Content uploaded by Aresh Dadlani
Author content
All content in this area was uploaded by Aresh Dadlani on Dec 26, 2017
Content may be subject to copyright.
Adaptive Generalized Minimum Variance Congestion
Controller for Dynamic TCP/AQM Networks
R. Barzaminia,b,∗, M. Shafieea, A. Dadlanib
aAmirkabir University of Technology, Tehran, Iran
bGwangju Institute of Science and Technology, Republic of Korea
Abstract
Generic generalized minimum variance-based (GMV) controllers have been
adopted as efficient control mechanisms especially in presence of measure-
ment noise. However, such controllers exhibit degraded performance with
change in process dynamics. To overcome this problem, a novel congestion
controller based on active queue management (AQM) strategy for dynami-
cally varying TCP/AQM networks known as adaptive generalized minimum
variance (AGMV) is proposed. AGMV is the combination of the real-time
parameter estimation and GMV. The performance of the proposed scheme is
evaluated and compared with its adaptive minimum variance (AMV) coun-
terpart under two distinct scenarios: TCP network with unknown parameters
and TCP network with time varying parameters. Simulation results indicate
that, in either case, AGMV is able to keep the queue length around the de-
sired point. In addition, the superior performance of the proposed controller
has been shown with regard to the PI controller, which is well-known in the
AQM domain.
Keywords: Transmission control protocol (TCP), Congestion control,
Active queue management (AQM), Adaptive generalized minimum variance
(AGMV)
∗Corresponding author. Tel.: +82-62-7153123; Fax: +82-62-7152274.
Email addresses: barzamini@gist.ac.kr (R. Barzamini), mshafiee@aut.ac.ir
(M. Shafiee), dadlani@gist.ac.kr (A. Dadlani)
Preprint submitted to Computer Communications June 26, 2011
1. Introduction
In recent years, with the rapid growth of throughput-demanding applica-
tions, congestion control has emerged as a major issue in computer and com-
munication network design. Congestion of packets at the outgoing queues
in routers results in poor performance and low reliability of the network.
Several congestion control approaches have been proposed that are mainly
of intuitive nature [1]. In addition to these approaches, the application of
control theory to solve congestion problem has been considered since late
90’s [2]. In this approach, the main idea is to use the available tools to de-
sign, engineer and analyze suitable congestion controllers for communication
networks as a closed loop system [3].
Generally, to solve this problem with a systematic approach using control
theory, closed loop data transfer processing structure in computer networks
can be considered. So, active queue management (AQM) is designed as a
congestion controller to be implemented in network routers, while the rest
of the components are altogether defined as a plant [3]. AQM algorithms
run on routers and detect incipient congestion by typically monitoring the
instantaneous or average queue size. When the average queue size exceeds
a certain threshold, AQM algorithms infer congestion on the link and notify
the end systems to speed down their transmissions by proactively dropping
some of the packets arriving at a router or by marking the packets. End
systems that experience the marked or dropped packets reduce their trans-
mission rates to relieve congestion and prevent the queue from overflowing.
With AQM, congestion is prevented before it actually occurs. Thus, the de-
ployment of AQM could lead to a high throughput, reduced packet loss and
low queuing delay network [4][5].
By introducing mathematical models for congestion control process in
transmission control protocol (TCP)/AQM networks, control theory-based
approaches are used either to analyze or to design the AQM schemes. Based
on these models, several conventional controllers such as P, PI [6], PD [7], PID
[8], sliding mode [9], coefficient diagram method (CDM) [10], and adaptive
schemes [11] have been designed as AQM methods in TCP networks.
Each of these controllers, however, has its own limitations and disadvan-
tages. For instance, if a PI controller is used, deviation of network param-
eters from nominal values would result in network fluctuation. PID con-
trollers are very sensitive to system parameter variations [8]. Therefore, a
new controller design with better performance and stability seems necessary.
2
q(t)
!
q
ref
AQM
TCP
Delay Buffer
( )
TCP Window
( )
( )
( )
(t)
Controller
Parameter
( )
(t)
Figure 1: The feedback control system for TCP/AQM.
