Modeling and Clustering Analysis of Broadband Convergence Networks

Modeling and Clustering Analysis of BroadbandConvergence Networks

Vladimir Denchev, Franz Pemkopf, Dimitar Radev

University of Rousse "Angel Kanchev", Studentska Str. 8, 7017 Rousse, Bulgaria2Graz University of Technology, Infeldgasse 16c, A-8010 Graz, Austria

{ vndenchev@yahoo&om, pernkopf@tu , dradev@abvbgI

Abstract. In contemporary telecommunication systems Markov processes areseldom observed, and the widely used Markovian models don't representprecisely the real system. In order to omit the need of modeling the system witha Markov chain we apply different clustering approaches for obtaining thesteady state probabilities, which are represented by the data clusters. Somewidely used data clustering methods are applied for performance evaluation ofdifferent telecommunication networks. However, in order to accomplish ourinvestigation, we conduct our research with Markovian models, so that we havea solid ground for comparison, although the benefits of applying clusteringtechniques lie in the domain of the non-Markovian processes.

Keywords: Telecommunication Systems, Queueing Networks, MarkovianModels, Clustering Algorithms, Rand Index

1. Introduction

A main problem in the field of telecommunication engineering is the need to definethe performance measures of communication networks when traffic data from thisnetwork is given. The widely used approaches, based on Markovian models are notoften applicable because the system, where this traffic data originates form can'talways be described by Markovian models, and even if that is possible with someassumptions, the number of the obtained steady-states will be too big and this yieldsthe solution of a system with linear equations with too many unknowns.Under these circumstances, the usage of data clustering algorithms can be verybeneficially and good results can be obtained. The problem is that the simpleapplication of clustering algorithms to original data cannot give an accurate estimateof the obtained precision, since there is no ground for comparison.In our work we investigate open queueing networks represented by Markov chainsand apply to them clustering analysis in order to obtain the steady-state probabilityvector because we can derive from it most of the important performance measures ofthe investigated system. The aim that is set to this paper is to define the applicabilityof some widely used clustering algorithms for 3D networks of parallel queues,representing various telecommunication systems with different entrance flows andlevels of overlap.

2 Vladimir Denchev, Franz Pernkopf, Dimitar Radev

2. Simulation Modeling Approach

A big number of data clustering algorithms exist and the conducted experiments giveuseful information about their behavior and applicability, but in the field oftelecommunications no such research to our knowledge has been done so far.Therefore it is not clear what their domain of application is for steady-state vectorestimation in telecommunication networks - some heuristics can be applied but still itis important what algorithms should be applied for different network configurations,or different telecommunication systems, or different traffic flows. It is known, thatartificial traffic flows can be generated with various statistical distributions andvarious parameters.The simulation modeling approach for steady-state probability vector evaluation ofbroadband convergence networks is considered in fig. 1.

Broadbanid netwoirkQueueing mode

Simuflation mo)dellingCMarkov c ain

CSampling=

Xustering analy

teady-state ve

Performance measuires

Fig. 1. Simulation modeling approach

The approach starts from a telecommunication system, that we want to investigate.Telecommunication systems can usually be presented by means of networks ofqueues. Of course contemporary communication networks can be complex with a lotof service stations and a lot of customers, but they can usually be simplified and bereduced to some well-known queueing models, which can be presented by Markovchains. To do so, the memoryless (Markov) property must be present in the system,that is, the future development of the process depends only on the current state, butnot on the previous states. This doesn't always hold, especially for contemporarytelecommunication systems, where long range dependences and heavy-tailed trafficwith deterministic service times are observed. But some assumptions can always bemade, which allow us to construct a Markov chain that corresponds to the queueingmodel with a given precision.The constructed Markov chain is used for generation of samples. For this purpose wederive the generator matrix Q of the Markov chain and use it to obtain the transitionprobability matrix P, which on its part is used to obtain the cumulative probability

Modeling and Clustering Analysis of Broadband Convergence Networks 3

matrix cumProbP. This matrix is used for the generation of a given number ofsamples with several different statistical distribution laws and parameters,representing various traffic flows. The sampling procedure is explained with a simpleexample. In figure 2 is depicted a part of a Markov chain, representing two parallelqueues.

