Intelligent traffic control on internet-like topologies

Book TitleBook EditorsIOS Press, 2003

1

Intelligent Traffic Controlon Internet-Like Topologies

DATRA - Dynamically Adjusted Traffic Rate Alterations

Antonia Katzourakia,1, Philippe De Wildea,2 and Robert Ghanea Hercockb,3

a Imperial College London and Heriot Watt Universityb BTexact Future Technologies

Abstract. The growing scale and importance of technological net-works throughout the world has highlighted the devastating conse-quences of catastrophic network failures. In this paper, we addressthis crucial issue through a detailed analysis of network traffic dis-tribution across network nodes, with the aim of developing an in-telligent traffic control model. Specifically, we develop and demon-strate the Dynamically Adjusted Traffic Rates model, which aims tofairly distribute traffic amongst network nodes according to theirnetwork characteristics. Our model is independent of topology andis based on dynamically adjusted traffic rates and properties sim-ilar to those observed in real Internet traffic. In this paper ourmodel is numerically analyzed over a variety of network topolo-gies to display its chaotic features leading to self-similarity. Themodel is inspired by biological evidence.

Keywords. Dynamic Adjustments, Networks, ODEs, Intelligent Traffic Control,Equilibrium

1. Introduction

Catastrophic network failures are one of the major problems arising from resource over-loading and the potential propagation of network deficiencies. In this work, we attemptto prevent such catastrophic failures by strategically controlling the distribution of trafficto the available network resources according to their observed characteristics.

Cascading failures of networks have proved to result in devastating consequenceson everyday human life. A characteristic example is that of the 10th August 1996, when

1Department of Electrical and Electronic Engineering, Imperial College London, London, SW7 2AZ, UKE-mail: [email protected]

2Department of Computer Science, Heriot Watt University, Edinburgh, EH14 4AS, UKE-mail: [email protected]

3BTexact Future Technologies Group Adastral Park, Martlesham Heath, Ipswich, IP5 3RE, UKE-mail: [email protected]

2 A. Katzouraki et al. / Intelligent Traffic Control

a voltage-based failure occurred in two power lines. Major power disturbances werepropagated through a series ofcascading failures, ultimately leading to blackouts in 11US states and 2 Canadian provinces, leaving 7 million people without power for up to 16hours [1]. Similarly, on the 4th May 2000, the VBS/Lovelet-A virus (also known as theLove Bug or ILOVEYOU worm) caused considerable damage as it infected computersworld-wide. The virus fooled computer users into thinking they had received a love letterin their email. However, when the attached file was opened, the virus would forward itselfto other email addresses found on the infected computer [2]. Such events demonstrate theimportance and necessity of intelligent traffic control, in mitigating the effects of thesefailures.

In this work, we develop a biologically inspired deterministic model, through whichwe aim at controlling the distribution of traffic to the available network resources withthe goal of fairly re-distributing traffic and preventing node overloading. In our modeltraffic rates are dynamically adjusted according not only to the link costs but also to eachnode’s buffer capacity, so that all network nodes will behave in a synchronized mannerto achieveequilibrium(provided that the exogenous parameters allow equilibrium to bereached). This strategy not only distributes the traffic load amongst network nodes, butalso synchronizes the traffic and link cost changes with buffer and topology variations.

In contrast to previous approaches (e.g. [3,4,5,6]) our model isindependentof topol-ogy, and focuses on controlling traffic and cost rate dynamics rather than improving thenetwork’s topology. The majority of the previous research studies the influence of net-work structure on the dynamics of the system, whereas through our model we aim tocombinenumerical analysiswith observedempirical studieson real networks appliedon diverse Internet like topologies.

In an approach different to that commonly used in research studies on multi-agentsystems, we provide cost strategies designed to achieve competitiveness by utilizing non-linear functions (e.g. [7,8,9,10]). In this way our model controls traffic and cost ratedynamics to avoid overloading of nodes and deliver global load balancing.

2. Previous Work

Recently there has been much interest in studying real-world networks and using themto investigate various phenomena, such as network resilience in random or deliberateattacks and various spreading. Real-world networks can be characterized accurately byPoisson-distributed, Scale-Free, Watts-Strogatz graphs as well as other more recently in-vented graphs [11,12,13,14,15]. E–R graphs are the oldest type of graphs used to rep-resent real-world networks. Whilst E–R graphs do not possess important properties ofreal-world networks [16,17], they can be used in order to investigate various phenomena,such as network resilience in random failures, deliberate attacks and virus spreading.