Among the conventional AQM algorithms, random early detection (RED)
is another well-known approach that applies queue-based management by
randomizing packet dropping [7]. In [9], the authors have designed a self-
regulating AQM controller and have compared its efficiency with RED and
PI controllers. Nevertheless, it is known that self-regulatory control performs
weakly in the presence of noise (measurement noise and error modeling in
the system model). To overcome this problem, the minimum variance con-
trol strategy is presented in [10]. Generalized minimum variance (GMV)
controllers are designed to nullify the effect of noise through output variance
minimization. Simulation results indicate that minimum variance controller
outperforms the other proposed controllers (RED and PI) in subduing the
congestion problem [12].
The GMV approach is a suitable control method especially in the pres-
ence of measurement noise; however, in the presence of process dynamics
changes, the performance of the controller degrades significantly [13]. There-
fore, in order to fill this gap, we present a novel adaptive generalized minimum
variance (AGMV) control method as an AQM controller. Results obtained
through simulation justify that the proposed method further improves the
performance of the GMV controller.
The rest of the paper is structured as follows. A brief introduction on
TCP/AQM congestion control is presented in Section II. A dynamic model of
TCP/AQM in congestion avoidance phase is discussed in Section III. Section
IV investigates the design of our proposed AGMV controller for congestion
control. Section V presents numerical and parametric results on the effec-
tiveness of the proposed method. Finally, Section VI concludes the paper.
2. TCP/AQM Congestion-Avoidance Flow Control
In this approach, the TCP/AQM flow dynamics are modeled and analyzed
in terms of feedback control theory. AQM algorithms are designed to reduce
the response time (short-term performance) and to improve the stability and
3
P
d
(t)
AQM
TCP Window
Dynamics
u
d
(t)
l(t)
Queue
Dynamics
q(t)
(t)
Controller
Parameter
( )
(t)
Figure 2: Block diagram of a TCP/AQM system.
robustness (the long-term performance) of TCP/AQM congestion control.
This is usually obtained by regulating the queue around a desired value
[3]. As shown in Figure 1, TCP congestion control dynamics with an AQM
algorithm can be modeled as a feedback control system. In this model, the
feedback control system comprises of:
1. A desired queue length at a router (i.e. reference input) denoted by qref .
2. The queue length in a router as a plant variable (i.e. a controlled variable)
denoted by q.
3. A plant which represents a combination of sub-systems such as TCP
sources, routers, and TCP receivers which send, receive, and process TCP
packets, respectively.
4. An AQM controller that controls the packet arrival rate of the router
queue by generating the probability of packet drop, Pd, as a controlling
signal.
5. A feedback signal as the sampled system output (i.e. queue length) is
used to determine the error term, qref −q[3].
Several mathematical models of AQM schemes supporting TCP flows
in communication networks have been reported in the literature [8][14][15].
These control theory-based models can be used to analyze or design AQM
schemes. The authors of [6] have derived a delayed differential equation
model of TCP congestion avoidance mode using fluid-flow and stochastic
differential equation analysis. A simplified version of this approach is pre-
sented in [16]. The model consists of some delayed differential equations and
is developed based on the fluid-flow and stochastic differential equation anal-
ysis. The model is then modified in [17] to consider the effect of unresponsive
4
flows. Figure 2 shows the block diagram of the model for a single bottleneck
queue fed by long-lived, homogeneous TCP connections, l(t), and some un-
responsive flows, ud(t). Note that it is also possible to have unresponsive
flows which are mainly composed of short-lived TCP flows, Markov on-off
UDP flows, and traffic with long-range dependencies. In [3], a few models
for these types of traffic have been derived. However, the diagram in Figure
2 highlights the fact that the long-lived flows, l(t), are under direct AQM
control while unresponsive flows are treated indirectly.
