Improving Network Performane

Management of Nonlinear Dynamics

Seyed Shahrestani

School of Computing and Mathematics, University of Western Sydney, Sydney, Australia

Keywords: AQM/RED, Bifurcation Analysis, Delay Control, Internet, Nonlinear Dynamics.

Abstract: To manage the huge amount of traffic that is to be carried using the limited bandwidth and other resources,

large networks and the Internet are heavily dependent on the use of protocols, and in particular, on TCP/IP

protocol suite. While the utilization of TCP/IP is of significant practical value, for most large complex

networks it can be inefficient, as it fails to fully take into consideration the importance of the major parts of

the system. To overcome this, more complicated congestion control mechanisms, such as AQM/RED are

widely utilized. However, these complex mechanisms exhibit nonlinear dynamics, which are not well

understood and are usually unaccounted for. As a result of this, to avoid oscillatory behavior or loss of

stability, the parameters of these systems are often set too conservatively. In turn, this will lead to

unnecessary underutilization of the network resources. On the other hand, through the analysis and

management of nonlinearities, the operability regions for the networked systems can be expanded, while its

performance is also improved. This paper presents our visionary works of applying these ideas to networked

systems, resulting in higher loading and throughput, and avoiding oscillatory or unstable behavior.

1 INTRODUCTION

The explosive growth of the Internet has provided

modern societies with many new opportunities and

challenges. Apart from the two end-systems,

invariably for communications outside a LAN,

routers play dominant roles in establishing the

communication paths. In general, given the store-

forward nature of the Internet, queuing and delay of

packets are inventible.

Obviously, the utilized transport protocols play a

major role in delay and reliability of communications

of the packets. However, it is well established that

for most large networks this simplistic utilization of

TCP/IP can result in severe inefficiencies (Fan,

2010). Such approaches tend to ignore to fully take

into consideration the importance of the major parts

of the system, namely the routers and the links or the

network itself.

To overcome such inefficiencies, many

approaches have been proposed and implemented. Of

the well established approaches, among these, is the

so-called Active Queue Management (AQM).

Random Early Detection (RED) is the most widely

used AQM scheme (Hollot, 2002). While the

concepts behind RED mechanism are very

straightforward, its interfaces with TCP can lead to

nonlinear dynamics that are not well understood. The

rather complex Internet behavior arising from the

existence of inherent nonlinearities can cause

instability and oscillatory dynamics. To avoid such

undesirable dynamics, in practice the parameters of

AQM/RED mechanisms may be set more

constrained, compared to what the system is actually

capable of. In other words, underutilization of the

networks and resources has been a common practice

as the system is set to operate below its operation

limits.

Nonlinearities of TCP/AQM and the drastic

changes of the system behavior that they can cause,

even with small loading changes for example, are

well established (Chen, 2005). Such variations in

behavior and the existence of chaos can in turn be

linked to bifurcations in the networked systems (Liu,

2007). Consequently, rather than ignoring the

nonlinearities, it is advantageous to exploit them and

employ strategies to manage and control the chaotic

behavior and bifurcations. This must be based on a

global view of the system and can improve

performance of the underlying networks, leading to

higher loading possibility of the Internet without the

oscillatory or unstable behavior (Shahrestani, 2000).

Shahrestani S..

Improving Network Performane - Management of Nonlinear Dynamics .

DOI: 10.5220/0004129900520057

In Proceedings of the International Conference on Data Communication Networking, e-Business and Optical Communication Systems (DCNET-2012),

pages 52-57

ISBN: 978-989-8565-23-5

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

To facilitate the discussion of these points, this

paper is organized as follows. The bifurcation and

structural stability analysis are introduced in the next

section. Such analysis is used to study the

performance of the AQM/RED based networks. The

results along with strategies to manage such

networks to improve their performance are presented

in Section III. The conclusions and potential future

works are given in the last section.

