Congestion control in packet switched wide area networks using a feedback model

International Journal of Computer Science & Engineering Survey (IJCSES) Vol.4, No.5, October 2013

Congestion Control in Packet Switched Wide Area
Networks using a Feedback Model
Ogbimi, E.F.1, A.O. Oluwatope1, and G.A. Aderounmu1
1

Department of Computer Science and Engineering
Obafemi Awolowo University,
Ile-Ife, Osun State, Nigeria

ABSTRACT
In a complex world, where networking expands very rapidly, the network stability of flow of bandwidth
played a vital role in transmitting packets. Hence, it was imperative to find solution to the problem of
congestion especially in the follow of bandwidth stability.
Congestion in computer networking is caused by so many factors. Some of the signs are packet loss,
queuing delay resulting from overloading the buffer, faulty hardware devices, intermixing of old and new
technologies and unstable flow of bandwidth resulting from positive feedback
A negative feedback mechanism was added to the network. Feedback was developed using both control and
queuing theory. The congested signalling was tested for stability using control theory. Kendall notation was
used to determine both arrival and service rates with all the queue disciplines. Matlab was used to simulate
the model. Bode plots and impulse responses were used to analyse the performance of the improved model.
The results showed that flow of bandwidth was stable with increase in service rate of packets and decrease
in the probability of routers not accepting packets. This shows that adopting dynamic explicit feedback
model reduced congestion by minimal packet drops, balanced flow of bandwidth, converges to fairness and
minimized amplitude of oscillation.

KEYWORDS
Feedback, Packet-Switched, Wide Area Networks, Bandwidth Stability and Congestion control

1

INTRODUCTION

The rapid growth of WAN in recent years is part of the struggle for network administrators to
monitor the network for demand resources such as voice, video and data. Most users continue to
add more networking devices of higher capacities to satisfy the needs of network users. Yet,
network remains congested and application software continue to run slowly and experience poor
performance of networks. New breeds of software applications and information technology
drivers like academics, industrialists, and business tycoons contribute to this problem by
installing new software which may not be favourable to the networking environment [1].
Technology drivers, demand for bandwidth at the same time leads to the flow of bandwidth
instability, where the bandwidth demand for resources is higher than the capacity of the network.
In view of the various efforts of the network providers to sustain network performance,
congestion and poor performance are still common features of networks. Hence, there is need to
continue to conduct research on how to improve the performance of computer networks.

DOI : 10.5121/ijcses.2013.4501

1


Congestion is a fundamental communication problem that occurs in shared networks [2]. When
the network users collectively demand for more resources than the network is able to offer. In
typical packet switched networks, this occurs quite easily when the output links are slower than
the input and multiple traffic sources competing for same output link at the same time. Some of
the features of congestion are packet loss resulting from overflow of packets from the buffer,
faulty hardware devices which may dissatisfy the user and slow down the network; mismatching
of different speeds clogging the path with higher and lower speeds of network links and feedback
delay in transmitting packets from source to destination. When message is transmitted through a
link and passed through the feedback controlled loop, the network is said to be experiencing
positive feedback [2]. The output of the positive feedback fails when the input packet transmits
faster than the system can handle. The packets eventually get lost due to overcrowding and packet
delay. This affects flow of bandwidth stability and have negative implications for routing.
The objective of the paper is to develop a TCP control theoretic model for congestion control in
packet switched wide area networks.