3. System Model for TCP Dynamic Queue Behavior
In this section, we overview a system model for TCP and queue dynam-
ics based on non-linear differential equations. The TCP dynamic model is
then, represented by a linearized fluid model under the congested network
condition. A stochastic differential equation that describes a sample path of
each long-lived TCP connection (implementing an additive increase and mul-
tiplicative decrease (AIMD) strategy) has been derived in [21] and is given
as:
dWl(t) = dt
R(q(t)) −Wl
2dη(t),(1)
where Wl(t) is the congestion window size, R=q/C+Tpis the round trip time
(in seconds), and ηis th packet loss modeled by Poisson process. Over here,
q,C, and Tpdenote the queue length (in packets), link capacity (in packets
per second), and propagation delay (in seconds), respectively. Suppose that
the number of long-lived TCP connections (N) is large. Then, the aggregate
rate of all long-lived TCP connections is N·¯
Wl(t)/R(q(t)), where ¯
Wl(t) is
the average window size of the TCP connections. This model shows the
dynamic behavior of the TCP window size and the queue length of a router
with stochastic non-linear differential equations which can be represented by
equations (2) and (4) as given below:
¯
Wl(t)
dt =1
R(q(t)) −¯
Wl(t)
2
¯
Wl(t−R0)
R(q(t−R0))Pd(t−R0),(2)
where Pdis the probability of packet drop. The sample path of the bottleneck
queue length is [11]:
dq
dt =−C+N(t)
R(q(t)) ¯
Wl(t) + ud(t).(3)
5
If we are only interested in the averaged behavior of the queue on a time
scale coarser than that of ud(t) variations, we can replace ud(t) in equation
(3) by its mean, ud0. Hence,
d¯q
dt =−(C−ud0) + N(t)
R(q(t)) ¯
Wl(t),(4)
where the link bandwidth is reduced from Cto Cef f =C−ud0. The
equilibrium point (Wl0, Pd0, q0, ud0) of equations (2) and (4) is defined by:
W2
l0Pd0= 2; Wl0=R0Ceff
N;R0=q0
C+Tp.(5)
Linearizing the model in this equilibrium results in the following:
δWl(s) = −Pwin (s)e−sR0δPd(s),
δl(s) = N
R0
δWl(s),(6)
δq(s) = Pque (s)(δl(s) + δud(s)),
where δWl, δl, δq, δPdare the corresponding perturbations. The window and
the queue transfer functions in equation (6) are defined as:
Pwin(s) = C2
ef f R0/2N2
s+ 2N/R2
0Ceff
;Pque(s) = 1
s+Cef f /CR0
.(7)
Therefore, the small signal transfer function of the system can be repre-
sented by:
Pdc =δq(s)
δPd(s)=N
R0
Pwin(s)Pq ue(s).(8)
Note that, in order to design the controller, the effect of the transfer delay
can be neglected assuming that R0>> N C/C2
eff [17]. In this model, since
Ceff =C−ud0, UDP flows have been taken into account as well. Now,
we present the continuous system model and then derive the discrete model
parametrically.
Using equation (8) and assuming R0>> N C/C2
eff (which allows us to
ignore the delay), the open loop transfer function of linearized system can
be written as follows:
Pdc(s) = δq(s)
δPd(s)·C2/2N
(s+ 2N/R2
0Cef f )(s+ 1/R0).(9)
6
Here, the effect of ud0over bandwidth reduction is ignored due to the
assumption that there is no previous information about the existence in the
designing, and nature of non-responsive traffic. Discretizing equation (9)
with sampling period Tsusing the zero order hold (zoh) method [18], the
equivalent discrete system model of the linearized TCP/AQM is obtained as:
Pdc(z) = m1z−1+m2z−2
1 + n1z−1+n2z−2,(10)
in which n1,n2,m1, and m2depend on Tsand the system parameters. Thus,
we have:
m1=e−Ts
R0R3
0C4
2N(−2N+R0C2)+e−2NTs
R2
0C2−R4
0C6
4N2(−2N+R0C2)+
+R3
0C4
4N2,(11)
m2=e−Ts
R0−R4
0C6
4N2(−2N+R0C2)+e
−2NTs
R2
0C2R3
0C4
2N(−2N+R0C2)+
+e
−Ts
R0e
−2NTs
R2
0C2R3
0C4
4N2,(12)
n1=−e
−Ts
R0+e
−2NTs
R2
0C2,(13)
n2=e
−Ts
R0e
−2NTs
R2
0C2.(14)
4. Adaptive Generalized Minimum Variance Controller
In this section, we present the proposed controller. First, we describe the
GMV controller taken into consideration. Then, we introduce our proposed
adaptive GMV controller.