Recently scientists have investigated the network resource allocation problem to de-liver more reliable networks by applying market-based control mechanisms on multi-agent systems. According to their architectures agents represent various network re-sources to coordinate their resource allocation decisions without assuminga priori coop-eration. The design of the agents’ trading behavior utilizes either simple linear functionsor concepts from game theory [7,8,9,10].

In other research, multi-scale networks were designed to construct a framework forexploring organizational robustness with respect to a range of environmental stresses.

A. Katzouraki et al. / Intelligent Traffic Control 3

As a result, they produced a class of networks to reduce the likelihood that individualnodes will suffer congestion-related failure and ensure that the network as a whole willdisintegrate when failures do occur [6].

Another hierarchical design of interest is the Hypergrid topology, which also seemsto offer considerable advantages in respect of robustness. It appears to have the best con-nectivity robustness of all recent topologies. The absence of just a few highly connectednodes (homogeneous degree) makes the topology resilient enough to intentional attackswithout propagating propagating deficiencies of individual nodes [3].

It is only recently that scientists have started to study the behavior of network re-silience using models defined on random graphs and ideas from percolation theory. Toaddress the resilience of the network, scientists have studied the consistency of thegiantcomponent[12,15] on different types of topologies.

One of the first approaches to study node congestion is Benohamed’s control algo-rithm, which utilizes a non linear system, whilst applying a saturation function to imposebounds to traffic rate. It describes network load that consists of traffic corresponding toall source-destination pair – connection – flows. They study a single congested node,where the congestion is due to saturation of one of its outgoing links [18].

An alternative approach of studying node congestion control utilizes determinis-tic modeling which is based on the analytical study of Ordinary Differential Equations(ODE). Frank Kelly provided a flow control model and described how the flows operateas a result of the fine operation of the TCP-software at the packet-level [19] and how hismodel is extended utilizing concepts from game theory, in order to provide Quality ofService [20,21].

Additionally, TCP-Vegas is congestion control algorithm, based on a dual method,utilizing a discrete-time non linear system with a continuous update function. The mainconcept of this method is to treat source rates as primal variables and congestion mea-sures as dual variables looking for the global optimization [22].

In this paper, we propose a source control algorithm to adjust the sources’ ratesin order to fairly re-distribute traffic and prevent node overloading and thereby nodes’congestion. Various network topologies have been simulated by a numeric solution of thecoupled ODEs to prove that the system always reachesequilibriumand even when thetotal demand is larger than the network capacity the input source’s rates continue to adjustto avoid the overloading whereas the packets continue circulating across the networkreachingstable limit cycle. Therefore, our source algorithm dynamically adjusts sourcerates bases on both topology’s and sources’ characteristics. It monitors the nodes/linkscapacities via the link cost and traffic equations, which adjust their rates automaticallywhen the buffer tends to being full.

3. Model Formulation

We implement dynamic traffic distribution by assigning costs to the links connectingnetwork nodes. The link cost will depend on the amount of packets inside the node. Theload of each link depends on the cost. That is,the higher the cost, the fewer packets aresent through the link. Our goal is that all packets will receive the least possible congestedservice through theDATRAmodel depending on network conditions. On the contrary,packets willing to use an almost congested link would face ahigh cost. Thus, assuming


that packets choose the least congested path (denoted by the lower cost of the specificlink), we can prevent pathological conditions, such as congested links or failing nodesby intelligent, adaptive cost management and load balancing on the links.

Besides the obvious benefits (e.g. avoiding overloaded nodes) of smart cost man-agement, our model can offer other less obvious advantages such as Quality of Service(QoS). According to the model the packets are selected to be sent to an almost congestednode only when the packet is willing to pay the high link-cost. Our model identifieswhere there is link capacity, but does not specify which individual packets can use theavailable link capacity. Each packet’s header has a flag where the type of service it needsto receive is indicated. In other words, this flag indicates the packet cost/ quality it re-quires. Enabling packets to choose their link according to their flag indication results inQoS.

Here we outline our model, the parameters that it is dependent on, the concept it isbased on and the scope of its applications.

