumulated different between the incoming rate and the
outgoing link capacity, i.e.,  r t C . In addition, the
round trip time,  R t , is the sum of propagation delay
pT and queuing delay  q t C . That is,
   pR t T q t C (2)
As well as, the aggregate incoming rate  r t can be
represented as a function of  W t and  q t as follows:
 
 
 
 
 
 
 
p
W t W t
r t N t N t
R t T q t C
 

(3)
where  N t is the load factor, i.e., number of TCP ses-
sions. As shown in Fig. 1, Equation (1) denotes the net-
work topology which was assumed to be a single bottle-
neck with TCP sources that share the bottleneck link and
the same round-trip time.
Figure 1. Structure of TCP network systems.
In order to construct the affine T-S fuzzy model for the
TCP dynamic model (1), we ignore the dependence of
the time-delay argument  t R t on window size  W t
and queue length  q t , and assume it is fixed to ct R ,
where cR is a constant value. The delay in  p t is
omitted for the same reason. As a result, a simplified
dynamic of TCP control model (1) can be described as
follows:
 
 
   
 
 
P
P
W t RW t1 c
W t p t
T q t C 2 T q t R C
c
-
-
   
 
 (4a)

Wen-Jer Chang et al.: Robust Fuzzy Congestion Control of TCP/AQM Router via Perturbed Takagi-Sugeno Fuzzy Models 205
 
 
 
 
P
W t
q t = N t -C
T +q t C
 (4b)
The construction of the affine T-S fuzzy model is
achieved by applying the small signal linearization
method [27]. In system (4), let (W, q) be the system
state to be controlled, p be the input, and  0 0 0
W , q , p
be the equilibrium point of the system. Then, the equi-
librium point  0 0 0
W , q , p can be obtained as follows by
solving  W t =0 and  q t =0 with the assumption
   0 0
c dW t-R W t-R W  and    0 0
c dq t-R q t-R q  . Note
that     0
t
lim q t-R -q t =0

and     0
t
lim W t-R -W t =0

, i.e.,
0 0
dW W and 0 0
dq q when t   . Hence, one can
obtain the equilibrium point as follows:
 0 0
p
C
W = T +q /C
N
(5a)
   
0
p0
0 2 2
0 0
p
T +q /C2 2
p = =
T +q /C W W
 (5b)
0
0
p
q
R =T +
C
(5c)
In order to shift the equilibrium point to origin, let us
define new state variables as follows:
    0
1 t =W t -Wx (6a)
    0
2 t =q t -qx (6b)
     0 0 0
1 1d dt-R t W t-R -wx x  (6c)
     0 0 0
2 2d dt-R t q t-R -qx x  (6d)
    0
t =p t -pu (6e)
Then, the new equilibrium point can be stated as
 0
1 t =0x ,  0
2 t =0x ,    0 0
1d 1t = t-R =0x x ,    0 0
2d 2t = t-R =0x x
and  t =0u . Hence, the simplified TCP fluid-flow dy-
namic model (4) can be represented as
 
  
  0
1
1 0
p 2
t +W1
t =
2T + t +q /C
x
x
x

 
  
  
0 0
1 0
0 0
p 2
t-R +W
t +p
T + t-R +q /C
x
u
x
  (7a)
 
    
  
0
1
2 0
p 2
t +W N t
t = -C
T + t +q /C
x
x
x
 (7b)
According to the nonlinear dynamic model (7) with
new state variables defined in (6), the corresponding
perturbed time-delay affine T-S fuzzy model can be con-
structed of the following IF-THEN form by using the
small signal linearization method [27].
Rule i: IF  1 tx is i1M and  2 tx is i2M THEN
            i i id i i it t t t tx x x u         A A A B a a
   t tx   , 0
t -R 0,    , i 1 2 r, , ,  ,   it i Iˆx , X
(8)
where   2
tx  is the state vector and  tu  is the
control input vector. The matrices 2 2
i

A , 2 2
id

A ,
2 1
i

B , and 2
i a are constant for i =1, 2, …, r and r
is the number of IF-THEN fuzzy rules. Let us define
            