6

7

4 5

Fig. 2. Markov chain model, representing two parallel queues.

States 2, 3 and 6 are depicted in the figure. The Markov chain is represented in termsof a graph - the states are depicted as circles and the edges are the arrows between thefeasible states. The transition intensities are written next to the arrows and in thecircles is the number of calls in the two parallel queues. The number of the state isrepresented with a digit in the bottom of each circle. From the Markov chain wecreate the Generator matrix, which we use to write down the transition probabilitymatrix, from where we obtain the Cumulative transition probability matrix (table 1):

Table 1. Cumulative transition probability matrix for a system with two parallel queues1 2 3 4 5 6 7 8 9

1 0.52174 0.69565 0.69565 1 1 1 1 1 12 0.21739 0.52174 0.69565 0.69565 1 1 1 1 13 0 0.21739 0.69565 0.69565 0.69565 1 1 1 14 0.26087 0.26087 0.26087 0.52174 0.69565 0.69565 1 1 15 0 0.26087 0.26087 0.47826 0.52174 0.69565 0.69565 1 16 0 0 0.26087 0.26087 0.47826 0.69565 0.69565 0.69565 17 0 0 0 0.26087 0.26087 0.26087 0.82609 1 18 0 0 0 0 0.26087 0.26087 0.47826 0.82609 19 0 0 0 0 0 0.2609 0 0.2174 0.5217

Each row corresponds to one of the nine possible states for this system. The possibletransitions, when we are in a given state are represented with a change of the numbersin each row. As we see from table 1, there are three possible actions when we are instate three - we can move either to state 2 or 6, or stay in state 3. The transition tostate 2 is achieved by serving of a service request in server 2 and as a result atransition to state (0,1) is performed. The transition to state 6 is achieved by entranceof a new service request in server 1. For the purpose a new call is generated from oneof the following statistical distributions: Uniform, Gaussian, Pareto, Poisson,Exponential, Binomial or Gamma - these are the statistical distributions that mostoften describe the processes of arrival of service requests in telecommunicationnetworks. In this way the transition probabilities are followed and the states of theMarkov chain are visited. This process continues till a given number of service


requests are generated, which defines the precision of the steady-state probabilityvector, obtained during the sampling process.While sampling the targets, the bandwidth of the different jobs is also taken intoaccount. We assume that the size of a job from a given service class can be within agiven interval, i.e. a maximum and minimum bandwidth for the service requests fromthe different classes is introduced. This allows the simulation of different types oftraffic flows, and during the sampling this is achieved by truncation of the differentstatistical distributions. Recommendations for the characteristics and requirements forthe traffic flows in the various types of telecommunication networks can be found inthe documents of ITU-T.Another important issue in sampling is the physical nature of the generated samples.In practice, the size of a service request is the number of busy cells in the queues ofthe server in a multimedia gateway, or a WATM hub, which is represented by anatural number (i.e. a positive integer). Often the cells are not well filled by theentering calls, which lead to losses. The multiplication of these losses with a bignumber of service requests leads to big errors and therefore the use of real numbers intraffic flow sampling is imposed. The real numbers represent the exact bandwidth,used by a given call. The calls, generated with a given bandwidth are led into thesystem in terms of a flow of arrival service requests. When a given number ofsamples are reached the process is terminated and a data cloud is obtained, which isrelated to the Markov chain, describing the investigated telecommunication system.The belonging to the corresponding state of the Markov chain is known for eachgenerated sample, which allows the calculation of the steady-state probability vectorof the corresponding Markov chain (and the precision of this vector depends on thenumber of the generated samples).The next step in the Simulation modeling algorithm is to apply clustering analysis butat this point we do not use the knowledge of the samples' assignment, but applyunsupervised clustering. We use the following algorithms: k-means (with Euclidianand Manhattan distances), Fuzzy c-means, Expectation-Maximization (EM) and Self-organizing Map (SOM) [1], [3], [6].We obtain new assignments for the samples after performing the clustering algorithmsto the different data sets. Having the old assignments, obtained from the sampling, wecan apply the Rand index, which is a measure for correctness of the obtainedclustering, which compares the old and new assignments of the samples [4]. We alsouse another measure for evaluation of the obtained results - the relative error betweenthe steady-state probabilities of the generated samples and the one obtained by theclustering algorithms. The computed comparative measures allow us to define whichtype of algorithms are preferable and showing better results for different types ofcommunication networks under different overlap conditions and different types ofservice request flows. This information is very useful for solution of the presentedproblem with a real-world data.The last step in the Simulation modelling algorithm is to derive different performancemeasures from the obtained steady-state probabilities. For differenttelecommunication systems, various performance measures can be obtained, which isnot a subject of this investigation.