Consider a directed graph with bidirectional links and a set of nodes. A graph withn nodes is completely specified by ann × n matrix, which is calledadjacency matrixA = [aij ] consisting of zeros and ones. Each entryaij = 1 represents a connectionbetween the nodes i and j, whereas an entry such asaij = 0 implies that there is noconnection between the nodes i and j.

Consider single nodesi=0,1,2,3,...,nwhose number of packets at timet is denotedby xi(t). To keep the notation simple we denotexi(t) by xi wherexi ≥ 0, since thenumber of queued packets at any given node in the network is at least zero. The amountof packets at each node is specified by an1 × n matrix, X = [x1j ]. Finally, the initialnumber of packets at each node must be larger than its service rate (xi[0] > si).

Link cost from node i to node j is specified by ann × n matrix,C = [cji]. It partlycontrols the rate and the way in which packets flow through the network.

The arrival rates are specified by ann × n matrix,Λ = [λji], which represents thefraction of traffic relayed to each node. It is the routing probability that the traffic flowsfrom the same node through different links. We assume that the input of each node isdescribed in terms of the outputs from the nodes that are connected to it. The aforemen-tioned routing probability not only depends on the link cost, but also on the input andoutput of this link. In order to calculate this routing probability, it is first assigned valueswhich are produced by the formula:

Λ =1C

(1)

Then the routing probabilities are normalized and thereby their sum becomes equalwith unity. Finally, those routing probabilities are multiplied with the amount of trafficthat leaves the node producing the fraction of traffic that traverses each link, respectively.

The service rates are specified by an1 × n matrix, S = [s1j ] and depend on thenodes’ processing power. The change of the service rate depends on both the routingprobability (λij) and the link cost (cij).

The buffer spaces for each nodei are represented by an1×n matrix,B = [B1i]. B1i

controls how many packets are injected at each node in the network and at which pointHysteresis (immediate change) [23] takes place.

At nodei, scheduling and queue management mechanisms are responsible for thepacket processing and forwarding. Therefore, the loss caused can be avoided by prevent-


ing packets from entering that node. In the event that a network node overloads or fails,the network paths can be changed to compensate utilizing the DATRA model. As a resultthere are no fixed paths through the network.

3.1. Dynamic Traffic Rate Alterations

A major problem is the competitive demands that applications make for network re-sources, such as link bandwidth and buffer spaces in routers or switches [9,7,8,10]. Re-lated competitive interactions have also been studied in the context ofbiological popu-lation models – A. Lotka (1925) and Voltera (1926), the dimensionless system form ofwhich we partially utilize in this research.

We assume that once a node’s almost full buffer has emptied, the traffic rate thenincreases exponentially. This assumption plays a vital role in theburstinessof the net-work traffic. For instance, packets from NNTP, FTPDATA and WWW connections areclustered in bursts. These large bursts are likely to completely dominate traffic dynamicswhilst their inter-arrival times are regulated by network factors, such as shortage of essen-tial networking supplies [24]. We express resource competitiveness utilizing non-linearfunctions. These incoming traffic alterations are illustrated by the first part of Eq.(2),which is called the Logistic Differential Equation and which is commonly used in pop-ulation models, where the population growth rate decreases with increasing populationdue to various factors such as limited food supply, overcrowding and disease [25,26].Similarly, users and applications compete for limited network resources.

Conversely, when the buffer approaches being full, the traffic rate is taken to be zerootherwise the node will become congested. The node has to employ internal mechanismsto avoid congestion, otherwise the node will start dropping packets and the buffer willoverflow. In order to avoid the frequent use of those mechanisms, the arrival rate needsto be reduced to zero value at the appropriate times.

Traffic alterations at each node are modeled by nonlinear dynamic equations:

x′i =∑

j

λij · xj · sj ·(

1− xj

Bj

)− si ·

(1− xi

Bi

)· xi (2)

Particularly, the terms inside the brackets in the above equation decrease the trafficrate for large values of traffic, whilst the summed term represents the arrival rate of trafficfrom nodej to the nodei.

3.2. Dynamic Link Cost Alterations

Each link cost alteration is modeled by nonlinear dynamic equations, in order to deliverthe least congested service possible. The link cost strategy we apply provides a robustway to distribute the traffic amongst the nodes. In this way we prevent node overload-ing by dynamically adjusting the traffic and link cost rates. The link-cost to each nodedepends on its traffic.