2

LITERATURE REVIEW

2.1

Existing schemes for Congestion control in Packet Switched WAN

Several schemes of managing congestion have been used in the past. Among these are three
prominent feedback schemes, which are active networking also called active congestion control
technique, convention feedback technique and explicit feedback scheme. Active congestion
control scheme makes the feedback congestion control more responsive to the network [3]. In
active congestion control, congestion is detected at the router which also immediately begins
reacting to congestion by changing the traffic that has already entered the network. Active
networking with idea of reprogramming routers with data was used to address this shortcoming of
feedback control.
Conventional feedback schemes detect congestion when they receive notification from the
congested router or when they deduce the existence of congestion due to packet loss or excessive
delays. The congestion relief starts from the sender and moves to the congestion node as the
sender’s rate is reduced. At the congestion nodes, the congestion relief changes in state and
sustains that relief that propagates out to the end points[4].
In explicit Binary feedback scheme, a router that is congested sets a congestion indication bit in
the network layer header of a data packet that is flowing in a forward direction. When the data
reach the head of the acknowledgement packet, the acknowledgement packet is then transmitted
from the destination to the source. In this case, users are expected to adjust the window size[5].

2.2

Drawbacks in the Existing Schemes

The drawbacks of Binary Feedback scheme were: the protocol TCP/IP is not efficient enough to
transmit data of higher data rate between nodes and propagation delay of the network is very
large due to indicated congested data by a bit. The current protocol which is TCP was not
effective to handle high sophisticated bandwidth transmission [6]. The current protocol which is
TCP was not effective to handle high sophisticated bandwidth transmission. TCP used a binary
congestion signal. Oscillation occurred always at the packet level. The dynamics of the network
were unstable leading to severe oscillation can only be reduced by accurate estimation of packet
loss probability and a stable design of flow dynamics [6]. The drawback of conventional feedback
model is that the propagation delay in sending packet was too large and therefore congestion sets
2


in and increases. The drawback for active congestion control was that it was limited to servicetime distribution. The model also could not be expanded and it was not robust. The drawback of
explicit scheme was that it lacked scalability with respect to network bandwidth and there must
always be signals of congestion before the scheme is adopted. It is not flexible to all kinds of
traffic conditions. Delay increased in quantity along the congestion site and endpoints. The larger
the delay goes from one endpoint to another, the longer it took the packets to be transmitted.
Congestion then sets in. The higher the bandwidth, the larger the amount of data, the more
congested is the endpoints.
In this paper, a method called Dynamic Explicit Feedback Scheme was adopted to reduce
congestion in packet switched WAN.

2.3

The Improved Feedback scheme

The improved feedback scheme being proposed was called dynamic explicit feedback model
which kept the bandwidth stable. It reduced packet drops to its barest minimum. It reduced
bandwidth delay in packet switched WAN. It balanced the flow of transmission through paths. A
feedback mechanism was added to monitor the acknowledgement of packets from receiver to
sender.

3

METHODOLOGY

The main concepts adopted in the research were packet switching, feedback mechanism and
dynamic systems.

3.1

Model Development

Figure 1 shows the block diagram of the improved feedback model. The message is broken into
packets by means of packetization. Packets are transmitted from source to destination(s). A
message is transmitted to the first router which acted as a buffer. The message remains in the
buffer for a while until there is an available router that packets can be routed on the way to their
destinations. The packets are then distributed to the second, third and many other routers. A
feedback indicates that the packet had been sent to the receiver. If some of the packets were lost
or delayed in the queuing system, the feedback controller passes a feedback message to any of the
routers that there was congestion somewhere along the link. The router therefore adjusts the
speed of transmitting packets. The stability of flow of packets is then regulated.

3


Figure 1:Block Diagram for Improved Feedback Model for Managing congestion in Packet Switched Wide
Area Networks

3.2 Model Construction
Queuing model was applied to construct the model [7]. The parameters of the queuing model
were modified and applied to the situation of packet switched WAN. The queuing model for
packet flow in packet switched WAN was then modelled by the following modified parameters:
= rate of arrival of packets
µ = service rate (rate of transmitting packets)
Ls = Average number of users on the network
Lq = Average number of users requesting for web resources from a router in the
network (in the queue).
c = number of routers
ρ = network utilization
Po = Probability of routers that do not accept packets
∞

=

np

(1)