2 MANAGEMENT OF COMPLEX

NETWORKS

Many complex systems, such as large networks,

exhibit multiple equilibrium points leading to several

potential steady-state operating states. Such systems

require a management scheme capable of

administration over a wide range of anticipated

operating conditions. In developing such schemes,

the qualitative changes that occur in the behavior of

the system in different operating regions must be

taken into account.

The limits, at which qualitative changes in

complex system behavior occur, may be related to

the structural stability of the system. The structural

stability limits of a nonlinear system can in turn be

related to bifurcation points in the mathematical

model of the system. The bifurcations refer to

qualitative changes in the system behavior as some of

the system parameters vary quasi-statically (Seydle,

2010).

Complex nonlinear dynamical systems can

generally be described by a number of coupled

differential-algebraic equations

(1)

Where t is time, x(t) and a are the dynamic and

instantaneous states of the system and w represents

external influences, such as system input, time

varying parameters, disturbances, and the like. Now,

consider a single-output case where the variations of

the control input u(x,a,t) and another one of the

system parameters

min max

( ) [ , ]t

  



, dominate other

variations in the system. For example, in networked

systems there is a rather clear separation between the

time-frames involved in the analysis (and design of

the required management actions) for transient

congestion and delay and longer-term bandwidth and

capacity management consideration (Shahrestani,

2011). To emphasis these points, the model (1) can

be put in the following form

(2)

A typical network management problem can be

considered as identification and setting of system

parameters such that a point

is a secure operating

point of the system (2). Additionally, there may be

constraints on the manageable parameters or the

specification of transient characteristics of the

network. Naturally, if with

xx

no proper

manageable and controlled parameters satisfying (2)

can be identified, then the network will not be

operable steadily at that point.

More generally though, the system state may be

considered to be constrained to a certain operating

region

containing steady operating points (or in

some cases, to some operating region containing

secure oscillatory solutions with restrained

amplitudes). Consequently, with properly identified

parameters at the operating points, the state space

region of interest for system (2) can be considered as

min max

{ },

d e e e e

X x x x x  

(3)

while



varies slowly with time within a certain

range of interest

{



min max

e e e e

   





(4)

The value of the managed parameters with the

network operating at some steady-state, is to be

chosen such that the existence of some

xX

for

some





is assured. Therefore, with a particular

and proper value of

say

,e e j

uu

, the network

operation constrained to region

may happen to

exist for only a range

  

defined by

min max

{ | }

   

   

(5)

That is, to cover the complete range of interest



in general, multiple sub-ranges may need to be

considered, with each sub-range corresponding to a

different set of managed parameters with the system

at steady-state. These points are illustrated in Figure

1. It can also be noted that each value of

,ej

( , , , ),

0 ( , , , ),

( , , , ); ( ) .

x f t x a w

t x a w

y h t x a w x t x







( , , , ),

0 ( , , , ),

( , ).

x f x a u v

x a u v

y h x a





Improving Network Performane - Management of Nonlinear Dynamics

Figure 1: An illustration of parameter dependent steady-state operation manifolds.

corresponds to some reference segment on the

parameter dependent steady-state operation manifold

of a nonlinear system such as (1).

As pointed out earlier, in general, the region

may contain steady-state operating points with

different characteristics. That is, as



varies within

its limits, the system with the managed parameters

being simply set as

uu

may exhibit

qualitatively different behavior for different levels of



. To account for the manager capability in handling

the system dynamics appropriately, the managed

parameters are now considered as

e j j

u u u





The critical values of the slowly changing parameter



that are associated with the boundaries of

operating regions with distinct dynamic behaviors,

correspond to bifurcation points of the system (2).

In some cases, the management of bifurcating

systems is achieved by ensuring that the system

operation is such that the bifurcation parameter is

always below its critical values, and the bifurcations

are ultimately avoided (Shahrestani, 2000). While this

approach solves part of the operation problems, it can

result in a conservative design with for example

loading margins larger than what is really required.

For a bifurcating system exhibiting regions with

non-identical structural stability behavior, the

bifurcation points may be used to establish the

bounds on segments of the state space of the system

with different management requirements.