Specifically, we assign a cost,cij to each link between the nodesi andj, which de-pends on the queue lengthxi at the nodej. The link cost increases rapidly with the outputqueue length and decreases rapidly with the input queue length, so that the packets areheld back when faced with a high link cost. Through the use of thesedynamically ad-justed traffic rate alterationswe ensure that all packets travel through the least congested


route and the packets should be forced to pay a very high cost if they want to use analmost congested link. Consequently, the link-cost function not only avoids an overloadin congested nodes but it also reduces the load of the relatively overloaded nodes:

c′ij = λij · sj ·[

x2i

1 + x2i

]− λij · si ·

[x2

j

1 + x2j

](3)

The term inside the squared brackets controls the flow of traffic. During busy periodsit allows the node to relay as many packets as possible, whereas during non-busy periodsit allows the node to relay the number of packets that are currently available.

3.3. DATRA Model on Multiple Nodes

By combining the non linear Ordinary Differential Equations (ODEs) of the traffic alter-ations coupled with the ODEs of the link cost alterations we achieve the following set ofODEs:

c′ij = λij · sj · x2i

1 + x2i

− λij · si ·x2

j

1 + x2j

x′i =∑

j

λij · xj · sj ·(

1− xj

Bj

)− si · xi

(1− xi

Bi

)

(4)

Over the next sections we observe, using two case studies, that our model’s resultsinclude extreme sensitivity to the initial conditions of the model, creating chaos, whichgenerates self-similarity along with odd periodicity.

4. Case Studies on Multiple Nodes

In this section we study the performance of our model through a series of simulationson multiple nodes forming diverse topologies. Solving the resulting nonlinear system ofODEs can be addressed through the use of numerical procedures especially developedfor solving ODEs. In the next section we utilize the classic R-K method of order fourapplied on a 64-bit computer to produce more accurate results. Even though our modelhas been tested for over 1000 nodes, for simplicity here we will illustrate our resultsthrough two case studies: the first based on a 3-node with 2-degree network; and thesecond based on a 20-node with 3-degree network.

4.1. Case study 1: Directed Graph of 3 Nodes With Degree 2

Consider a network of 3 nodes with degree 2

X = (250, 50, 100)S = (10, 10, 10)B = (450, 450, 450)


We assume that maximum number of packets to each node equal with their buffersizes, so that nodes do not overflow. If a node overflows, the program deactivates the nodeand its links. The deactivation of a node withholds the amount of packets it contains justbefore its deactivation. This phenomenon can be interpreted as adropping mechanism,in order to reduce the traffic in the network.

0100200300400

−2−1012

Core Node

0100200300400

Que

ue L

engt

h

−2−1012

Traf

fic R

ateNode 1

0 500 1000 1500 2000 25000

50100150200

0 500 1000 1500 2000 2500−1−0.500.51

Time (iterations)

Node 2

Figure 1. Solid line: Traffic alterations, Dashed line: Traffic rate alterations.The traffic rates reversals occur when the buffer approaches towards being full or idle, at which points it jumpsto a lower or higher value respectively. This phenomenon is well known in chaos theory asHysteresis

Figure 1. highlights the relationship between the traffic levels in each node and theirrate alterations. Each subgraph within Figure 1. consists of two phases –that of thetop-ping rateand that of thesinking rate, reflecting a two reversal pattern. Additionally, boththe total traffic and the rates oscillate rhythmically between upper and lower limits, whichare determined by the buffer capacity of the node. The traffic rates reversals occur whenthe buffer approaches towards being full or idle, at which point it jumps to a lower orhigher value respectively. This phenomenon is well known in chaos theory asHysteresis,which is the immediate jump that reduces the load of an overloaded node. After or beforeHysteresis both the traffic and the rate vary exponentially/logarithmically.

Figures 2. and 3. plot the phase portrait for traffic and its rates over time (iterations).Think of the trapezoid-shaped spiralas having four distinct phases: (i) a slow toppingup traffic phase; (ii) an immediate change of the traffic rate; (iii) a slow sinking trafficphase; and (iv) an immediate change of traffic rate.

Note that as time passes the trapezoid-shaped spirals are repeated in identical fashionother than the fact that their size becomes slightly smaller each time. Thisself-similarityis a reflection of the fact that our model is attempting to achieve dynamical balancebetween queue lengths and traffic rates among nodes and hence the trapezoid-shapedspirals tend towards the non-zero equilibrium. The self-similarity property applies acrossall nodes simultaneously so that they behave in a synchronized manner to achieve theequilibrium. Self-similarity is a well-known and established property of Internet trafficin the networking and modeling community [27,28].