T0
1 2f t , t R , t t x t x tx x u x         , then the con-
sequent system parameters can be obtained by [27] as
follows:
      
  i i i
d
0
i
x ,x ,u
f x t ,x t-R ,u t
=
x t


A (9a)
      
 
i i i
d
0
id x ,x ,u0
f x t ,x t-R ,u t
=
x t-R


A (9b)
      
 
i i i
d
0
i x ,x ,u
f x t ,x t-R ,u t
=
u t


B (9c)
with the affine terms
 i i i i i i
i d i id d i=f x ,x ,u - x - x - ua A A B (10)
where  i i i
d, ,x x u is the operating point of the ith fuzzy
rule. Besides, the ijM are fuzzy sets and  t is the
initial condition of the state defined on 0
R t 0   . The
region 2
i  X is assumed to be a fuzzy subspace and
iX is called as a cell. The set of cell indices is denoted
as Iˆ and the union of all cells   ii I
t ˆx 
 X is referred
to as the whole fuzzy space. Let 0I Iˆ ˆ be the set of in-
dices for the fuzzy rules that contain the origin and
1I Iˆ ˆ be the set of indices for the fuzzy rules that does
not contain the origin. The origin is an equilibrium point
of the time-delay affine T-S fuzzy models and it is as-
sumed that i 0a for 0i Iˆ . The quantities iA , idA , iB
and ia are constant matrices. Besides, iA and ia
are time-varying matrices with appropriate dimensions
and they are structured in the following norm-bounded
form:
    i i i i 1i 2it  A a D Δ Q Q (11)
where iD , 1iQ and 2iQ are known real constant ma-
trices of appropriate dimensions, and  i tΔ is an un-
known matrix function with Lebesgue-measurable ele-
ments and satisfies    T
i it t Δ Δ I .
Given a pair of     x t , u t , the final outputs of the
fuzzy model (8) are inferred as follows:
        
r
i i i
i 1
t h t t{x x x

   A A
      id i i it t t }x u      A B a a (12)
where
     
T
1 2t x t x tx     (13)
     
2
i ij j
j 1
ω t M tx x

  (14)

  
  
  
i
i r
i
i 1
ω t
h t
ω t
x
x
x



(15a)
  ih t 0x  (15b)
  
r
i
i 1
h t 1x

 (15c)
The perturbed time-delay affine T-S fuzzy model of
TCP network with AQM routers has been constructed in
(8) or (12). In the next section, the stability conditions
for the proposed fuzzy controller design are derived
based on the above perturbed time-delay affine T-S fuzzy
model.
3. Designing Robust Fuzzy Congestion
Controller
For the perturbed time-delay affine T-S fuzzy model
represented by (12), a fuzzy controller is designed based
on the PDC concept [2] as follows:
Rule i: IF  1 tx is i1M and  2 tx is i2M THEN
   it tu x F , i =1, 2… r, for   it i Iˆx , X (16)
where 1 2
i

F are constant. The output of the
PDC-based fuzzy controller is determined by the sum-
mation such as
       
r
i i
i 1
t h t tu x x

  F (17)
Substituting (17) into (12), one can obtain corre-
sponding closed-loop system as follows:
           
r r
i j i i
i 1 j 1
t h t h t t{x x x x
 
   A A
      id i j i it t t }x x      A B F a a (18)
Lemma 1 [28]: For any two real matrices n m
X and
n m
Y , one has
T T T T 1
  X Y Y X X NX Y N Y (19)
where 0N is a constant matrix (or scalar).
The asymptotical stability analysis issue to
closed-loop perturbed time-delay affine T-S fuzzy model
(18) is discussed based on Lyapunov stability criterion.
Besides, the Lemma 1 and S-procedure [27] are used to
derive the stability conditions. In the following theorem,
sufficient conditions for ensuring delay-independent sta-
bility of the perturbed time-delay affine T-S fuzzy model
(18) are introduced.
Theorem 1: The perturbed time-delay affine T-S fuzzy
model (18) is quadratically stable in the large if there
exist common positive-definite matrices 0P , 0S ,
control gains iF and scalars ijq 0  and 0  such
that
T T
11 i j i j i j id jdα
0 0 0 0 0
2
0 0 0 0
0
0 0 0
0 0
0
*
* *
* * *
* * * *
* * * * *
* * * * * *
         
 
 
 
 
 
 
 
 
 
  