For a single server model (c =1), there is no limit on the maximum number of users in the
network. The model assumes a capacity source. Arrivals occur at rate user per unit time. Under
these conditions, ρ < 1 and < µ. If arrival rate is higher than the service rate, then the geometric
series will not converge and the steady-state probability will not exist. The queue length will
continually increase and no steady state will be reached in the network.
Letting ρ =

, the expression for pn in the generalised model then reduced to

4


Pn = ρn po

, n = 0, 1, 2, -------, n

(2)

To determine the value of po we use the identity
po = (1+ ρ + ρ2 + ----- ρn) = 1
(3)
Assuming ρ < 1,

) the Sum to infinity of

(

a series

where a = 1 and r = ρ

=
=

(4)

=1

For steady state probability
∞

= 1

(5)

= [

= (

]

)

= (1 − )

(6)

For multiple server models which is the case of interest, there are c parallel servers (routers). The
arrival rate is and the service rate per server is µ. There is no limit on the number of routers in
the network. With Kendall’s notation (K∣M∣C):(GD∣∞∣∞), K is the arrival distribution, M is the
service distribution and C is the number of servers in the network. The queue discipline is a
generalized distribution that satisfies all conditions [8]. The parameters considered were arrival
rate ( ), service rate (µ), number of servers (c), probability of number of routers not accepting
packets (Po). Ls and Lq are computed as:
If ρ =

μ and assuming

< 1, the value determined from

=1
Lq was determined in MIMO systems

If ρ =
Then

=

×

(n = 0, 1, 2....., n)

μ
×

×

×

(8)

5


Pn =
=

Lq =

* P0

!

n<c

* Po

!

ρc+1
2

(c−1)!(c−ρ)

n≥

* Po

(9)

Ls = Lq + ρ
Ls =

P0

(

)!(

= P0
(

)!(

+ρ

)

+ ρ (c − 1)! (c − ρ)

)

(10)

These are the measurements of performances in a multiple server and multiple queue in a packet
switched wide area networks.
The mathematical models for packet switched WAN were subjected to fluid flow analysis.
For the equation of number of users requesting for web resources,
Lq =

(

)!(

(9)

)

Assuming c = n for many servers to many number of users
L =

Pρ
(n − 1)! (n − ρ)

Which on further substitution
L

=

(

)!(

)

Network utilization = ρ =

L =

; insert ρ = λ μ in equation (11)

Pλ μ
(n − 1)! − 2λμn(n − 1)! + λ (n − 1)!)
μ n

U= P λ

μ

(11)

(12)

and V = μ n (n − 1)! − 2λμn(n − 1)! + λ (n − 1)!

For the equation of the number of users in the network:

6


L s = Po

(

)!(

)

+ ρ ( − )! ( − )

(10)

For many servers to many users
Assuming c = n

L =

(

(

L =

)!(

+ρ

)!(

)

)
(

=

(

+ρ

)!(

)

)!(

(13)

)

Substituting ρ = λ µ in equation (13) gives

P oλn+1μ2−n + λμ2n2(n−1)!−2λ2μn(n−1)!+ λ3(n−1)!

=

(14)

μ 3n2(n−1)!−2λμ2 n(n−1)! + λ2μ (n−1)!

λ

U=

+ λμ n (n − 1)! − 2λ μn(n − 1)! + λ (n − 1)!

μ

V = μ n (n − 1)! − 2λ μn(n − 1)! + λ μ(n − 1)!
The equations of both the number of network users requesting for web resources (Lq) and number
of network users in the system (Ls) were differentiated with respect to the arrival rate of packets
( ). Using the quotient rule of differentiation.
L
Pλ μ
μ n (n − 1)! − 2λμn(n − 1)! + λ (n − 1)!)

=

=

=

(

)

(

)!(

(

)

)!(

)

(15)

P oλn+1μ2−n + λμ2n2(n−1)!−2λ2μn(n−1)!+ λ3(n−1)!
μ 3n2(n−1)!−2λμ2 n(n−1)! + λ2μ (n−1)!