In Figure 4., it appears that the nodes have reached equilibrium after roughly 2,000iterations. However, after closer investigation we observe that the link cost function con-tinues to vary even after 2,000 iterations. Note that the magnitude of the oscillationssteadily decreases as we approach the steady-state equilibrium.


0100

200300

−2

0

20

1

2x 10

4

Queue Length

Spiral of core node as it convergesto equilibrium

Traffic Rate

Time (

iterat

ions)

Figure 2. The trapezoid-shaped spiralhas four distinct phases: (i) a slow topping up traffic phase; (ii) animmediate change (Hysteresis) of the traffic rate; (iii) a slow sinking traffic phase; and (iv) an immediate change(Hysteresis) of traffic rate.

0100

200300

−2−1

010

1

2x 10

4

Queue Length

3D spiral of the three nodes as they converge to equilibrium

Traffic Rate

Tim

e (it

erat

ions)

Figure 3. Note that as time passes the trapezoid-shaped spirals are repeated in identical fashion, other than thefact that their size becomes slightly smaller each time. This self-similarity of traffic applies across all nodessimultaneously so that they behave in a synchronized manner to achieve the equilibrium. Self-similarity is awell-known and established property of Internet traffic in the networking and modeling community.

Recall that the number of packets that each node sends to other nodes affects thequeue length which influences the cost function. The cost function affects the traffic ratewhich describes the number of packets that each node receives. Given that thetrafficrate is inversely proportional to the link cost, thechange in the traffic rateis negativelyproportional to the link cost and hence the graph of Sparky DCost (change of the linkcost) in Figure 4. is a mirror image of the graph of the change in the traffic rate. Suchbehavior as the one described by this mirror image graph of the change in the traffic ratehas been observed in modeling studies of Internet backbone traffic where the cumulativenumber of packet arrivals appears to closely follow a piecewise-linear function [29]. Thespikes in Figure 4. highlight the points at which the rate immediately changes (Hystere-sis), whilst the height and direction of each spike represents the relative magnitude ofchange in the slope. Finally, we have identified the Sparky DCost pattern through graph-


2500 3000 3500 4000 4500 5000

−5

0

5

x 10−3

Time (iterations)

Delta

Cos

tSparky DCost

From node 1 to core nodeFrom core node to node 1

Figure 4. Sparky cost change over time. This mirror image graph of the change in the traffic rate has been ob-served in modeling studies of Internet backbone traffic where the cumulative number of packet arrivals appearsto closely follow a piecewise-linear function [29]. The spikes highlight the points at which the rate immediatelychanges (Hysteresis), whilst the height and direction of each spike represents the relative magnitude of changein the slope.

ing the change in cost, since the spikes are not distinguishable from simply graphing thecost (due to the large scale involved).

4.2. Case study 2: Directed Graph of 20 Nodes With Degree 3

The conclusions from our first case study hold for large number of nodes and large de-gree. Through both case studies we aim to show that the DATRA model is independentof topology. Consider a homogeneous tree topology of 20 nodes with degree 3.

X = (250, 30, ..., 50)S = (10, 10, ..., 10)B= (1450, 1450, ..., 1450)

Unlike the first case study where there were enough links to ensure that all nodescould potentially be allocated with packets at the first iteration, in this case study thereare only enough links to allow three first neighbor nodes to be allocated with packetsat the first iterations. Consequently, the traffic at the three first neighbor nodes increaseexponentially over time, whereas the traffic at the more distant nodes from the core nodeoscillate at lower magnitude since much of the distribution of packets has already takenplace at earlier node levels. This is illustrated in Figure 5.

The first graph in Figure 5. illustrates the traffic and its rate alterations at the corenode, whereas the second graph illustrates the traffic and its rate alterations at the 17thnode. The curves in this case study exhibit similar phenomenon as the previous casestudy. Any differences are simply attributable to the different input, such as networkdynamics.