3. Rand Index for Evaluation of Performed Clustering

In data clustering the accuracy of the clustering algorithms is searched and by theterm "accuracy" of a clustering algorithm, we assume the similarity of the obtainedclustering to a known labeling of the data. Such labeling is available from theartificially generated data sets. In order to use data sets with known class labels, wehave to make sure that the structure of the Markov chain corresponds to the clusters indata. Many authors have used real benchmark data sets with known class labels toevaluate clustering algorithms, but we follow another approach - we use the sampledassignments from the generated transitions following the transition probabilities of theMarkov chain and compare them to the assignments, obtained from the unsupervisedclustering algorithms.There are many indices evaluating the match between two partitions, from amongwhich we selected the adjusted Rand index [4]. This index takes value 1 if thepartitions are identical and has an expected value of 0 if they are drawn independentlyof one another, regardless of the number of clusters.Let A and B be partitions of Z with kA and kB clusters, respectively. Let n, be thenumber of objects in cluster i in partition A and mj be the number of objects in clusterj in partition B. Denote by nii the number of objects which belong simultaneously tocluster i in partition A and cluster j in partition B. The adjusted Rand index iscalculated as

AR(A,B)= 'i=l ~j=\K I/ (1)

where

l =- 2 t2 =K, JJ and t3 = N(N-1) (2)

4. Model of Investigation

A typical feature of digital broadband networks is the simultaneous realization ofvarious different services, ranging from ordinary telephony to interactive multimediateleconferences in the infrastructure of the same network. Thus, ATM is wellrecognized as an ideal network technology for support of integrated services, whichsuggests that an integrated services network is capable of observing the given QoSrequirements, while maintaining permissible blocking probabilities. The variousservices, proposed to the customers have different bandwidth requirements, whichmay vary from some Kbit/s for voice data to hundreds of Mbit/s for videoapplications, as well as different arrival and service time intensities.


An important aspect in integrated services networks management is the bandwidthaccess control at connection level. Different access control schemes have beencreated that aim to manage the access of new calls to the network resources and try toavoid one of the traffic classes to predominantly use the system capacity and thusdepriving other traffic classes from being served. For integrated services applicationstwo bandwidth access schemes are used - complete sharing (CS) and partial sharing(PS). CS allows unlimited sharing of the whole bandwidth between all the trafficflows so that a new call is accepted if there is available bandwidth. In partial sharing apart of the bandwidth is dedicated for all the traffic classes, while the rest is dedicatedfor specific classes. For the sake of proper modeling, modified variations of these twobandwidth access schemes are used.The Partial Overlap bandwidth access scheme is defined in the following way: thetraffic of service i receives rim, units of bandwidth and all of the rest services competefor the remaining capacity of C-nlml-n2m2-n3m3 units (for a 3D case). The enteringtraffic is accepted for service i, if there is mi available units in the dedicated capacityof rim, units or in the shared capacity of C-n1m1-n2m2-n3m3 units, otherwise theconnection is blocked and the call is lost.