=

(

)

(

)!(

)

(

)!(

)

+

(16)

Substitute n = 4 for number of servers in the following equations (17) and (18)
When n = 4 for dLq

=

(
(

)
)

(17)

7


When n = 4 for dLs
(

=

)

(

(18)

)

The input and output functions of the differentiated equations with respect to the arrival rate of
packets are not linear and could not be used to find transfer functions. However, one of the
prominent processes for stability analysis in non-linear systems is linearization of equations [9].
The non linear equations were made linear by the process of linearization. Linearization assesses
the local stability of the equilibrium point of non linear equations of dynamic systems .
when y = 0,
Linearization of dLq and dLs for n = 4
y = f(x) + f '(x)(x – )

(19)

substitute f(λ) = 0 in equation (19)
y = f '(x)(x – )

y=

(
(

)

(x – )

)

(20)

when y = 0
x = , At equilibrium point
y = f(x) + f '(x)(x – x0)

(21)

when f(x) = 0
y = f '(x)(x – x0)
when x0 = 0
y = f '(x)(x)
y=

y=

(
(

)
)

(
(

x

)
)

(22)

(x – ), which implies that

x = λ at equilibrium point

y = f(x) + f '(x)(x – x0)

(21)

when f(x) = 0

8


y = f '(x)(x – x0)
when x0 = 0
y = f '(x)(x)
(

y=

)

(

(x)

)

(23)

The Laplace’s transforms for the differentiated linear equation of the number of network users
requesting for web resources and number of users in the system were computed [10]. The
transfer functions of equation was computed from the two Laplace’s transforms of number of
servers n = 4.

Laplace’s transform for
ℒ

=

( ) − ( 0)
(

Y(s) =

)

(

s

)
(

SY(s) =

ℒ

for n = 4

)

(
(
(

=

)
)

)

(24)

Laplace’s transform for
ℒ

=

Y(s) =
SY(s) =

ℒ

=

for n = 4

( ) − (0)
(

)

(

)
(

)

(
(

)

(

)

)

(25)

Transfer function for n = 4
ℒ
ℒ

=
=

(
(

)
(

(

)
(

)

)
)

(

)

(26)

9


3.3 Analytical validation of Bode Plots and Impulse responses using matlab
Two types of graph of Bode plots were produced from the transfer functions; one is the Bode plot
and the other is impulse response. Bode plots has graph has two parts, one part measures
magnitude in decibels and the other part is the phase angle in degrees which are both y-axis and
both have frequency in Hertz as x-axis.
Based on the aforementioned simulations, each flow of bandwidth scenario was simulated twice
for consistency. One set used probability of number of routers not accepting packets with 10
variations of service rate of packets. The service rate of packets was varied 10 times per set of
data. The service rate of packets were in multiples of 100Mb/s which is from 100 to 1000 Mb/s
while the probability of routers not receiving packets were in multiples of 0.10. The simulation
set was done using the same set of service rate values and taking value of probability of number
of routers not accepting packets of 0.1.
Cut-off frequencies, magnitude, phase angles were produced from the Bode plots while amplitude
at 0.0001 seconds was produced from impulse responses.
For n = 4
Table 1: Results from Bode plots and impulse responses of the service rate of packets received with the
probability of no of routers not accepting packets (Po) of 0.10 for 4 servers.

Service rate
(Mb/s)

Magnitude
(dB)

Cut off
Frequency
(Hz)

Phase angle
(°)

Amplitude at t = 0.0001
secs

100
200
300
400

2.35
2.35
2.35
2.35

12.00
25.00
38.00
50.00

74.60
74.60
74.60
74.60

-19.40
-75.70
-166.00
-286.00

500
600
700
800
900
1000

2.35
2.35
2.35
2.35
2.35
2.35

62.00
75.00
87.00
102.00
109.00
120.00

74.60
74.60
74.60
74.60
74.60
74.60

-433.00
-607.00
-796.00
-1000.00
-1230.00
-1460.00

Table 2: Results from Bode plots and impulse responses of the probability of routers not accepting packets
with the service rate of 1000 Mb/s for 4 servers.