Figures 6 and 7 plot the phase portrait for traffic and its rates over time (iterations).Think of a spiral consisting of trapeziums that are repeated in an identical fashion otherthan the fact their size becomes slightly smaller or larger each time step. The differencesin their sizes are attributable to the different input, such as network dynamics. This self-similarity is a reflection of the fact that our model takes into consideration network fac-


0

500

1000Qu

eue L

ength −20

0

20

Traff

ic Ra

te

Core Node

0 0

40

80

120

160

0 200 400 600 800 1000−2

−1

0

1

2

Time (iterations)

Node 17

Figure 5. Solid line: Traffic alterations Dashed: Traffic rate alterations

4050

6070

−10

12

1500

2000

2500

3000

Queue Length

Order in disorder (Chaotic Spiral)

Traffic Rate

Tim

e (it

erat

ions

)

Figure 6. The trapezoid-shaped spirals are repeated in identical fashion other than the fact that their sizebecomes slightly smaller or larger each time step

tors (e.g. available bandwidth) which leads to a process quite different from memorylesspacket arrivals. Moreover, in Figure 7. nodes behave in a synchronized manner, similarto that observed in modeling studies of TCP/IP [30].

In Figure 8., it appears that the nodes have reached equilibrium after roughly 5,000iterations. The spikes in Figure 8. highlight the points at which the rate immediatelychanges, whilst the height and direction of each spike represents the relative magnitudeof change in the slope.

5. Conclusions and Future Work

‘No man is an island’ (John Donne). We make our way in the world through humannetworks, and thereby networks play a vital part in our lives. Failing networks can causesevere adverse consequences.

Our overarching goal is to develop a traffic-control model. According to our model’sstrategy, we control the distribution of traffic to available network resources and thereby


4060

80

−10

12

1500

2000

2500

3000

Queue Length

Synchronised Chaotic Spirals

Traffic Rate

time (i

teration

s)

Figure 7. Nodes behave in a synchronized manner, similar to the one observed in modeling studies of TCP/IP[30]

2000 3000 4000 5000−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Time (iterations)

DCos

t

Change of Cost

Figure 8. Sparky Cost change over time (last 3000 iterations). The spikes highlight the points at which therate immediately changes (Hysteresis), whilst the height and direction of each spike represents the relativemagnitude of change in the slope

minimize network failures by disallowing resource overloading. Traffic rates are dynam-ically adjusted based on link costs so that all network nodes behave in a synchronizedmanner. Synchronized reconfiguration ensures that deviations from equilibrium are pre-vented (provided that the exogenous parameters allow equilibrium to be reached). Thisstrategy not only distributes the traffic load amongst network nodes, but also synchro-nizes the traffic and link-cost changes with buffer and topology variations.

Our model currently takes into consideration only the node’s buffer capacity. In thefuture we intend to extend our analysis to consider other factors such as link capacity.Additionally, throughout our model’s description, we use the concept of packets havingneither a destination nor a source, and all having equal size. In particular, these packetscirculate around the network until the network reaches equilibrium. If there is not enoughbuffer space for them, then the packets will continue circulating for ever reaching a limitcycle, instead of equilibrium. In future work, the model will be extended to allow us touse a variety of routing strategies, where the packets not only will have a appropriate


size, but also they will be generated by a source node sending them to a destinationnode. Taking into consideration that networks change in time and thereby are productsof evolution, we intend to incorporate this evolving feature of real-world networks intoour model.

Achieving intelligent traffic control in today’s networks depends on our ability tocorrectly model the mechanisms involved in the dynamics of complex networks. As nu-merous studies over the last decades have shown this is an overwhelming intellectualchallenge. We believe, our approach is a first step towards this direction.

6. Acknowledgments

The research was funded by General Dynamics and BTexact. The authors would like tothank Prof. Patrick Purcell, Matthew Johnson and Thomas Karagiannis for constructivetechnical discussions and comments.

References

[1] Western Systems Coordinating Council (WSCC): Disturbance report for the Power SystemOutage that occured on Western Interconnection at 1548 PAST,http://www.wscc.com, (1996).