Λ P P D D P B B F F PA PA
I
I
I
I
S
S
for 0i Iˆ (20)
T T
11 12 i j i j i j id jd
22
α
0 0 0 0 0 0
0 0 0 0 0
2
00 0 0 0
0 0 0
0 0
0
*
* *
* * *
* * * *
* * * * *
* * * * * *
* * * * * * *
         
 
 
 
 
  
 
 
 
 
 
 
  
Λ P Λ P D D P B B F F PA PA
Λ
I
I
I
I
S
S
for 1i Iˆ (21)
and > 0P S for 0 1i I and Iˆ ˆ (22)
where * denotes the transposed elements or matrices for
symmetric position and
T T
11 i j i j 2    Λ A P A P PA PA P
T T
T T T T T T T T
ij ji ij ji 1i 1j 1i 1j
               E E E E Q Q Q Q
     T T T T T
ij i j i j ij ji j i     E B P F PB F E E B P F
 
2
T
j i ji ijq ijq
q 1
2 T

   PB F E (23)
 
2
12 i j ijq ijq
q 1
2

   Λ P a a n (24)
2T
T T T T
22 2i 2j 2i 2j ijq ijq
q 1
2

         Λ Q Q Q Q v , (25)
T
ij i j E B P F (26)
Besides, the S-procedure weighting parameter ijqT , ijqn ,
ijqv are defined such that
        T T
ijq ijq ijq ijqt t t 2 t v 0x x x x   T n ,
q 1 2, and i 1 r  (27)
for all  tx which activates rule i (i.e.,   ih t 0x  ).
Proof:
Select a Lyapunov function as
      T
t t tV x x x P (28)
The derivative of the Lyapunov function   tV x along
the trajectories of (18) is
  tV x       
r r
i j ij
i 1 j 1
h t h tx x V
 
  (29)
where

   
T
ij ji ij jiT
ij t t
2 2
V x x
      
      
     
G G G G
P P
   
T
i j i jT
t t
2 2
x x
    
    
   
a a a a
P P
         T T T
id idt t t t t t
2
x x x x    

A P PA
         T T T
jd jdt t t t t t
2
x x x x    

A P PA
 
   
 
T T
i i j jT
t t
2
x x
        
  
  
A P P A A P P A
   
T
i j i jT
t t
2 2
x x
        
    
   
a a a a
P P (30)
and ij i i j G A B F . From the Lemma 1, one has
         T T T
id idt t t t t tx x x x    A P PA
         T T 1 T
id idt t t t t tx x x x
     S PA S A P ,
for i 1 r, ,  (31)
where 0S is a constant matrix. Using    T
i it t Δ Δ I
and Lemma 1, one can obtain
       T T T
i it t t tx x x x  A P P A
           T T T T T
1i i i i i 1it t t t t tx x x x Q Δ D P PD Δ Q
           T T T T T
i i 1i i i 1it t t t t tx x x x PD D P Q Δ Δ Q
       T T T T
i i 1i 1it t t tx x x x PD D P Q Q (32)
and
   T T
i jt tx x  P a a P
       T T T T
i i 2i 2i i it t t tx x PD Δ Q Q Δ D P
   T T T
i i 2i 2it tx x PD D P Q Q (33)
From (30)-(33), one has
ijV     
T
ij ji ij jiT
t t
2 2
x x
      
     
     
G G G G
P P
   
T
i j i jT
t t
2 2
x x
    
    
   
a a a a
P P
     
   T 1 T
id idT
t t
t t t t
2
x x
x x

     
PA S A P
S
   
   
T 1 T
jd jd T T
i i
t t
t t
2
x x
x x

 
PA S A P
PD D P
       
T
T T T 1i 1i
j jt t t t
2
x x x x 
Q Q
PD D P
   
T TT
1j 1j 2j 2jT 2i 2i
t t
2 2 2
x x  
Q Q Q QQ Q
(34)
If the condition (22) holds, the following relationship
can be obtained.
         T
t t t t t tV x x x      P
     T
t t t tx x    S (35)
Based on the Lyapunov-Razumikhin theorem [29], if the
inequality      t tV x V x   holds for all time, the
stability condition is undoubtedly asymptotically stable.
So, it is necessary to check the stability for the case of
     t tV x V x   only. Hence, if there exits a real
number 1  such that      t tV x V x    for
0
0 R    , then (34) can be replaced as follows due to
(35).
 
