Magyar Kutatók 1. Nemzetközi Szimpóziuma 1 th International Symposium of Hungarian Researhers on Computational Intelligene and Informatis Gyula Max Budapest University of Tehnology and Eonomis, Department of Automation and Applied Informatis H-1521 Budapest, Po. Box. 91. Hungary e-mail: max@aut.bme.hu Abstrat. Dataflow in omputer networks an be desribe as a ompliated system. To simulate behaviours of these systems are also diffiult, beause these systems show nonlinear features. Before implementing network equipments users wants to know apability of their omputer network. They do not want the servers to be overloaded during temporary traffi peaks when more requests arrive than the server is designed for. Aording to the non-linear harater of network traffi, a system model is established to exam behaviour of the network planned. This paper presents setting up a non-linear simulation model that helps us to handle dataflow problems of the networks. Keywords: non-linear system analysis, omputer networks, data ongestion, stability 1 Introdution In a large-sale omputer network data pakets may hoose among many routes to arrive their destination address. These seletions are based on knowledge of reent nodes of the omputer networks. In most ases parts of the omputer networks do not know all levels of the fully network. Though almost all nodes take part in the proess of forwarding pakets, nodes of the omputer network know more on neighbours than far ones. In this paper, we firstly summarize the ontributions and limitations inluded in the existing works. In seond session the non-linear model and its behaviours are established foused on ongestion problems and stability. The third session explains the estimation proess of model parameters and riteria, while in the fourth session an ase study is given. These system parameters and riteria are used to analyze quantitatively behaviour and stability of system as well as to investigate the impat of system parameters, suh as data density, load level, link apaity on system stability. The simulation results validate our analysis. Last session shows tendeny of future works. 155
Gy. Max 2 Related Works To study network behaviour, traffi parameters are needed to measure and analyse, and to find their statistial laws, suh as was done by Y. Bhole and A. Popesu [1]. After having known the statistial rules, some traffi models are built, suh as the establishment of a model for Novel network traffi by J. Jiang and S. Papavassiliou in [2] or we try to understand why buffers are overflowed like S. Sidiroglou et al. did in [3]. Congestion ontrol methods. We also set up ongestion models. Model of Du Haifeng et al. [4] performs an effetive network ongestion ontrol method for multilayer network. If operational ost is important read I. Chuang s artile [5] on Priing Multiast Communiation. If the time-sale of network traffi is onsidered, the network traffi behaviour will be different in different time-sale. Paxson and Floyd [6] showed that the traffi behaviour of milliseond-time sale is not self-similar by the influene of network protool. Due to the influene of environment, the traffi behaviour whose time-sale is larger than ten minutes is not also self-similar and is a non-linear time-series. Only the traffi behaviour in seond-time sale is self-similar. This problem beomes known when transations of the omputer networks have to be desribed. As it was presented before, the system shows non-linear features, i.e. transations. It means no tools to desribe transations by mathematis of linear ontrol systems. In this paper, a new nonlinear traffi behaviour model is set up and by the aid of this model, the non-linear features of dataflow are desribed. Traffi simulation. The most extensive and modern researhes present diretions of network parameter estimations [7], analyses of traffi generators [8, 9], elasti network nodes [1] et. The non-linear analyses of network nodes [11, 12, 13, 14] is a separate important researh area. Therefore, very important the optimal funtion of the nodes in the system. In terms of the data traffi that would either be the ideal if fewer nodes existed in the network, or in the ase of huge number of nodes these all were linked to all the others. This apparently an absurd approah, already if we look at the eonomi sides of the solution only. A question beomes known, that neessary, that let the nodes be on the entral plae of the examinations. The orret answer is in this diretion, that in terms of the data traffi, the whole network is neessary to put onto the examined entral plae. Stability ontrol. Many papers have theoretially analyzed omputer networks in framework of feedbak ontrol theory based on the ontinuous time model (e.g. [15-19] et.) or disrete-time model (suh as [2] and [21]) after having made some neessary simplifiation and assumption, and finally provided some very revelatory and signifiant onlusions and judgments. In [16], a general nonlinear model was developed, then after linearisation it around equilibrium and making several simplifying assumptions, the suffiient stability ondition was obtained. The main ontribution in [16] was to replae the single-link idential-soure model in [15] with a general model with heterogeneous soures, but omponents 156