Probability
of routers
not
accepting
packets
0.10
0.20
0.30
0.40

Magnitude
(dB)
2.350
1.420
1.050
0.812

Cut off
Frequency (Hz)
120.00
4.00
7.00
4.00

Phase angle
(°)
74.60
62.50
56.30
52.30

Amplitude at t = 0.0001
secs
-1460.00
-744.00
-498.00
-375.00
10


0.50

0.659

4.00

49.40

-300.00

0.60
0.70

0.566
0.495

2.00
8.00

47.10
45.10

-251.00
-215.00

0.80
0.90
1.00

0.440
0.396
0.360

8.00
6.00
3.00

43.80
42.50
41.40

-188.00
-168.00
-150.00

4 RESULTS AND DISCUSSIONS
4.1 Introduction
This section presents the findings from tables 1 and 2 the investigations carried out in to achieve
the objectives of the paper. The order of presentation is the experiment, data, graphical analysis
and discussion of the results. The software used was Matlab. The purpose of using MatLab 7.0
was to compute transfer functions and plotting Bode plots arising from the Laplace transforms of
the equation of both the number of users assessing the web and the number of network users in
the system.
2.5

Magnitude(dB)

2
1.5
1

Mag (dB)

0.5
0
0

200

400

600

800

1000

1200

Service rate of packets (Mb/s)
Figure 2: Service rate of packets against Magnitude

Figure 2 produced from table 1 shows the graph of the service rate of packets against magnitude.
The graph produced a uniform plot. The magnitude of flow of bandwidth was uniform at 2.35dB
with increasing rate of packets from 100 Mb/s to 1000 Mb/s. The higher the service rate of
packets, the uniform the magnitude becomes. When the magnitude of the flow of bandwidth is
uniform, the flow of bandwidth is stable and the feedback becomes responsive to the network.

11

140

Cut-off Frequency (Hz)

120
100
80
60

Cut off

40
20
0
0

200

400

600

800

1000

1200

Service rate of packets (Mb/s)
Figure 3: Graph showing service rate of packets against cut-off frequency

Figure 3 produced from table 1 shows the graph of cut-off frequency against cut-off frequency.
The cut off frequency produced a positive slope which rose from 12 Hz to 120 Hz. The higher the
service rate of packets, the higher the cut off frequency of the flow of bandwidth. When the cut
off frequency of flow of bandwidth is high, the flow of bandwidth is stable and the feedback
becomes active to the network.
80

Phase angle (°)

70
60
50
40
30

Pha ang(°)

20
10
0
0

200

400

600

800

1000

1200

Service Rate of Packets (Mb/s)
Figure 4: Graph showing service rate of packets against phase angle

12


Figure 4 produced from table 1 shows the graph of service rate of packets against phase angle.
The phase angle was uniform at 74.6° increasing service rate of packets from 100 Mb/s to 1000
Mb/s. The higher the service rate of packets, the uniform the phase angle becomes. When the
phase angle of stability is uniform, the flow of bandwidth is stable, the feedback becomes active
to the network.

Ampltude at 0.0001 secs

0
-200 0

200

400

600

800

1000

1200

-400
-600
-800
-1000

Amp at
0.0001

-1200
-1400
-1600

Service Rate of Packets (Mb/s)

Figure 5: Graph showing service rate of packets against Amplitude at 0.0001 seconds

Figure 5 produced from table 1 shows the graph of service rate of packets against phase angle.
The amplitude at 0.0001 seconds produced a negative curve which fell from -14.60 to -1460.00.
The higher the service rate of packets, the lower the amplitude at 0.0001 seconds. When the
amplitude of the flow of bandwidth is low, the flow of bandwidth is stable and the feedback
becomes responsive to the network.