T
ij ji ij jiT
ij t
2 2
V x
     
    
   
G G G G
P P
1 T1 T
jd jdid id
2 2

  
PA S A PPA S A P
P
   
T T
1i 1i 1j 1jT T
i i j j t
2
x
 
   

Q Q Q Q
P D D D D P
   
T T T
i j i j 2i 2i 2j 2jT
t t
2 2 2
x x
     
     
   
a a a a Q Q Q Q
P P
(36)
Applying the S-procedure [27], the matrix inequality
(36) becomes
 
T
ij ji ij jiT
ij t
2 2
V x
     
    
   
G G G G
P P
1 T1 T
jd jdid id
2 2

  
PA S A PPA S A P
P
   
T T
1i 1i 1j 1jT T
i i j j t
2
x
 
   

Q Q Q Q
P D D D D P
   
T T T
i j i j 2i 2i 2j 2jT
t t
2 2 2
x x
     
     
   
a a a a Q Q Q Q
P P
  
2
ijq ijq
q 1
tx

   (37)
Let us define
   
T
ij ji ij jiT
δ t
2 2
x
     
     
   
G G G G
P P
1 T1 T
jd jdid id
2 2

  
PA S A PPA S A P
P
   
T T
1i 1i 1j 1jT T
i i j j t
2
x
 
   

Q Q Q Q
P D D D D P
   
T
i j i jT
t t
2 2
x x
    
    
   
a a a a
P P
  
T T 2
2i 2i 2j 2j
ijq ijq
q 1
t
2
x


   
Q Q Q Q
(38)
If the condition  1 0  holds, then by continuity, there
exists a 1    with sufficiently small 0  one can
obtain   0   for all i. Hence, if  1 0  one has
  0   and   t 0V x  . Thus, the stability of system
(18) can be guaranteed due to   t 0V x  for all
  i 1t i Iˆx , X .
In order to derive the condition (21), let us define the

following matrix.
        T
T
ij ij ji ij ji2 1 tx     L G G P P G G
1 T 1 T
id id jd jd2  
  P PA S A P PA S A P
    T T T T
i i j j 1i 1i 1j 1j2 tx   P D D D D P Q Q Q Q
       
T
T T
i j i j 2i 2it tx x    P a a a a P Q Q
  
2
T
2j 2j ijq ijq
q 1
2 tx

   Q Q (39)
From Lemma 1, one can obtain
 T T T
ij i j i jtx   L A P A P PA PA
1 T 1 T
id id jd jd2  
  P PA S A P PA S A P
T T T T T T
i i j j i i j j ij ij ji ji     F F F F PB B P PB B P E E E E
     T T T T T
ij i j i j ij ji j i     E B P F PB F E E B P F
   T T T
j i ji i i j j2   PB F E P D D D D P
  T T
1i 1i 1j 1j tx Q Q Q Q
       
T
T
i j i jt tx x   P a a a a P
  
2
T T
2i 2i 2j 2j ijq ijq
q 1
2 tx

    Q Q Q Q
=  T T T 1 T
i j i j id idt 2x 
    A P A P PA PA P PA S A P
T
1 T T T
jd jd i i j j i j i j

          PA S A P F F F F P B B B B P
   
T
T T T T T T T
ij ji ij ji ij i j i j ij
          E E E E E B P F PB F E
   T T T
ji j i j i ji   E B P F PB F E
T
i j i j2        P D D I D D P
  
T
T T T T
1i 1j 1i 1j tx       Q Q Q Q
       
T
T
i j i jt tx x   P a a a a P
  
2
T T
2i 2i 2j 2j ijq ijq
q 1
2 tx

    Q Q Q Q
 T
t 1x   
 
 
2
ij i j ijq ijq
q 1
2T
T T T T
2i 2j 2i 2j ijq ijq
q 1
2
t
1
2
x
*


 
   
                 
  


Γ P a a n
Q Q Q Q v
(40)
where ijE is defined in (26) and
T T 1 T
ij i j i j id id2 
     Γ A P A P PA PA P PA S A P
T
1 T T T
jd jd i i j j i j i j

          PA S A P F F F F P B B B B P
   
T
T T T T T T T
ij ji ij ji ij i j i j ij
          E E E E E B P F PB F E
   T T T
ji j i j i ji   E B P F PB F E
T
i j i j2        P D D I D D P
2T
T T T T
1i 1j 1i 1j ijq ijq
q 1
2 T

         Q Q Q Q (41)
It is obvious that if the following inequality is satisfied,
then ij 0L and   0   can be obtained.
 