4.1. The GMV Controller
Now, the system equation with input u(k), output y(k) and the system
noise e(k) can be written as:
A(z−1)y(k) = z−dB(z−1)u(k) + e(k),(15)
7
where drepresents the system delay, A(z−1) is a single polynomial of order
nwith coefficients ai, and B(z−1) is a general polynomial of order mwith
coefficients bi. Moreover, A(z−1) and B(z−1) are assumed to be known and
represented by:
A(z−1) = 1 + a1z−1+a2z−2+···+anz−n(16)
B(z−1) = 1 + b1z−1+b2z−2+···+bnz−n.(17)
The main purpose of designing the generalized minimum variance con-
troller is to minimize the following performance index [19]:
JGMV =E{(P(z−1)y(k+d)−R0yr(k) + q0u(k))},(18)
where y,uand yrare the output, control, and input reference signals, re-
spectively. R0,q0, and P(z−1) are the design parameters which should be
determined by the user. GMV controller has been designed to overcome
the noise effects by minimizing the variance of the output [19]. Design of
the controller for a minimum phase system requires solving the following
Diophantine equation [19]:
A(z−1)E(z−1) + z−dF(z−1) = 1,(19)
where E(z−1) and F(z−1), the controller transfer functions, are as follows:
E(z−1) = e0+e1z−2+···+ed−1z−(d−1) (20)
F(z−1) = f0+f1z−2+···+fn−1z−(n−1) .(21)
In this controller, the control signal can be obtained from the following
closed loop transfer function:
u(k) = R0yr(k)−F(z−1)y(k)
q0+E(z−1)B(z−1).(22)
The following lemma provides a necessary and sufficient condition for the
stability of this GMV-based controller.
Lemma 1 ([20]).The necessary and sufficient condition for the closed loop
system to be stable under the GMV control is that all the roots of the polyno-
mial Gd(z−1) = A(z−1)Q(z−1) + B(z−1)C(z−1)belong to the open unit disk,
and the polynomial pairs (Q, C),(A, C ), and (B, Q)have no common zeros
outside the unit disk.
8
qref (t)TCP/AQM
Controller
Design
GMV
Controller
Parameter
Estimation
q(t)
Pd(t)
GMV
Adaptive
T
Figure 3: Block diagram of the AGMV controller for TCP/AQM systems.
4.2. The Proposed AGMV Controller
Although, the GMV is a suitable control method, especially in the pres-
ence of input-output measurement, when the process dynamics changes, the
controller degrades in performance. To overcome this problem, the adaptive
GMV controlling approach is recommended. AGMV is the combination of
on-line (real-time) parameters estimation and GMV [19]. The TCP/AQM
parameters are estimated on-line, using the real-time measured input-output
data. The estimated system parameters will then be used to update the
controller parameters. Figure 3 depicts the block diagram of the AGMV
controller. As seen in the figure, in this approach, a parameter estimator
block becomes necessary. The three lines connecting the GMV and Adaptive
block in Figure 3 are θT,Pd(t), and q(t) which represent the estimated pa-
rameter, control signal, and the system output, respectively. In what follows,
we discuss the real-time parameters estimation process in more details.
In order to estimate the on-line parameters, we consider the recursive least
square (RLS) method with forgetting factor. For this purpose, the system
transfer function should be first discretized using the zoh method. To do so,
we first rewrite the system equation in (15) as follows:
y(k) = θTϕ(k),(23)
9
where,
θT= [a1a2. . . anb0b1. . . bm]and (24)
ϕ(k) = [−y(k−1) −y(k−2) · · · − y(k−n)...
. . . u(k−d)u(k−d−1) . . . u(k−d−m)]T.(25)
We also write the estimated system model as:
ˆ
y(k) = ˆ
θTϕ(k),(26)
where ˆ
θ(k) and ˆ
y(k) are the estimations of θand yparameters at each time
instant k, respectively. In the RLS method with forgetting factor, the un-
known parameter ˆ
θ(k) is estimated in such a way that the estimated total
square error, J=Pk
j=1 λk−1ε2(j), is minimized with λ(0 < λ ≤1) as the
forgetting factor [20]. Therefore, the estimation error can then be written as:
ε(k) = y(k)−ˆ
θT(k−1)ϕ(k).(27)
Implementation of RLS control algorithm with forgetting factor includes
the following steps which are repeated every kseconds [20]:
Step 0: Setting initial values P(0) and ˆ
θ(0).
Step 1: Reading data input-output per k.
Step 2: Calculating the estimated error according to equation (27).