Magnitude (dB)

2.5
2
1.5
1

Mag (dB)

0.5
0
0

0.2

0.4

0.6

0.8

1

1.2

Probability of number of routers not accepting packets
Figure 6: Graph showing the probability of the number of routers not accepting packets against magnitude
13


Figure 6 produced from table 2 shows the graph of the probability of number of routers not
accepting packets against magnitude. The magnitude of the flow of bandwidth produced a
negative curve which fell from 2.35 dB to 0.396 dB. The higher the probability of number of
routers not accepting packets, the lower the magnitude. When the magnitude of the flow of
bandwidth is low, the flow of bandwidth is stable and feedback model becomes responsive to the
network.
140

Cut off Frequency (Hz)

120
100
80
60

Cut off

40
20
0
-20

0

0.2

0.4

0.6

0.8

1

1.2

Probability of Number of routers not accepting packets
Figure 7: Graph showing the probability of the number of routers not accepting packets against cut-off
frequency

Figure 7 produced from table 2 shows the graph of the probability of number of routers not
accepting packets against cut-off frequency. The cut off frequency fell from 120 Hz to 4 Hz, rose
to 7 Hz, fell to 4 Hz, became uniform and fell further to 2 Hz, rose to 8 Hz and finally fell to 3
Hz. The lower the probability of number of packets, the higher the cut-off frequency of the flow
of bandwidth. When the cut off frequency of the flow of bandwidth is high, the flow of bandwidth
is stable and the feedback becomes responsive to the network.

14

80

Phase Angle (°)

70
60
50
40
30

Phase (°)

20
10
0
0

0.2

0.4

0.6

0.8

1

1.2

Probability of number of routrs not accepting packets
Figure 8: Probability of the number of routers not accepting packets against phase angle

Figure 8 produced from table 2 shows the graph of probability of number of routers not accepting
packets against phase angle. The phase angle of the flow of bandwidth produced a negative curve
which fell from 74.6° to 41.4°. The higher the probability of the number of routers not accepting
packets, the lower the phase angle of stability. When the phase angle of stability is high, the flow
of bandwidth is stable and the feedback becomes responsive to the network.

Amplitude at 0.0001 secs

0
-200 0

0.2

0.4

0.6

0.8

1

1.2

-400
-600
-800
-1000
-1200
-1400
-1600

Probability of Number of routers not accepting packets

Figure 9: Graph showing the probability of the number of routers not accepting packets against amplitude
at 0.0001 seconds

Figure 9 produced from table 2 shows the graph of probability of number of routers not accepting
packets against amplitude at 0.0001 seconds. The amplitude of the flow of bandwidth produced a
positive curve which rose from -1460 to -150. The higher the probability of number of routers not
15


accepting packets, the higher the amplitude at 0.0001 seconds i.e. the lower the probability of
number of routers not accepting packets, the lower the amplitude. When the amplitude of the flow
of bandwidth is low, the flow of bandwidth is stable, the feedback becomes responsive to the
network.

5 CONCLUSION
In this paper, a scheme was proposed for improved feedback model for reducing congestion using
TCP protocols at the network layer. The scheme used variable amount of feedback from the
network, with a router to indicate congestion. Each network server that is congested sets the
router and sends message to the source by the user. The message goes back to the source from the
destination which receives packets. The network was modelled as a negative feedback control
system and identified various component in the scheme in the model. The policies that needed to
be used in each of these components were studied through analysis and simulation.
The network servers detect their state as being congested and sets the congestion indication router
when the time to deliver packets the receiver is longer than expected. The arrival rates and
service rate of packets were described. The users receive their packets and determine the flow of
bandwidth stability. The probability of routers accepting packets too also determine the stability
of flow of bandwidth and it is between 0 and 1.
It was also demonstrated that the scheme is distributed, adapts to any state of the network,
converges to fairness, minimizes packet loss and amplitude of oscillation.
The performance of the scheme was addressed under transient changes in the network. The
scheme operates the network at a stable point when the network is overloaded even, when the
users try to cause bottleneck in the path

ACKNOWLEDGEMENTS
The authors would like to thank my parents and my lecturers and colleagues for contributing this
paper in all forms of the other for publication.