2
ij i j ijq ijq
q 1
2T
T T T T
2i 2j 2i 2j ijq ijq
q 1
2
0
2*


 
   
  
 
         
  


Γ P a a n
Q Q Q Q v
(42)
Using the Schur complement, the above inequality can
be represented as follows:
T T
11 12 i j i j i j id jd
22 0 0 0 0 0 0
0 0 0 0 0
2
00 0 0 0
0 0 0
0 0
0
*
* *
* * *
* * * *
* * * * *
* * * * * *
* * * * * * *
        
 
 
 
 
  
 
 
 
 
 
 
  
Λ Λ P D D P B B F F PA PA
Λ
I
I
I
I
S
S
(43)
where 11Λ , 12Λ and 22Λ are defined in (23) , (24) and
(25), respectively.
If the condition (21) is satisfied for a scalar 0  ,
then the inequality (43) can be obtained. In this case, one
has   0   and   t 0V x  . The proof of stability
condition (21) is completed for all   i 1t i Iˆx , X .
Moreover, for the case of   itx  X , 0i Iˆ , the stability
condition (20) can be obtained by ignoring the
S-procedure from the similar proof procedure.
Q.E.D.
Theorem 1 provided the stability conditions for the
closed-loop perturbed time-delay affine T-S fuzzy model
(18). However, these conditions are of the BMI formulas
which cannot be solved by the LMI technique [26]. In
order to solve this problem, an ILMI algorithm is pro-
posed to find the feasible solutions for conditions
(20)-(22). The purpose of this algorithm is to interac-
tively search for P , S , iF , ijq ,  and to update the
auxiliary variables until  becomes negative. Based on
the stability conditions of Theorem 1, the solutions of
the robust fuzzy control problem for TCP/AQM routers
are solved via the following ILMI algorithm.
Fuzzy Controller Design Algorithm-ILMI Algorithm
Step 1:
Solve the initial matrix  0
P from the following Ri-
catti equation.
       0 0 0 0T T
0ˆ ˆ ˆ ˆ   A P P A P BB P Q (44)
where
r
i
i 1
1
r
ˆ

 A A ,
r
i
i 1
1
r
ˆ

 B B and 0Q . The matrix
Q is assigned by the designers. Besides, choose the ei-

genvalues of  0
i i iA B F and solve the initial gains  0
iF
by standard pole placement technique. Denote k as the
iteration index and set k 1 for the initial conditions.
Step 2:
Find the iterative auxiliary variables  k
ijE by the fol-
lowing equation:
     k k-1 k-1T
ij i j E B P F (45)
Using the auxiliary variables  k
ijE to solve the optimiza-
tion problem for  k
P ,  k
S ,  k
iF ,  k
ijq from (20)-(22)
subject to minimizing  k
 . If  k
0  then  k
P ,  k
S ,
 k
iF and  k
ijq obtained in Step 2 are the feasible solu-
tions and stop the iterative manner. Otherwise, if  k
0 
then go to Step 3.
Step 3:
Resolve the optimization problem for  k
P ,  k
S ,  k
iF
and  k
ijq from (20)-(22) subject to minimizing
 