4.1. Characteristics of Internet Traffic

The vast majority of traffic on the Internet relates to the transfer of jobs (web pages,audio/video downloads, file transfers, etc.) coordinated by TCP. Trafficdifferentiation helps to lower the variance in the connection throughput and maintaina more uniform throughput [5]. This is important because the service becomes morepredictable and steadier. Therefore we decide to discriminate between four differenttypes of TCP traffic flows - video, VoIP, FTP and WWW, as it is shown in fig. 3.

J ackground

|Integrated servicles |/I Video|TCP traffic flow Real time ¢

) on-real time, P-

Fig. 3. Types of traffic flows in TCP

Two main groups of traffic flows can be discriminated - elastic and real time.Elastic applications - The elastic applications benefit from increased bandwidth butthey are able to operate with only a minimal amount of network resources. It is a well-known fact that the Internet traffic is heavy tailed which means that Internet traffic iscarried by a small number of long lasting connections (elephants) while a largeportion of the connections is short in lifetime (mice). Typical elephants are P2Papplications and FTP file transfers. The applications using short connections areusually bursty and interactive in nature (they need their data within a certain timelimit). A typical application with interactive requirements is the Word Wide Web.


Real-time applications can be categorized into two main groups: hard real-time andsoft real-time. Applications with hard real-time requirements need their data within astrict delay period and data with larger delay is useless and is discarded. Typicalapplications have conversational properties - telephony and video confererencing.Applications with soft real-time requirements are more robust to changes in delay andbandwidth. Typical examples are streaming media applications that do not requiretwo-way communication. The encoding schemes of Video traffic can be divided intotwo categories: 1) constant bit rate (CBR) and 2) variable bit rate (VBR). CBR videomaintains the transfer rate during the transmission on the same level varying onlylittle over time, while VBR video may have peak values that differ in many Mbpsfrom the average rate. In that sense CBR video encoding is more predictable andfacilitates the network resource management.

4.2. Traffic Flows Generation

Traffic generation aims to produce a realistic mix of applications found in the modernInternet that covers the basic traffic types. Several traffic generators have been createdto generate samples simulating a wide range of applications. The generated trafficflows are as follows: Voice over IP; Video streaming; World Wide Web and FileTransfer Protocol. It should be noted that a small part of the link bandwidth (usually2%) is reserved for the background traffic, which is used for clocks synchronizationand other official information, but since this is a constant value, this traffic flow is notmodeled [2]. The traffic flows are generated with certain probability distribution laws(Table 2):

Table 2. Statistical distribution laws, used for TCP traffic flows generation

Distribution Parameters Traffic flows TCPUniform [0, 1] CBR streams (G.71 1) Voice over IPPoisson X = 2 Short TCP requests World Wide WebPareto a =1.8 Real-time Video Video streamingGeometric p=0.I Long TCP requests File Transfer ProtocolBinomial n=1.8; p=0.6 ON/OFF sources (G.723.1, Voice over IP

G729 B, GSMFR)

These distributions describe various types of entrance flows of calls and are amongthe most common statistical distributions, observed in telecommunications. Theirparameters have been chosen to fit well the traffic flows that they characterize. All thedistributions are normalized in compliance with the job size that they simulate. Itshould also be noted that Uniform and Pareto are continuous distributions and need tobe multiplied with a discrete uniform distribution in the interval [0,1] On the otherhand, Poisson, Pareto and Geometric distributions are characterized with long-rangedependences, so they had to be truncated, in order to eliminate some undesired effectsduring the simulation process.


4.3. Internet Traffic Model with Partial Overlap Bandwidth

For modeling of systems with integrated services a Markov chain is used. The classeswith guaranteed bandwidth need to be modeled, i.e. the constant, adaptive and elasticand the stochastic model is presented by a network of three parallel queues in the statespace - fig. 4.