REFERENCES
[1]
[2]
[3]

[4]
[5]
[6]
[7]

Hicks, M., (2004), Managing Gigabit Networks, Applications and Services – Compuware; Questnet
2204 Conference Networking Far and Wide Held at Cairns International Hotel, Cairns, Austrailia.
Patternson, D.A., (2001), Network, Spring Lecture Slides; Department of Computer Science,
University of California, Berkeley, U.S.A.
Janarthanan, Y., G. Minden, and J. Evans (2003), Enhancement of Feedback Congestion Control
Mechanisms by Deploying Active Congestion Control; The University of Kansas Centre for
Research, Inc., 2235 Irving Hill Road, Lawrence, KS 66044-7612. U.S.A.
Mamatas L., T. Harks, and V. Tsauossidis, (2007), Approaches to Network Congestion in Packet
Switched Networks; Journal of Internet Engineering, Volume 1, Number 1.
Ramakrishnan, K.K. and R. Jain., (1990), A Binary Feedback Scheme for Congestion Avoidance in
Computer Network; ACM Transactions in Computer Systems, Vol, 8. No. 2. Pages 158-181.
Katabi, D., M. Handley and C. Rohrs, (2002). Congestion Control for High Bandwidth delay Product
Networks, ACM Special Interest Group on Communication, Pittsburgh Supercomputing Center,
U.S.A.
Shor, M.H., Li Kung, J. Walpole, D.C., Steere, Pul Calton., (2003), Application of Control Theory to
Modelling and Analysis of Computer Systems; Department of Electrical and Computer Engineering,
Oregon State University, Corvallis, U.S.A.
16

[8]