Step 3: Calculating the estimation gain, g(k), as follows:
g(k) = P(k−1)ϕ(k)
λ+ϕTP(k−1)ϕ(k).(28)
Step 4: Calculating ˆ
θ(k) according to:
ˆ
θ(k) = ˆ
θT(k−1) + g(k)ε(k).(29)
Step 5: Updating P(k) as follows:
P(k) = 1
λ[I−g(k)ϕT(k)]P(k−1).(30)
10
5. Implementation and Simulation
In this section, the case study is simulated under two distinct scenarios
namely, unknown parameters and time-varying parameters, and its perfor-
mance is justified in comparison with its adaptive minimum variance (AMV)
counterpart [19] in the case of time-varying parameters scenario.
By on-line recursive estimation of n1,n2,m1, and m2and regulating GMV
controller parameters in each sampling period, an adaptive GMV controller
is implemented. Hence, we can derive the following equations:
θT= [n1n2m1m2] (31)
ϕ(k)=[−δq(k−1) −δq(k−2) δPd(k−1) δPd(k−2)]T,(32)
where Pd(k) and q(k) are the input and output of the controller system.
By applying the recursive least square algorithm in sub-section 4.2, the un-
known parameter θis estimated on-line. It is worth mentioning that the
sampling time selection is of great importance. Selecting a small sampling
time could lead to an unstable closed-loop system, while a large sampling
time would result in prohibitive long delays in the response time of the con-
troller and, therefore, wrong control process. It has been suggested in [21]
that an appropriate sampling time is R0.
Henceforward, the AGMV controller performance for the purpose of ac-
tive queuing management is studied. Having the system model, the controller
described in the preceding section is numerically designed. The efficiency of
the designed controller is then studied using MATLAB simulation under two
different scenarios: unknown parameters and time-varying parameters.
5.1. Unknown Parameter Scenario
Consider the benchmark network topology as shown in Figure 4. The
following numerical values are considered for the system parameters: N=
50 TCP sessions, TP= 0.24 secs,qref = 90 packets and buffer size is
300 packets. Also, the outgoing link bandwidth is 1.2Mb/sec with an aver-
age packet size of 500 bytes this results in C= 300 packets/sec. Using these
parameters, one can calculate other parameters such as R0= 0.54 secs and
W0= 3.2packets. Therefore, the transfer function of the open loop model
can be derived as:
Pdc(s) = 900
s2+ 1.856s+ 0.007056.(33)
11
Figure 4: Network topology considered for simulation.
By discretizing the above system using zoh approach and considering the
sampling time Ts= 0.54 sec, the following discrete system model can be
derived:
A(z−1)q(k) = z−dB(z−1)Pd(k) (34)
(1 + a1z−1+a2z−2)q(k) = z−1(b0+b1z−1)Pd(k),(35)
where a1=−1.365, a2= 0.367, b0= 96.4750, and b1= 69.2482. The
measurement noise and modeling errors are modelled as a white noise process
with signal-to-noise ratio (SNR) of 20dB:
A(z−1)q(k) = z−dB(z−1)Pd(k) + e(k) (36)
(1 + a1z−1+a2z−2)q(k) = z−1(b0+b1z−1)Pd(k) + e(k).(37)
The parameters of the model are considered unknown. We only pick
L= 500 data samples from the signals at system input and output. Using
these samples, the system parameters are estimated at each sampling instant
and are used to update the controller parameters. To estimate the model
parameters by RLS method, the initial values of parameters are chosen as
λ= 0.995, θ(0) = [0 0 0 0], P(0) = 106T4×4with C(z−1) = 1, Q(z−1) = q0=
0.5, and R(z−1) = R0= 1. The polynomials for the GMV controller method
in Section IV can be written as:
F(z−2) = f0+f1z−1(38)
E(z−1) = e0= 1.(39)
12
0 5 10 15
−1.5
−1
−0.5
0
a1
time (sec)
Actual value
0 5 10 15
0
0.2
0.4
a2
time (sec)
Actual value
0 5 10 15
0
50
100
b1
time (sec)
Actual value
0 5 10 15
0
50
100
b2
time (sec)
Actual value
Figure 5: Estimation of the system model parameters a1,a2,b1, and b2.