Real timeConstant/'Variable b1

22

Non-real time 23b3 Elastic brin , /13

Fig. 4. Model for partial overlap of the main traffic classes

The model state is characterized by the arrival intensities (4J, i 2' i 3) and service timeintensities (j'I, U 27 U I)4 the bandwidths (bl, b2, b3) and their minimum bandwidthrestrictions (Sm I ,0mi) and also by the corresponding number of flows (nl, n2, n3).The simultaneous presence of constant traffic classes with constant bit rate CBR,adaptive traffic flows with variable bit rate VBR, or available bit rate ABR and elastictraffic flows with available bit rate ABR doesn't allow an equivalent distribution ofcapacity in the wideband channel. Considering the above model of partial overlap andthe results in table 2, four different TCP teletraffic systems that are often met inpractice are introduced in table 3.

Table 3. Traffic flows in 4 investigated TCP systemsTraffic c1las System 1 System 2 System 3 System 41

Voice over IP - CBR T * l_l 1Voice over IP -ON/OFF T I_._ 1Video streamingWorld Wide WebFile Transfer Protocol

5. Simulation Results

An open queueing network with three servers is considered where service requestsfrom different independent traffic flows arrive with intensities 2 = 5,X2 = 7 X3 =4' X

and are served with intensities ,uA =~8,2= 6, /13= 5 correspondingly (fig. 5a). Six


possible actions exist for the described system - entrance of service requests in queue1, 2 and 3 and services form queue 1, 2 and 3. The state transition diagram is depictedin fig. 5b. There are 27 steady states in total since we introduce a limitation of amaximum of 2 customers in each queue. A new service request can enter in a queueonly if it is not full.

-4- _ 6

4 6 6(3 @ _ 6 X 4 w2 1 3 6 ~~~~~~~~~7-)a~~~~~~~~~~~~

Al =5 ul IT5 5';sFig. 5. Teleommunicatio 5 swa)t ia (8

_ -_ _ _ 2 1 1 7 > 5 1 +=S 6 4

wakn ston an t6 is r a u ad m u p_ _ _ _ 5 St5 8.8foras . We w w r t u 5

.3=4 3 =5 4X - -; - ---6+_L-

b 7~

Fig. 5. Telecommunication system with three parallel servers (a); State transition diagram (b).

Following the simulation modeling approach we generate 10 000 samples for thedifferent systems according to various statistical distributions with differentparameters, as it is shown in table 2.Three levels of overlap are introduced - tight (which coabesponds to no-overlap case),weak and strong and this is realized by giving a minimum and maximum permissiblebandwidth for a sampled call. We work with relative bandwidth units (BU) and bmin =5, 4, and 3 BU for the different levels of overlaps, while bmax =10 BU is kept fixed forall scenarios. The idea of overlap arises from the statistical multiplexing, whichallows the simultaneous transmission of different flows, whose total bandwidthexceeds the total available bandwidth, but still they can be transmitted because theirbandwidth varies between a minimum and a maximum margin.Using the results from the sampling procedure, the steady-state probabilities arecomputed. Applying the clustering algorithms: K-means (Euclidean and Manhattandistances), EM, Fuzzy C-means and SOM enables us to compute the steady-statevectors obtained by the different algorithms (fig. 6). These results allow us to evaluatethe Rand index and the relative error which is computed according to a standardformula. The obtained results allow us to give recommendations for appropriateapplication of the clustering algorithms under different circumstances.We apply this procedure for four different telecommunication systems and threelevels of overlap, which is a total of 12 times. The obtained steady-state vectors for 12cases and 5 clustering algorithms are informative but an estimate of proximity to theactual steady-state vector needs to be done (table 4).


Case: 1 Overlap: 1 k-means clustering (euclidean distance) k2means clustering (cityblock distance)

20202

Nl 2 t Nl20

EM c0lustering uzzy E-mesns clustering SOM Ilustering

20 2

Fig. 6. Results of the obtained clustering with different algorithms for system 1 and tightdistribution of the teletraffic flows.