 k
trace P by using  k
 and the corresponding auxil-
iary variables  k
ijE obtained in Step 2. Given a prede-
termined small value . If      
  k k kT
ij i j
ij
    E B P F ,
then the conditions (20)-(22) may not be feasible and
stop the iterative manner. Otherwise, one can set
k k 1  and go back to Step 2 to update the auxiliary
variables  k
ijE using  k-1
P ,  k-1
iF , where  k-1
P and  k-1
iF
are determined in Step 3.
By applying the above ILMI algorithm, one can find
suitable matrices  k
>0P ,  k
>0S ,  k
iF , and scalars ijq 0 
to satisfy stability conditions (20)-(22). The ILMI algo-
rithm provides a useful scheme to find the feasible solu-
tions of robust fuzzy controller for the TCP network with
AQM routers which is modeled by the perturbed
time-delay affine T-S fuzzy model (18). In the next sec-
tion, a numerical example is provided to illustrate the
applicability and effectiveness of proposed fuzzy con-
troller design procedure that can be efficiently employed
to design AQM router for the TCP network systems.
4. Numerical Simulations
In order to verify the feasibility, usefulness and appli-
cability of proposed fuzzy controller design approach,
this section looks at an experiment with computer simu-
lations. Let us consider a TCP network made up of a
router connected to a single link with some sources and
one destination. Referring to the simplified TCP
fluid-flow dynamic model (4), one can obtain its
time-delay affine T-S fuzzy model described in (8).
Based on this time-delay affine T-S fuzzy model, one can
find a stable T-S fuzzy controller by using the design
procedure developed in Section 3.
Let us consider a TCP network with system parame-
ters: link capacity C 3750 packets/sec., propagation
delay pT 0 2. seconds. In order to obtain the time-delay
affine T-S fuzzy model, it is necessary to determine the
numbers of IF-THEN fuzzy rules and the coordinate op-
erating points. Here, we use three fuzzy rules to con-
struct the time-delay affine T-S fuzzy model. First, we
fix N 120 to choose three queue length values, i.e.,
575q
 , 175*
q  , 0q
 packets, to be the operating
points. Substituting q
, *
q and q
into (5), one can
obtain the operating points as follows:
 T T
d d
oper1
, ,W q W q p    
      
      T T
11.04 575 , 11.04 575 , 0.0164 (46a)
 T T
d d
oper2
, ,* * * * *
W q W q p      
      T T
7.7083 175 , 7.7083 175 , 0 0337. (46b)
 T T
d d
oper3
, ,W q W q p    
      
      T T
6.25 0 , 6.25 0 , 0 0512. (46c)
Now, we choose oper2 in (46b) as the new equilib-
rium point for the state transformation. That is,
0
175*
q q  , 0
7.7083*
W W  , 0
0 0337*
p p .  and
0
R 0 2467. . Then, the coordinate new operating points
for (46) can be obtained from (6) as follows:
        T T
d oper1
, , 3.3 400 , 3.3 400 , 0.01725x x u  
  (47a)
        T T
d oper2
, , 0 0 , 0 0 , 0* * *
x x u  (47b)
 d oper3
, ,x x u  
      T T
1.458 175 , 1.458 175 , 0.01754     (47c)
Note that  d oper2
, ,* * *
x x u is the new equilibrium point
for state variables. Applying the small signal lineariza-
tion method to linearize nonlinear dynamic system (8)
for three operating points described in (47), one can ob-
tain three coordinate linear subsystems. Define the
membership functions for state variable  2 tx as Fig. 2.
Then, the time-delay affine T-S fuzzy model can be ob-
tained by blending the three linear subsystems as fol-
lows:
12M32M 22M
0175  2 tx55 400
Figure 2. Membership functions of  2 tx .

Rule 1: IF  2 tx is 11M THEN
        1 1 1dt t t tx x x     A A A
   1 1 1tu   B a a (48a)
        2 2 2dt t t tx x x     A A A
   2 2 2tu   B a a (48b)
        3 3 3dt t t tx x x     A A A
   3 3 3tu   B a a (48c)
where 1
-0.2563 -0.0021
339.6226 -2.8302
 
  
 
A ,
2
-0.5259 -0.0044
486.4865 -4.0541
 
  
 
A , 3
-0.8 -0.0067
600 -5
 
  
 
A ,
     1
0 0
2.83 N t -120 -0.0235 N t -120
 
   
   
A
 
 
N t -120
0
40 0 0 040
0 40 2.83 -0.0235N t -120
0
40
 
             
 
 
,
     2
0 0
4.05 N t -120 -0.0338 N t -120
 
   
   
A
 
 
N t -120
0
40 0 0 040
0 40 4.05 -0.0338N t -120
0
40
 
             
 
 
,
     3
0 0
5 N t -120 -0.0338 N t -120
 
   
   
A
 
 
N t -120
0
40 0 0 040
0 40 5 -0.0338N t -120
0
40
 
             
 
 
,
1d
-0.2563 0.0021
0 0
 
  
 
A , 2d
0 5259 0 0044
0 0
. . 
  