The polynomials coefficients should satisfy the following Diophantine equa-
tion:
A(z−1)E(z−1) + z−1F(z−1) = I. (40)
Therefore, the controller signal based on equations (20) and (21) at each
instant is determined as:
u(k) = R0yr(k)−ˆ
f0(k)y(k)−ˆ
f1(k)y(k−1) ...ˆ
b1(k)y(k−1)
(q0+ˆ
b0(k)) ,(41)
where, {ˆ
f0(k),ˆ
f1(k),ˆ
b0(k),ˆ
b1(k)}are the estimated versions of parameters
{f0, f1, b0, b1}in each sample time k. Recall that the controller should be
designed such that the output of the closed loop system follows reference
input, qref = 90, while minimizing the noise effects. Then, by choosing
P(z−1) = 1 + 0.4z−1,q0= 90, and R0= 1, the design of the adaptive GMV
controller is complete.
Figures 5 and 6 show the system model parameters and controller es-
timation with respect to time, respectively. As depicted in Figure 5, the
estimated parameters namely, a1,a2,b1, and b2converge to the actual value
in less than 5 seconds. During the estimation transient time, the change
in the estimated parameters is around the actual value. Similarly, Figure 6
illustrates the estimated GMV parameters in terms of time. The parameters
f0and f1reach the actual value in less than 5 seconds.
In Figure 7, the output q(t), control signal Pd(t) , and signal W(t) are
shown. The first figure shows that our controller is able to converge the
queue length, q, to the expected length, qref , in less than 6 seconds. The
13
0 5 10 15
0
0.5
1
1.5
f0
time (sec)
Actual value
0 5 10 15
−0.8
−0.6
−0.4
−0.2
0
0.2
f1
time (sec)
Actual value
Figure 6: Estimation of the GMV parameters f0and f1.
0 2 4 6 8 10 12 14 16 18 20
0
200
400
time (sec)
q
qr=90
0 2 4 6 8 10 12 14 16 18 20
−10
0
10
20
time (sec)
pd
Figure 7: Performance of adaptive GMV control in the unknown parameters mode.
second figure reveals that all the results are obtained with a good controller
performance. Similarly, Pdtakes less than 8 seconds to bring the queue length
to the desired value, which is quite appreciable.
The mean, variance, and standard deviation of the output signal q(t),
resulting from 100 Monte Carlo simulation runs, are 90.7203, 208.8887, and
14.4530, respectively.
5.2. Time-varying Parameter Scenario
In this scenario, the number of TCP (N) and bottleneck line capacity
(C) are considered to be time-dependant assumptions as shown in Figure 8.
The controller and network topology used in this sub-section is the same as
that in sub-section 5.1. As shown in Figure 9, the queue length reaches the
expected queue length, qref in less than 8 seconds. It can be seen that under
14
0 50 100 150
30
40
50
60
70
time (sec)
N
0 50 100 150
280
300
320
340
360
time (sec)
C
Figure 8: Variation of time-varying parameters Nand C.
0 2 4 6 8 10 12 14 16 18 20
0
200
400
time (sec)
q
qr=90
0 2 4 6 8 10 12 14 16 18 20
−10
0
10
20
time (sec)
pd
Figure 9: Performance of adaptive generalized minimum variance controller (AGMV) in
the case of time-varying parameters.
the time-varying conditions, the AGMV controller is be able to keep queue
around the desired value.
Similar to Figure 9, Figure 10 depicts the performance of AMV controller
for the time-varying parameters. By comparing Figure 9 and 10, we can
infer that in the transient state, AGMV outperforms AMV in terms of queue
length criticality. By comparing the controller parameter Pdin these two
figures, it can be seen that AGMV shows better performance when compared
to AMV. Since TCP/AQM networks are dynamic in nature, it is preferable
to adopt AGMV for such networks. The results of these two controllers after
100 Monte Carlo simulation runs are summarized in Table 1. As seen in this
table, the statistical parameters of AGMV are better than those of AMV.
Even at times when parameters such as Nand Care constant and invariant
in time, the AGMV controller exhibits better performance than its AMV
15
0 2 4 6 8 10 12 14 16 18 20
−200
0
200
400
600
time (sec)
q
qr=90
0 2 4 6 8 10 12 14 16 18 20
−20
0
20
time (sec)
pd
Figure 10: Performance of adaptive minimum variance (AMV) controller in the case of
time-varying parameters.
Table 1: Mean, variance and standard deviation of the signal q(t).