The algorithm with best performance for each particular case is high lightened and thedata can be used as a guideline for application of a given algorithm in one of the fourconsidered systems under different conditions for the entrance distributions and levelsof overlap. The Rand index is considered to be the more informative measure forperformance evaluation of the clustering algorithms. Considering the mean value ofthe rand index, which is the last column in table 4, best is algorithm 4, followed byalgorithms 1 and 5. Algorithm 1 shows best results in 4 out of 12 cases, next arealgorithms 2 and 5 with 3 out of 12 best performances, while the best overallalgorithm 4 is at the top, just twice. This fact demonstrates the high performancehomogeneity of this algorithm, which suggests that it is the most reliable one.

Table 4. Rand index and relative error, computed for 4 telecommunication systems, 5clustering algorithms and 3 levels of overlap.

_ I 11~~tght weaktron0g1 tight weaktron0g tight weaktrnti041ght weak tron 0gMaK-means Eci. 0,941 0,826 0,705 0,979 0,831 0,691 0,977 0,828 0,704 0,887 0,705 0,622 0,8080K-meansMMnh 0,943 .0,816 0,703 0,953 0,837 0,677 0,978 0,834 0,676 0,843 0,708| 0,614 0,7984EM 0,932 0,735 0,641 0,875 0,676 0,596 0,900 0,721 0,573 0,719 0,637 0,549 0,7129Fuzzy Cmeans; 0,947 0,833 0,698 0,977 0,833 0,69 0,976 0,823 0,684 0,922 0,710 0,628 (081005SeM 0,950 0,846 0,659 0,967 0,828 0,687 0,976 0,8261 0,18 0,913 0,236 0,623 080427AE MW 00943 0,21 0 1 0122 0,8 1 07668 0963 01,06 0,59 0,852 0 0465 0,3863

Fuzzy CEan1. 0,048 0,115 0,189 0,027 0,108 0,228 0,024 0,102 0,217 0,066 0,158 0,16 0710361](4indd MW 0,048 0, 124 0, 189 0,036 0, 107 0, 153 0,025 0, 1 11 0, 184 0,J138 O0,236 O0,245 1 lMU 0,059 0,265 0,37 1 0, 122.0,28 1 0,47 0,083. 0,306 0,55 O0,3 82 O0,443 O0,46541Wz Ge# 0,045 ,1JO7 0,166 0,025 0,094 0, 132 0,024 0, 1 01 0, 188 %066 07158 oJ612 Oi 65SOM 0,043 0,159 0,199 0,023 0,064 0,067 0,015 0,079 0117 0,071 0,160 0,171 0,092e iiivalu 0,047 0154 0,223 _0,046 013 1 0,210 0,034 04 02 1 0,158 0,238 244 01565


The rand index takes the biggest value for cases with no-overlap of traffic flows, orwhen we observe tight distribution of the traffic flows. The best result is obtained forsystem 3 - 0.9613. When the overlap increases, the value of the rand index decreasesand we obtain the worst result for system 4 in the strong overlap case - 0.6072.The results obtained for the relative error show best performance of algorithm 5,followed closely by algorithm 4. The rest of the algorithms fail to dominate in any ofthe considered cases.We perceive an estimation procedure, which takes into account the top threeperforming algorithms in all systems and levels of overlap. The best algorithm isawarded with 3 points, the second best takes 2, while for the third best algorithmremains 1 point. We observed that algorithm 4 performed 7 times best, 13 timessecond-best and 3 times third-best, which mean that it is one of the top-threealgorithms in 96% of all cases. The results are presented in table 5.

Table 5. Overall performance of five clustering algorithms according to the investigatedsystem, the level of overlap, Rand index, relative error and in total, measured in points.