Consequently, depending on the ranges of the

bifurcation parameter, several regional management

schemes will be needed, while each scheme may

pursue a different objective. For example, in some

range of the bifurcation parameter the manager may

force the system to track the existing steady-state

operation points while in some other range, the

stabilization of the bifurcated solutions may be the

primary objective of the managing scheme. These

points are further illustrated through management of

delay and congestions to improve the network

performance, in the next section.

3 SHAPING THE NETWORK

BEHAVIOR

To reduce the delay and to achieve improved

throughput, many adaptive Random Early Detection

(RED) algorithms have been developed and studied.

As discussed before, to overcome the shortcomings

associated with the linear dropping probability

functions originally used by adaptive RED, the

utilization of nonlinear adaptive approaches have

found widespread acceptance. These approaches are

mainly based on Active Queue Management (AQM).

As mentioned before, utilization of AQM/RED

introduces complex nonlinearities. Such

nonlinearities can in turn induce several bifurcated

solutions.

Effectively, the RED controller output that

provides the feedback to sender, is a probability of

drop rate p(t). This probability is a function of

average queue length q(t). The nonlinearities of the

AQM/RED model are essentially a consequence of

the multiplicative characteristics of packet loss and

are represented through describing TCP window

control mechanism (Rezaie, 2007).

Ignoring secondary effects and with D denoting

the propagation delay, it can be shown that R(t), the

round trip delay, will be ((ω/c) + D). The packet drop

probability, as a function of queue length q(t), can

then be put in the following form (Raina, 2005).

(6)

( ) 0, ;p q q qMin

()

( ) ( ), qMin<

q t qMin

p q pMax q qMa

qMax qMin







( ) 1, p q q qMax

DCNET 2012 - International Conference on Data Communication Networking

Figure 2: Unstable oscillatory behavior, at network loading

near but after a bifurcation point.

The TCP/AQM and the router queuing dynamics can

be described by the following set of differential

equations (Misra, 2000) and (Ranjan, 2004).

(7)

In this model, p(t) is the probability of packet drop

within the closed interval [0, 1], R is the round trip

delay in seconds, ω(t) is the average TCP window

size in packets, q(t) is the average length of the queue

in packets, , c is the link capacity in packets per

second, and n is the load factor or the number of TCP

sessions.

In most network management analysis, it is

assumed that the load n(t) and the round trip delay

R(t) are time-invariant. Given their slow variations

compared to other network parameters, this is usually

a reasonable assumption. But it is more realistic to

consider them varying slowly with time. This is in

line with bifurcation analysis approaches.

With these considerations, the second set of

differential equations (7) can be normalized by noting

that the dropping probability p(t) is proportional to the

queue length. The normalization can be carried out by

using the following substituting in those equations

(Rezaie, 2007).

(8)

These substitutions will result in the following

normalized equations.

(9)

The system described by the normalized

differential equations in (9) has a unique equilibrium

point,

( ,2/( . )).c K c

Due to space limitations, only

the results of bifurcation analysis of this model are

presented here. The eigenvalue analysis of this system

shows that a zero eigenvalue occurs for

0.K 

This

corresponds to one of the bifurcation points, at which

the system behavior changes and bifurcated solutions

are expected to emerge.

It can be noted that generally speaking, most of

the RED parameters are set based on network

manager experiences or at best based on experiential

data. This may lead to very conservative set-ups to

avoid instability or cyclic behavior similar to those

discussed so far and portrayed in Figure 2.

The choice of the RED parameters, selected by

the network administrator, dictates the value of K. In

other words, if the network manager sets the RED

parameters in a way that K is close to zero, a small

disturbance, such as small variations in network

traffic can destabilize the network through

disappearance of a stable steady-state operating

point. This means that under such conditions, even

vey small variation in network traffic can result in

the collapse of the whole network operation.

Figure 3: Growing oscillatory behavior, at network loading

near a bifurcation point.