 
A ,
3d
-0.8 0.0067
0 0
 
  
 
A ,
1
-172.526
0
 
  
 
B , 2
120 4427
0
. 
  
 
B , 3
-97.6562
0
 
  
 
B ,
1
-0.8083
0
 
  
 
a , 2
0
0
 
  
 
a , 3
-2.4992
0
 
  
 
a ,
  1 2 3
0
31.25 N t -120
 
       
  
a a a
 
 
N t -120
0
40 0 040
0 40 31.25N t -120
0
40
 
             
 
 
and  N t -120
1
40
 with  80 N t 160  .
According to the membership functions defined in Fig.
2, the S-procedure is presented as follows. For Rules 11,
i.e.,  2 t 5x  , the matrices of S-procedure are given as
follows:
111
0 0
0 0
 
  
 
T , 111
0
1
2
 
   
  
n and 111v 5 (49)
For Rules 33, i.e.,  2 t 5x   , the matrices of S-procedure
are given as follows:
331
0 0
0 0
 
  
 
T , 331
0
1
2
 
 
 
  
n and 331v 5 (50)
For Rules 12, i.e.,  25 t 400x  , the matrices of
S-procedure are given as follows:
121
0 0
0 1
 
  
 
T ,
 121
0
1
5 400
2
 
   
  
n and 121v 5 400  (51)
For Rules 23, i.e.,  2175 t 5x    , the matrices of
S-procedure are given as follows:
231
0 0
0 1
 
  
 
T ,
 231
0
1
175 5
2
 
    
  
n and    231v 175 5   
(52)
Note that the notation Rules ij means the correlation be-
tween Rule i and Rule j of the plant part bounding region.
The bounding region of S-procedure on the membership
function is shown in Fig. 2.
For the time-delay affine T-S fuzzy model (48), the
T-S fuzzy controller can be designed by applying Theo-
rem 1 and ILMI algorithm developed in Section 3. In
Step 1 of ILMI algorithm, the eigenvalues of  0
i i iA B F
are chosen as  1 0 . Then, the initial  0
iF can be ob-
tained by applying standard pole placement technique as
follows:
 (0)
1 0 0121 0 0001. . F , (53a)
 (0)
2 0 0297 0 0002. . F , (53b)
 (0)
3 0 0492 0 0003. . F (53c)
Let us assign the matrix ˆ Q I . Then, the matrix  0
P
can be obtained as follows by solving Riccati equation
(44).
(0) 0.0219 0.0075
0.0075 0.0059
 
  
 
P (54)
In this example, we can get a feasible solution after
112 iterations of the ILMI fuzzy controller design pro-
cedure. The final decay rate  is -0.3804 and the fea-
sible solutions are obtained as follows:

0.3741 0.4412
0.4412 0.6433
 
  
 
P , 0.3430 0.4045
0.4045 0.5897
 
  
 
S , (55a)
 1 27 9496 33 1886. .  F , (55b)
 2 88 7028 107 2815. .  F (55c)
 3 75 6525 92 8659. .  F (55d)
111 331 121 231ξ 390.6225, ξ 794.5773, ξ 1.4141, ξ 4.739    , (55e)
Thus, the T-S fuzzy controller has the following form:
     t -27.9496 -33.1886 tu x (56a)
     t -88.7028 -107.2815 tu x (56b)
     t -75.6525 -92.8659 tu x (56c)
In order to demonstrate the effectiveness and applica-
bility of proposed design methodology, the fuzzy con-
troller (56) is employed to control the TCP network sys-
tem (4) for the simulations. In the simulations, the num-
ber of TCP sessions  N t is assumed as a random inte-
ger function with range  80 N t 160  . The simulation
results of window size  W t and queue length  q t
are shown in Figs. 3-4. From the simulation results of
Figs. 3-4, one can find that the present designed fuzzy
controller (56) can drive the TCP network system (4) to
successfully achieve the desired equilibrium point
0
175q  and 0
7.7083W  . In general, the TCP flows are
varying in the practical network; hence, the T-S fuzzy
controller design approach proposed in this paper pro-
vides a better choice for the designers in superintending
the congestion control for the TCP networks.
Figure 3. The responses of window size  W t .
In order to compare the results of the proposed T-S
fuzzy controller with other AQM router design scheme,
a PI controller described in references [20-21] is applied.
The PI controller has a transfer function of the following
form.
  PI
s
1
z
C s K
s
 