[2] http://www.pewinternet.org/pdfs/LoveBug.pdf.[3] Fabrice Saffre: Adaptive Security and Robust Networks,Information Security Bulletin, 7, 11,

(2002).[4] Yiong Chen, Zhongshi He: Bounds on the Reliability of Distributed Systems with Unreliable

Nodes and Links,IEEE Transactions on Realibility, 53, 2, (2004).[5] Karp, Koutsoupias, Papadimitriou, Shenker: Optimisation problems in congestion control,

41st IEEE Symposium on Foundations of Computer Science, (2000), 66–74.[6] S. H. Strogatz: Information exchange and the robustness of organizational networks,Nature,

100, 21, (2003), 12516–12521.[7] N.Haque, N.R. Jennings, L. Moreau: Resource allocation in communication networks using

market-based agents,Knowledge-Based Systems, 18, (2005).[8] M.A. Gibney, N. R. Jennings: Dynamic Resource Allocation by Market-Based Routing in

Telecommunications Networks,2nd Int. Workshop on Multi-Agent Systems and Telecommu-nication, (1998), 102–117.

[9] E. Altman, H. Kameda: Equilibria for Multiclass Routing in Multi-Agent Networks,Ad-vances in Dynamic Games, 7, (2005), 343–368.

[10] M.A. Gibney, N.R. Jennings, N.J. Vriend, J.M. Griffiths: Market-Based Call Routing inTelecommunications Networks using Adaptive Pricing and Real Bidding,Third InternationalWorkshop on Intelligent Agents for Telecommunication (IATA), (1999).

[11] D. J. Watts and S. H. Strogatz: Collective dynamics of "small-world" networks,Nature, 393,(1998), 440–442.

[12] M. E. J. Newman: The structure and function of complex networks,SIAM Review,45, 2,(2003), 167–256.

[13] Albert-Laszlo Barabasi and Reka Albert: Emergence of Scaling in Random Networks,Sci-ence, 286", (1999), 509–512.

[14] M. E. J. Newman,S. H. Strogatz, D. J. Watts: Random graphs with arbitrary degree distribu-tion and their application,Santa Fe Institute, Working Papers, (2000)

[15] S.N Dorogovtsev, J. F. F. Mendes: Evolution of Networks,Advances in Physics, 51, 4,(2002),1079–1187.


[16] B. Bollobas: Random Graphs, 2nd Edition, Cambridge University Press (2001).[17] Michalis Faloutsos and Petros Faloutsos and Christos Faloutsos: On Power-law Relationships

of the Internet Topology,SIGCOMM ’99, (1999) 251-262.[18] Lotfi Benmohamed, Semyon M. Meerkov: Feedback Control of Congestion in Packet Switch-

ing Networks: The Case of a Single Congested Node,IEEE/ACM Transcactions of Netawork-ing, 1, 6, (1993), 693–708.

[19] Jacobson: Congestion avoidance and control,ACM SIGCOMM, 88, (1988), 314–329.[20] Peter B Key and Derek R McAuley: Differential QoS and Pricing in Networks : where flow-

control meets game theory,IEEE Software, (1999)[21] Frank Kelly: Mathematical Modelling of the Internet,Springer-Verlag, (2001), 685–702.[22] Steven H. Low, Larry L. Peterson, Limin Wang: Understanding TCP Vegas: A duality Model,

Journal of ACM, 49, 2, (2002), 207–235.[23] Steven. H. Strogatz: Nonlinear Dynamics and Chaos,WestView press, (2000).[24] Vern Paxson, Sally Floyd: Wide-Area Traffic: The Failure of Poisson Modeling,IEEE/ACM

Transactions on Networking, (1995).[25] B. Drossel, P. G. Higgs and A.J. McKane: The influence of Predator – Prey population dy-

namics on long–term evolution of food web structure,Theoretical Biology, 208, (2001), 91–107.

[26] V. Krivan, J. Eirner: Optimal foraging and Predator-Prey Dynamics III,Theoretical Biology,63, (2003), 269–279.

[27] W. E. Leland, M. S. Taqqu, W. Willinger, and D. V. Wilson: On the self-similar nature ofEthernet traffic (extended version),IEEE/ACM Trans. Networking, 2, (1994), 1–15.

[28] Mark E. Crovella, Azer Bestavros: Self-similarity in World Wide Web traffic: evidence andpossible causes,IEEE/ACM Transactions on Networking, 5(6), (1997), 835-846.

[29] Thomas Karagiannis, Mart Molle, Michalis Faloutsos, Andre Broido: A Nonstationary Pois-son View of Internet Traffic,IEEE INFOCOM, (2004).

[30] Andras Veres, Miklos Boda: The Chaotic Nature of TCP Congestion Control,INFOCOM, 3,(2000).

Intelligent traffic control on internet-like topologies

Documents

Transcript of Intelligent traffic control on internet-like topologies