Adaptive
Controller
Mean Variance Standard
Deviation
Accumulated Loss
of Control Signal
Ptf
k=1 P2
d(k)
GMV 90.3957 367.5154 19.1707 181.3771
MV 91.9347 657.5989 25.6437 413.0538
counterpart.
5.3. Comparison of AGMV and PI
In order to depict the stability and robustness of the proposed AGMV
scheme, we compare it with PI which is another well-known AQM controller.
In this comparison considered figure 4 as network topology. The number of
TCP flows (N) and capacity (C) are considered to be time-varying taking
values N= 60 TCP sessions and C= 3750 packets/sec corresponds to a
15 Mb/sec link bandwidth and average packet size 500 bytes. A Gaussian
noise with mean zero and variance 5 is added to N. Also, for a period of
20 seconds, Cis reduced from 3750 packets/sec to 3000 packets/sec. The
values of R0= 0.246 secs,Tp= 0.2secs,qr= 70 packets and buffer size is
800 packets as in [16]. Figures 11 and 12 show the simulation results of the
AGMV and the PI controllers, respectively.
As can be seen in these figures, the impact of the network changes on
the AGMV controller is less than the PI controller, as well as these changes
16
0 20 40 60 80 100 120 140 160 180 200 220
0
50
100
q
qr=70
0 20 40 60 80 100 120 140 160 180 200 220
−0.1
−0.05
0
0.05
0.1
time (sec)
pd
Figure 11: Performance of the AGMV controller in the unknown parameter mode.
0 20 40 60 80 100 120 140 160 180 200 220
0
50
100
q
q
qr=70
0 20 40 60 80 100 120 140 160 180 200 220
0
1
2
3
4x 10−3
time (sec)
pd
Figure 12: Performance of the PI controller in the unknown parameter mode.
converge to qrquickly. In general, the performance of the AGMV controller
against parametric changes is better than that compared with PI and other
non-adaptive controllers. It can be observed that the control signal of AGMV
is smoother than that of the PI controller.
In order to analyze the robustness of the controller under rapid network
condition changes, the controller has been simulated to study the stability
of the output queue length. A Gaussian noise with mean zero and variance
50 have been added to Nand Chas been altered several times as shown in
Figure 13. Like the preceding case, the impact of changes on AGMV con-
troller (Figure 14) is less than that on the PI controller (Figure 15) as these
changes converge to qrquickly. Therefore, under conditions with sudden
changes, AGMV outperforms the PI controller. Like the previous scenario,
the control signal of AGMV is smoother than that of the PI controller.
17
0 20 40 60 80 100 120 140 160 180 200 220
40
50
60
70
80
N
0 20 40 60 80 100 120 140 160 180 200 220
0
1000
2000
3000
4000
time (sec)
C
Figure 13: Variation of time-varying parameters Nand C.
0 20 40 60 80 100 120 140 160 180 200 220
0
100
200
300
400
q
0 20 40 60 80 100 120 140 160 180 200 220
−0.1
−0.05
0
0.05
0.1
time (sec)
pd
qr=70
q
Figure 14: Performance of the AGMV controller in the unknown parameter mode.
0 20 40 60 80 100 120 140 160 180 200 220
0
100
200
300
400
q
0 20 40 60 80 100 120 140 160 180 200 220
0
0.005
0.01
time (sec)
pd
q
qr=70
Figure 15: Performance of the PI controller in the unknown parameter mode.
6. Conclusion
In this paper, a new AQM method for TCP networks based on adaptive
generalized minimum variance (AGMV) technique has been proposed. The
proposed adaptive method can improve the performance of GMV-based con-
18
troller in dynamic networks. It is known that when the process dynamics
changes, the performance of the GMV-based controller degrades significantly.
To resolve this problem, an adaptive GMV control approach was proposed
in this paper. The method is based on minimizing the output variance to
overcome the effects of measurement noise and modeling errors. By using
the on-line parameters estimator, system parameters were identified. Next,
based on this identified parameters, controller parameters were updated. The
performance of the controller was evaluated in TCP network with unknown
fixed and time-varying parameters. We also compared the performance of
the AGMV controller with its adaptive minimum variance (AMV) counter-
part under time-varying condition. Simulation results indicate that the pro-
posed AGMV scheme is able to efficiently preserve the scheduler queue length
around the desired point by proper estimation of network and GMV con-
troller parameters adaptively. In addition, the superior performance of the
proposed controller with respect to the well-known AQM controller namely,
PI controller has been illustrated through results obtained via simulation.