 
  (57)
Figure 4. The responses of queue length  q t .
To meet the crossover condition  gL j 1  , we insist
that
g
0
PI g 2
1
j
R
K z
C
2N
 
  (58)
In [20-21], the authors set the unity gain crossover as
g 0 52.  rad/s, and set the phase margin to about 80
.
Thus, from (58) one can obtain 6
PIK 9 64 10. 
  . For ob-
taining a digital implementation, the z-domain transfer
function of the above PI controller is constructed as fol-
lows.
 
 
p z az b
q z z 1


 
(59)
where refq q q   with refq being the desired queue
length to which we want to regulate. The transfer func-
tion (57) can be converted into a difference equation of
the variables yielding, at time t kT , where sT 1/ f ,
         p kT a q kT b q k 1 T p k 1 T       (60)
In this simulation, we implemented the PI controller
with a sampling frequency of 160 Hz. The PI coeffi-
cients a and b that were implemented as 5
1 822 10. 

and 5
1 816 10. 
 , respectively. The refq for the PI con-
troller was chosen to be 175 packets. The responses of
the PI controller are shown in Figs. 5-6.
Figure 5. The responses of window size  W t with PI con-
troller [20-21].

Figure 6. The responses of queue length  q t with PI con-
troller [20-21].
Referring to Figs. 3-6, one can find that the responses
of the proposed robust fuzzy control approach are better
than PI control method introduced in [20-21].
5. Conclusions
Explosive growths in multi-media and end-to-end ap-
plications in the Internet have resulted in the traffic con-
gestion characterized by packet losses and delays. Re-
cently, considerable research has been undertaken on
AQM router. However the number of TCP sessions was
assumed to be constant in previous work. In this paper, a
perturbed T-S fuzzy model was employed to simulate the
behavior of varying numbers of session for nonlinear
TCP network systems. According to the perturbed T-S
fuzzy model, a robust fuzzy congestion controller design
for the TCP/AQM router has been developed in this pa-
per. The simulation results showed that the proposed
fuzzy controller has successfully provided robust per-
formance when the number of TCP sessions varies from
time to time.
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[2] K. Tanaka and H. O. Wang, Fuzzy Control Systems
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Wen-Jer Chang received the B.S. degree
from National Taiwan Ocean University,
Taiwan, R.O.C., in 1986. The Marine En-
gineering is his major course and the Elec-
tronic Engineering is his minor one. He
received the M.S. degree in the Institute of
Computer Science and Electronic Engi-
neering from the National Central Univer-
sity in 1990, and the Ph. D. degree from
the Institute of Electrical Engineering of the National Central
University in 1995. Since 1995, he has been with National Tai-
wan Ocean University, Keelung, Taiwan, R.O.C. He is currently
the Vice Dean of Academic Affairs, Director of Center for
Teaching and Learning and a full Professor of the Department
of Marine Engineering of National Taiwan Ocean University.
He is now a life member of the CIEE, CACS, CSFAT and
SNAME. Since 2003, Dr. Chang was listed in the Marquis
Who's Who in Science and Engineering. In 2003, he also won
the outstanding young control engineers award granted by the
Chinese Automation Control Society (CACS). In 2004, he won
the universal award of accomplishment granted by ABI of USA.
In 2005, he was selected as an excellent teacher of the National
Taiwan Ocean University. Dr. Chang has over 200 publications
including 98 journal papers. His recent research interests are
fuzzy control, robust control, performance constrained control.
Po-Hsun Chen was born on July 17,
1990 in Hsinchu, Taiwan, R.O.C.. He
received the B.S. degree in Marine En-
gineering from the National Taiwan
Ocean University. Now, he is a graduate
student of the Department of Marine
Engineering of the National Taiwan
Ocean University. His research interests
focus on fuzzy control, covariance con-
trol and robust control.
Cheng-Ting Yang was born on January
4, 1989 in New Taipei City, Taiwan,
R.O.C.. In June of 2011, she received
the B.S. degree in Marine Engineering
from the National Taiwan Ocean Uni-
versity, where she is working toward
the Master degree. Her research inter-
ests focus on fuzzy control and the ap-
plication of control for the TCP network
systems.

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