Magyar Kutatók 1. Nemzetközi Szimpóziuma 1 th International Symposium of Hungarian Researhers on Computational Intelligene and Informatis were approximated linearly. Under this assumption, some possible fators leading to the queue osillation have been suessfully identified, however, the others will be likely hidden. 3 Data Flow Model The omputer network models known by the literature trae bak the data traffi to the desription of a ommuniation happening between edges and nodes of a ommuniation graph. The models handle the nodes as an important element in this desription method. The outome of model is the ommuniation graph that faithfully imitates the physial arrangement of the omputer network, where the nodes, the ative elements of the system are the peaks of the graph, whih are onneted to eah other by the transfer mediums alled edges. This statement is important, beause in the network atually the nodes, the ative elements of the network ommuniate with eah other, and the nodes form the peaks of the network graph in the graph theory model of the omputer network. Figure 1 shows a plain example how nodes ommuniate: Figure 1 Internal and external elements of a network This desription gives bak that view on a natural manner. In this model the entral plae are oupied by nodes as the peaks of the network graph and the edges of the network graph show transations of the traffi in whih the peaks 157
Gy. Max ommuniate with eah other along the data lines onneting them. This proess separates our internal nodes, where swithes or bridges are loated with a losed urve from their input and output workstation nodes, as it is shown in figure 1. In the following number of internal, input and output nodes are marked by n, l and m respetively. Consider the network is known in a time t and numbers of bits stored in the nodes are marked with N(t). Now let us examine status of the network in time moment (t+ t). During time t if transfer speeds between nodes (v ij ) are known, the data (N i (t+δt)) stored in node i hange as (1): N i (t+ t) = N i (t) + N inter i + N input i - N output i, (1) where ΔN inter i = n j= 1; i j; C (v ), ij ij ΔN input i = l C ik k = 1; i k; (v ik ), ΔN output i = m i= 1; C (v ). ii ii In (1) C ij (v ij ) = ij *v ij is an elements of the ommuniation matrix depending on v ij and ij desribes features of ommuniation between node i and node j. The ommuniation matrix is desribed detailed in session 5. After summarizing data hanges in all nodes and supposed that node i has only one output, we get total data hange (2) in our network: N 1(t + Δt) N 1(t) N2(t + Δt) N2(t) +. =. Nn(t + Δt) Nn(t) 21 n1 21 n1 12 n2 12 n2 1n 2n 1n 2n (2) + 21 k1 21 k1 12 k2 12 k2 1n 2n 1 n 2n 11 11 22 22 nn nn or (2) in matrix form (3) N(t+ t) = N(t) + C int (v int )*Δt + C inp (v inp )*Δt - C out (v out )*Δt (3) Forming (3) and do t, the result is shown by (4): 158
Magyar Kutatók 1. Nemzetközi Szimpóziuma 1 th International Symposium of Hungarian Researhers on Computational Intelligene and Informatis N(t+ t) - N(t) lim = N (t) = C int (v inz ) + C inp (v inp ) - C out (v out ) (4) t t (4) beomes more simply (5) if v int = v inp = v out = v N (t) = v * ( int + inp - out ) (5) Complete alulation of (2) by Taylor series (6) N(t+Δt) = N(t)+N (t)* Δt+N (t)* Δt 2 /2!++ N (n) (t)* Δt n /n! (6) Comparing (2) to (6) we say that N (t)* Δt 2 /2!++N (n) (t)* Δt n /n!= (7) 4 State Variables In the network the transmission speed means how many bits an be transported in one seonds between two nodes. Transmission speed may not be equal in every part of the network, but from node j to node i the same value is supposed and marked with v ij. Eah node has some buffer for their messages. This apaity is measured by data density (8) that shows rate of reent data quantity in time and the maximal data quantity in node i. number of reent data bits in buffer of node i N i (t) x i (t) = = (8) maximal number of data bits in buffer of node i N imax In our model the data density of node i is a number without dimension between <= x <= 1. Traditionally, the numeration of data density shows the differene of outgoing and inoming bits in time t or in other words it shows the maximum number of data bits that an be transferred in the next time unit. Now data density is introdued as a rate of the length of messages stored in node i and the maximal message length. This property shows non-linearity, beause no hane to reeive more bits than the maximal size of the data buffer or node i an not reeive any bit, if node j has no any one to send. In our network data density of node i is marked with x i and data density of input node j that is loated outside of the network is marked with s j. In our model N(t) and x(t) are vetors of n elements. From (8) N i (t) = N imax * x i (t), onsequently N max is a matrix of nxn elements. Elements of N max represent the buffer size of nodes. Sine no buffer between node i and node j, N max is a diagonal matrix. 159