Taha, H.A., (2008), Operation Research: An introduction, Eight Edition, Upper Saddle River, New
Jersey, U.S.A. page 538.
[9] http://en.wikipedia.org/linearisation accessed 19th March 2009.
[10] http://en.wikipedia.org/nonlinear_control accessed 19th March 2009.
[11] www.sosmath.com/Linearization_Technique.
[12] Jain, R., K.K. Ramakrishnan, and D. Chiu (1998), Congestion Avoidance with Computer Networks
with a Connectionless Network Layer Part I: Concepts, Goals and Methodology; Digital Equipment
Corporation, 550 King Street (LKG 1-2/A19) Littleton, MA 01460.
[13] http://en.wikipedia.org/wiki/feedback accessed 7th July 2008.
[14] Jain, R., (1998), Congestion Management in High Speed Networks, Issues and Trends, IEEE
Magazine, pp 24-30.
[15] Dorf, R.C. and R.H. Bishop, (2004), Modern Control Systems; Pearson Education, Singapore.
[16] Ghosh, S., (2004), Control Systems: Theory and Applications; Pearson Education, Singapore.
[17] www.nationmaster.com/queuing_theory.
[18] Zabczyk, J., (1992), Mathematical Control Theory: An Introduction; Springer Series, Institute of
Mathematics, Polish Academy of Science, Sniadeckich, office 512, 00-956 Warsaw, Poland.
[19] Bosco, H.L. and D.C. Dowden, (2000), Evolution of Wide Area Networks, Bellab Technical Journals,
5:46 – 72.
[20] Bose, S.J., (2002) Chapter 1 – An Introduction to Queuing systems: Kluwer/Plenum Publishers.
[21] Che, X., D. Hunter, and I. Henning, (2008), Congestion Control over Multiple Time Scales for
Optical Packet Switched Networks; Department of Electronic systems Engineering, University of
Essex, CO43SQ UK.
[22] Constantinescu, D., D. Erman, D. Ilie, and A. Popescu, (2007). Congestion and Error Control In
Overlay networks, Research Report 2007:01, Department of Telecommunications System, School of
Engineering, Blekinge Institute of Technology, S-371 79 Kariskrona, Sweeden.
[23] Fang, Lu, (2006), ATM Congestion Control, http://www.cse.ohio-state.edu/~jaon/cis78895/ftp/tm_cong/index.htm assessed on 13th February, 2006.
[24] Gevros, P., J. Crowcroft, P. Kirstein and S. Bhantti., (2001), Congestion Control Mechanisms and the
Best Effort Service Model; University of London, London.
[25] Hellstein, J. and S. Parekh., (2008), An Introduction to Control Theory with Applications to Computer
Science, IBM; T.J. Watson Research Centre.
[26] http://en.wikipedia.org/packet_switcing accessed 26th May 2010.
[27] http://dsd.lbl.gov/TCP-turing accessed 3rd April 2009.
[28] Jain, R., (2006), Congestion Management in High Speed Networks, Viewed from
(http://www.cse.ohio.state.edu/~jain/cis788-95/fp/atm_cong/index.htm) accessed 13th February,
2006.
[29] Mahajan R., and S. Floyd, (2001), RED-with Preferential Dropping (RED-PD); University of
Washington , AT&T Centre for Internet Research at ICSI (ACIRI), U.S.A.
[30] Pang, A. (2003). Course Content, Spring 2011, Department of Computer Science and Information
Engineering, Graduate Institute of Networking and Multimedia, National Taiwan University, Tapei,
Taiwan.
[31] Penittinen A, (2003), Queuing systems, Lecture Notes: S-38. 145- Introduction to Teletraffic Theory.
[32] Tijms, H.C.,(2003), Algorithmic Analysis of Queues, Chapter 9 in A First Course in Stochastic
Models, Wiley, Chichester.
[33] Williamson C., (2001), Congestion Control and Traffic Management in High Speed Networks;
Department of Computer Science, University of Calgary, 2500 University drive, New Calgary, AB,
Canada T2N, 1N-1.
[34] Xia, Y., L. Subramania, I. Stoica and S. Kalyanaraman (2005). One More Bit is Enough, In
Proceedings of ACM SIGCOMM, Philadelphia, U.S.A.
[35] Yang, R.X., S.L. Lam., (2000), General AIMD Congestion Control; Department of Computer
Sciences, The University of Texas, Austin, Austin, U.S.A.

17


Authors
Ogbimi, E.F. is a staff of the Department of Computer Engineering of The
Polytechnic, Ile-Ife. He obtained a Bachelor of Science (B.Sc) degree in Computer
Technology from Babcock University, Ilishan Remo. He also obtained a Masters of
Science (M.Sc) degree in Computer Science from Obafemi Awolowo University, IleIfe. He worked in the area of Computer Communication Systems with exphasis on
bandwidth stability.
A.O. Oluwatope is a Senior Lecturer in the Department of Computer science and Engineering of the
Obafemi Awolowo University, Ile-Ife. He Obtained a Bachelor of Science (B.Sc) degree, Master of
Science (M.Sc) degree and Doctor of Philosophy all in Computr Science and Engineering from Obafemi
Awolowo University, Ile-Ife. His area of research includes design, analysis, modelling and simulation of
wireless ATM networks with respect to Qos management.
G.A. Aderounmu is a Professor in the Department of Computer Science and Engineering and Director of
the Information Technology and Communications Unit of the Obafemi Awolowo University, Ile-Ife. He
Obtained a Bachelor of Science (B.Sc) degree, Master of Science (M.Sc) degree and Doctor of Philosophy
all in Computer Science and Engineering from Obafemi Awolowo University, Ile-Ife. Aderounmu teaches
several courses including digital computer network, digital computer systems, computer graphics and
programming language. Area of specialisation is design, analysis and simulation of wireless ATM networks
with respect to switching protocol, error control and buffer management.

18

Congestion control in packet switched wide area networks using a feedback model

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Congestion control in packet switched wide area networks using a feedback model