Perhaps, more interestingly, it can be shown that

for each value of c, while K remains below some

certain limit, say K

, there exist exactly one stability

nterval that is a function of K. In that interval, the

network is operable, although at high levels of traffic,

oscillatory responses can come into picture. On the

other side of this point, with the RED parameters

chosen such that K>K

, growing oscillatory solutions

can be expected. Obviously the growing oscillations

1 ( ). ( ( ))

( ( ))

( ) 2( ( ))

t t R t

p t R t

R t t R t







  



()

( ), 0

()

max[0, ( )], 0.

()

() {

c t q



  

  



( ) ( 1)

1 ( 1)

KQ t







  

( ) , 0

max[0, ( ) ], 0

() {

t c Q









Improving Network Performane - Management of Nonlinear Dynamics

lead to unstable operation of the system and collapse

of the network. The expected system behavior for

values of K just after K

, will be similar to that shown

in Figure 2. K

corresponds to a subcritical Hopf

bifurcation of the original nonlinear system

describing AQM/RED. In the Hopf bifurcation a

branch of stable periodic solutions originates, which

again becomes unstable after a period doubling

bifurcation. The sequence of period doubling

bifurcations ultimately leads to chaos. For more

details see (Shahrestani, 2000). For the chaotic

situation, the unstable oscillatory behavior that the

network exhibits will be dependent on its initial state.

The behavior will be similar to those depicted in

Figure 4.

Clearly, more analytical approaches for selection

of RED controller parameters and analysis of their

effects on the network performance will be

advantageous. In our previous works, we have

developed a framework for design of management

schemes and control laws for a parameter dependent

complex nonlinear system (Shahrestani, 2000) and

(Shahrestani, 2008).

Figure 4: Unstable oscillatory behavior.

The result is a global multilevel management and

control scheme, where the first level depending on

signal and loading levels switches the parameters, so

that the system states are confined to the

neighborhood of some desired reference segment.

For each reference segment, the information gained

through bifurcation analysis is used for further

segmentation of the state space of the system, similar

to that shown in Figure 1.

To move the eigenvalues, λ,

to ( )

     



      

(10)

to locations,





corresponding to more desirable

behavior and operations, the so-called modal control

approach will require the feedback

( ( ) ) ;

i i i T

u k l x









(11)

where the gains are

ˆˆ

( )( )

(1 ),

( ) ( ) 2

()

i i i i

j j j k

i T i i

j j j

   



  



  



(1 ).

()







(12)

With such modal control, the response of the

system with the situation the same as that shown in

Figure 4 will be improved to the stabilized behavior

shown in Figure 5.

Figure 5: Stablized behavior through modal control,

corresponding to the response shown in Figure 4.

In our previous works we have also shown that

only the feedback of critical variables up to cubic

terms may have any effect on the existence of a Hopf

bifurcation or changing the stability behavior of the

bifurcated solutions. Obviously, even for a system

with controllable modes only linear terms have any

effect on the location of the eigenvalues.

Consequently, to change a subcritical bifurcation to a

supercritical one, quadratic and/or cubic (critical)

state feedback can be identified. These relate to

improving the network behavior, when the dynamics

exhibit oscillatory behavior. Figure 6 for instance,

shows the effect of cubic state feedback for the same

situation depicted in Figure 4.

Figure 6: Stablized behavior by cubic state feedback,

corresponding to the response shown in Figure 4.

DCNET 2012 - International Conference on Data Communication Networking

4 CONCLUDING REMARKS

In this paper, some approaches to improve the

performance of complex networks are proposed.

These are based on the analysis and management of

the system nonlinear dynamics and bifurcations. By

managing the bifurcations, performance of these

networks can be improved while their operability

region can also be expanded. We also reported the

works in-progress towards applying these ideas to

establish a more analytical management scheme for

networked systems. Choosing and regulating the

parameters, based on these types of analysis and

management and their utilization in communication

systems and the Internet can result in expanding their

stability and operability regions, for instance over a

wide range of loading, throughput, delay and

congestion levels over TCP connections. In our

future works, we aim to expand these ideas and

validate the analytical results through more

experimental works.

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Improving Network Performane - Management of Nonlinear Dynamics