Gy. Max 5 Communiation Matrix In (1) (5) the ommuniation matrix plays big role in our model. There are some funtions implemented in C ij (v ij ). Let us examine parts of this matrix. C ij (v ij ) must ontain at least the following: - internal onnetions of the network - environmental onnetions of the network - size of internal buffers in nodes - apabilities of nodes to send and reeive data The apability of node i shows when no data to send, no more room in its buffer as well as the probability of transmission in ase of more output hannels. An element of the Communiation matrix grants the onnetion when node j ommuniates with node i. Creating Communiation matrix is done olumn by olumn. We go through eah element of olumn j, and if a onnetion exists between nodes j and node i, that is node j works to node i, the ommuniation funtion marked C ij - is reated, where (i j, 1 i,j n ). All features are neessary to be taken into onsideration at the time of the forming of the ommuniational matrix by the aid of onnetion funtions C ij. The most important funtion is the model parameter of ij showing properties of the onnetion. Inside traffi regulations are also neessary to be taken into onsideration at the time of the forming of the ommuniational matrix (i.e. data link mehanisms depending on the density of the traffi). In our model, the inner traffi regulations depend on density three funtions of ij (t), S j (t), R i (t) and x i (t), where S j (t) defines properties of sender node, R i (t) delares features of the reeiver, while C ij are defined by produt of these four fators (9): C ij = ij (t) S j (t) R i (t) x i (t) (9) Struture of C ij (t): If there is the opportunity of the permanent ommuniation between two nodes, and the node j works for node i, then ij = e ij p ij, where e ij is the oeffiiene of effetiveness that shows the time rate of real data transfers and total working time, while p ij is the probability of data transmission from node j to node i. If there is not a physial onnetion between these nodes, then ij =. The probability of data transmission, p ij (t) presents the distribution proportion of given routes belonging to a node with relative weighting, where <= p ij (t) <= 1, if node j works for more than one node. The relative weightings of node j are shown in olumn j of the ommuniation matrix and Σp ij (t) = 1 for eah olumn. S j (t) is an automati internal self-regulation funtion (1) of transmitter node with values of 1 or. It shows whether node j has message to send or not. Connetion is disable if data density of node j (x j (t)) equals to, anyway 1. 16
Magyar Kutatók 1. Nemzetközi Szimpóziuma 1 th International Symposium of Hungarian Researhers on Computational Intelligene and Informatis 1, x j (t) > S j (t)= (1), x j (t) = R i (t) is another automati internal self-regulation funtion (11) of the reeiver node with values of 1 or. Connetion is enable if data density of node i (x i (t)) smaller, than 1, anyway. Value of 1 means that buffer of node i has been overloaded, so node i loses its ommuniation port to diretion of node j, therefore node i does not reeive any message from node j. 1, x i (t) < 1 R i (t)= (11), x i (t) = 1 Equation (9) transforms (3) into (12) N(t+ t) = N(t) + ij (t) S j (t) R i (t) x i (t) v ij Δt + + i,inpk (t) R i (t) x i (t) v i,inpk Δt - outm,i (t) S i (t) v outm,i Δt (12) In (12) we supposed that input node always an send data to and output nodes an reeive all data from our network. In longer time period x j (t) v ij shows real transmission speed. While S j (t) =1, i.e. v ij Δt <= x j (t) N imax, transmission works. Transmission stops if no more data to transfer. In other words, the average transfer speed for the total time period is x j (t) v ij. 6 Simulation That takes too muh simulation time if dataflow is simulated by bit by bit. We wanted to make a unit for our simulation. So in first step we measured frame lengths. We made data transfers in different time period (early morning, afternoon, midnight), with different data lengths (from some hundred bytes to hundred megabytes) and different transmission speed (from 24 Kbits/s up to 1Mbits/s). Measurements were made by the Wireshark Network Protool Analyzer. After having olleted more hundred thousands messages we analyzed inoming and outgoing frames. The result showed two signifiant figures among the values. One of them was harateristi of onfirmation messages, while the other one was typial data messages. Distribution funtion of data measured is presented in Table 1. It shows the most frequented values. Considering figures table 1 we hose 55 bytes as unit of frame length. Using this frame length an average data length is 26 units. 161