References
[1] S. Ryu, C. Rump, C. Qiao, Advances in Internet congestion control,
IEEE Communications Surveys and Tutorials 5 (1) (2003) 28-39.
[2] W.H. Kwon, H.S. Kim, A survey of control theoretic approaches in wired
and wireless communication networks, Korea-Japan Joint Workshop,
2000, pp. 30-45.
[3] M.A. Ardestani, M.T.H. Beheshti, A robust discrete-time controller for
delay sensitive applications, in: Proceedings of International Conference
on Information, Communications and Signal Processing (ICICS), 2009,
pp. 1-6.
[4] S. Ryu, C. Rump, C. Qiao, Advances in active queue management
(AQM) based TCP congestion control, Telecommunication Systems 25
(3) (2004) 317-351.
[5] R.T. Bhaskar, A. Ahammed, Performance comparison of active queue
management techniques, Journal of Computer Science 4 (12) (2008)
1020-1023.
19
[6] L. Tan, W. Zhang, G. Peng, G. Chen, Stability of TCP/RED systems in
AQM routers, IEEE Transactions on Automatic Control 51 (8) (2006)
1393-1398.
[7] J. Sun, K. Ko, G. Chen, S. Chan, M. Zukerman, PD-RED: to improve
the performance of RED, IEEE Communications Letters 7 (8) (2003)
406-408.
[8] C. Hollot, V. Misra, D. Towsley, W. Gong, On designing improved con-
trollers for AQM routers supporting TCP flows, in: Proceedings of IEEE
INFOCOM, 2001, pp. 1726-1734.
[9] M.F. Firuzi, M. Haeri, Active queue management in TCP networks
based on self tuning control approach, in: Proceedings of IEEE Con-
ference on Control Applications,2005, pp. 904-909.
[10] N. Bigdeli, M. Haeri, CDM-based design and performance evaluation
of a robust AQM method for dynamic TCP/AQM networks, Computer
Communications 32 (1) (2009) 213-229.
[11] V. Misra, W.B. Gong, D. Towsley, Fluid-based analysis of a network
of AQM routers supporting TCP flows with an application to RED, in:
Proceedings of ACM SIGCOMM, 2000, pp. 151-160.
[12] C. Wang, J. Liu, B. Li, K. Sohraby, Y.T. Hou, LRED: A robust and
responsive AQM algorithm using packet loss ratio measurement, IEEE
Transactions on Parallel and Distributed Systems 18 (1)(2007) 29-43.
[13] A. Patete, K. Furuta, M. Tomizuka, Self-tuning control based on general-
ized minimum variance criterion for auto-regressive models, Automatica
44 (8) (2008) 1970-1975.
[14] F.P. Kelly, A.K. Maulloo, D.K.H. Tan, Rate control in communication
networks: shadow prices, proportional fairness and stability, Journal of
the Operational Research Society 49 (1998) 237-252.
[15] S.H. Low, A duality model of TCP and queue management algorithms,
IEEE/ACM Transactions on Networking (TON) 11(4) (2003) 525-536.
[16] C.V. Hollot, V. Misra, D. Towsley, W.B. Gong, Analysis and design of
controllers for AQM routers supporting TCP flows, IEEE Transactions
on Automatic Control 47 (6) (2002) 945-959.
20
[17] C.V. Hollot, Y. Liu, V. Misra, D. Towsley, Unresponsive flows and AQM
performance, in: Proceedings of IEEE INFOCOM, 2003, pp. 85-95.
[18] K. Ogata, Discrete Time Control Systems, Prentice Hall International,
New Jersey, USA, 2002.
[19] K.J. Astrom, B. Wittenmark, Adaptive Control, 2nd Edition. Reading,
MA: Addison-Wesley, 1995.
[20] P. Ioannou, B. Fidan, Adaptive Control Tutorial, Cambridge University
Press, 2006.
[21] M.S. Herfeh, M. Haeri, Active queue management in TCP networks
based on minimum variance adaptive control, in: Proceedings of Inter-
national Symposium on Communications, Control and Signal Processing
(ISCCSP), 2006.
21