IK4&h (ada) 8 |8 |9 |5 9 |8 |13 20 |10 50)2 Kfmdhn Mahhj 7 5 9 1 5 9 8 14 8 363 0EM 0 0 0 0 0 0 0 0 0 04_ it ihmns 113 1 12 8 17 17 1 16 17 22 28 725 SOM 9 11 10 13 17 115 1 11 16 27 59

Table 5 consists of four main parts - the algorithms are appraised according todifferent communication systems, levels of overlap of the entrance distributions, therand index and relative error and in total. It is worth mentioning that the points fromthe rand index are double weighted in comparison with the relative error.The first part of the table investigates the performance of the clustering algorithmsapplied to the four various telecommunication networks. Generally, all the algorithmshave a homogeneous presentation with all the systems, except for algorithm 2, whichexhibits difficulties with system 4. Algorithm 4 performs best in systems 1, 2 and 4,while for system 3 is recommended algorithm 5.The second part of table 5 deals with the various levels of overlap. The trends,observed in the previous part of the table are valid here too. Best performance in allcases shows algorithm 4, but in the "tight" case, the dominant position is shared withalgorithm 5, which on the other hand decreases its performance in the strong overlapcase. Algorithm 1 improves its presentation with increased overlap.The results in the third part of the table show that according to the both parameters,the best algorithm is 4. Next are algorithms 5 and 1 and while according to the randindex, algorithm 1 is better, the relative error suggests that better is algorithm 5.Not surprisingly, the totally best clustering algorithm out of the 5 applied according toour research is algorithm 4, which collects 72 points. The second-best is algorithm 5with 59 points, followed by algorithms 1 and 2 with 50 and 36 points,correspondingly. Algorithm 3 doesn't demonstrate sufficient performance and is notawarded any points. The reason for this result can be found in the types of thegenerated traffic flows - it is known that the EM algorithm is constructed for


Gaussian mixture models and therefore shows good results for statistical distributionswith similar character like Poisson or Binomial, and shows bad results withdistributions with exponential character like Pareto and Geometric. The investigatedsystems have no suitable combinations and therefore, the application of this algorithmis limited to traffic models, characterized with suitable statistical distributions.

6. Conclusions

The obtained conclusions are useful, but we have to bear in mind that, theircorrectness depends on the accurate modeling and under various conditions we mayobtain slightly or roughly different results. The problem for clustering of 3D Markovchains is reduced to structure recognition in a data array, which defines the belongingto one ofM target classes S1,...,SM. Each entering stochastic sample can be presentedby a given cluster, so that to each steady-state of a Markov chain corresponds onedefinite cluster. The weight centers of the obtained data clusters correspond to thesteady-states of the embedded DTMC and therefore the clustering procedure isconnected with evaluation of a predefined number of weight centers, and thusobtaining the steady-state probability vector of the model. The use of clustering forstructure recognition in arrays of data contributes to the effective determination ofnodes in embedded DTMCs and as a result the marginal probabilities in the statespace can be easily obtained.

Acknowledgments. The authors would like to thank scientific project BU-TH105/2005 of the Bulgarian Science Fund.

References

[1] Buhmann J. (2002) Data Clustering and Learning, Handbook of Brain Theory and NeuralNetworks, M. Arbib (ed.), 2nd edition, MIT Press.

[2] Dang T. D., Sonkoly B., Molndr S. (2004). Fractal Analysis and Modelling of VoIP Traffic,NETWORKS 2004, Vienna, Austria, June 13-16.

[3] Figueiredo M., Jain, A.K. (2002). Unsupervised learning of finite mixture models. IEEETransaction on Pattern Analysis and Machine Intelligence, 24:381--396

[4] Kuncheva L.I., D.P. Vetrov, (2006). Evaluation of stability of k-means cluster ensembleswith respect to random initialization, IEEE Transactions on Pattern Analysis and MachineIntelligence, 28 (11), 1798-1808

[5] Lassila P., van den Berg H., Mandjes M., Kooij R. (2003). An integrated packet/flow modelfor TCP performance analysis, Proceedings of ITC-18, pp. 651-660, Berlin, Germany

[6] Radev, D., Lockshina, I. (2006). Performance analysis of mobile communication networkswith clustering and neural modeling. International Journal of Mobile Network Design andInnovation (IJMNDI), vol.1, No3/4, Inderscience Publichers, 188-196.

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Modeling and Clustering Analysis of Broadband Convergence Networks