Gy. Max Table 1 Number of piees of the most frequented frame length Length Number of Piees Total Length 54 165 535 8 938 89 66 11 773 777 18 1 82 1 73 11 69 86 1 434 271 383 389 163 222 1 514 43 127 65 294 278 An existing network was simulated that an be seen in Figure xxx. Two files were transmitted through the real network. The first file of 89 MB was transported from Inp1 and the seond one of 91 MB started from Inp2. First hannel used TCP protool, while seond file was transported by FTP protool. In our 1Mb/se hannels the alulated effiieny rate was about 72%, beause the transportations took appr. 2 se. Our model used the same files during the simulation. Figure 2 Struture of the network measured and simulated The seond file was started in 8 seonds after the first one. The third input hannel named Inp3 did not work. The throughput an be seen in Figure 3. Proess time of the simulation is a bit less than the real ommuniation. In the figure 3 we an se that the real ommuniation used different hannel speeds, while the transfer speed was the same in our simulation. During our simulation we supposed that speeds of transmission equal between nodes. This supposition was false as we ould see it in Figure 3. 162
Magyar Kutatók 1. Nemzetközi Szimpóziuma 1 th International Symposium of Hungarian Researhers on Computational Intelligene and Informatis Figure 3 Internal and external elements of a network In the simulation the data density of the internal nodes are also observable. Using different internal buffers in the network observed bottlenek effets an be tested. Figure 4 shows two simulation with different buffer sizes. On the left 5 MByte internal buffer was used, while in seond ase only 1 MByte buffer was alloated. On the left side inputs work linearly, while on the right side shows non-linear faes. Figure 4 Data transfers using different internal buffers in the simulated network Figure 4 helps us to find bottleneks of the network. In the right piture Inp2 was bloked by node Int1, beause there were no enough room for inoming frames. Conlusion and future works This paper looks for possibilities to desribe dataflow model of large-sale omputer networks. A model was presented that was appliable to simulation, planning and regulation of omputer network s traffi. At the time of the model's establishment, the partial differential equations were avoided in the mathematial model beause of the speially hosen state variables. The nodes have honoured 163
Gy. Max roles in this non-linear model beause storage apaities of the transmission medium are pratially zero. Nodes either ommuniate to eah other or not. In our model, the mean of the data density is the proportion of the size of data stored in the single node and the data quantity, whih an be stored maximally. Our model examines hange of data density ourred by data flow among the nodes in a region demarated by lose urve. Input and output data densities are regarded as known. At first sight, these proesses are the inputs and outputs of the model. Effetively, these proesses together form the atual inputs of the mathematial model. State variables present data densities arising in the internal nodes of the system. Our system applies a data traffi model involving n internal and m external nodes. To reate the mathematial model the ommuniation matries defining the network has fundamental importane. Our model applies four ommuniation matries. During simulation we supposed that speed of transmission equals between nodes. It must be hanged in the future. We have to work out a quik test to alulate possible ongestion points and gives some riteria to stability of the system. Finally, a simply example was presented to demonstrate usage of the model and how this model is appliable to the simulation, planning or regulation of omputer networks. Referenes [1] Y. Bhole, A. Popesu: Measurement and Analysis of HTTP Traffi. Journal of Network and Systems Management, Vol. 13, No. 4 (25) pp. 357-371 [2] J. Jiang, S. Papavassiliou: Deteting Network Attaks in the Internet via Statistial Network Traffi Normality Predition. Journal of Network and Systems Management, Vol. 12, No. 1, (24) pp. 51-72 [3] S. Sidiroglou, G. Giovanidis, A. D. Keromytis: A Dynami Mehanism for Reovering from Buffer Overflow Attaks, J. Zhou et al. (Eds.): ISC 25, LNCS 365, pp. 1-15, 25 [4] D. Haifeng, X. Yang, L. Lingyun: An Effetive Network Congestion Control Method for Multilayer Network, Journal Of Eletronis (China), Vol.25, No.4, July 28 [5] John C.-I. Change, Marvin A. Sirbu: Priing Multiast Communiation: A Cost-Based Approah, Teleommuniation Systems 17:3, 281-297, 21 [6] Paxson, V., Floyd, S.: Wide Area Traffi: The Failure of Poisson Modelling, IEEE/ACM Transations on Networking 3(3), pp. 226-244 (1995) [7] A. M. Galkin, O. A. Simonina, G. G. Yanovsky: Multiservie IP Network QoS Parameters Estimation in Presene of Self-similar Traffi, NEW2AN 26, LNCS 43, pp. 235-245, 26 164
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