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Multiservice Telecommunication Systems design with network’s incoming self-similarity flow Ageyev Dmytro, Evlash Dmytro Abstract - In this work is offered the determination method of communication channels should not exceed preset value Wmax. communication channels throughput for network’s incoming self-similarity flow. Research of efficiency of the offered III. A METHOD OF THE PROBLEM DECISION algorithm and the analysis of parameters influence of algorithm Earlier authors in article [4] offered a decision method of and dimension of a solved problem on algorithm convergence is carried out. the preset task. For the preset task decision the expression has Keywords - Throughput, self-similar flow, efficiency been received which is defining an average time delay of the probing, multiservice telecommunication network. message in a network f 1 21 H rs N N f f f f I. INTRODUCTION 1 λ rs n rs 1 λ rs n rs According to world tendencies of telecommunication's Tcp ∑∑ Λ ∑ r 1 s 1 c rs c rs systems development the main task of Ukrainian 2Hf 1 communication sector is creation of a multiservice rs telecommunication network. Among problems which are λfrs c rs 2 1 Hf , rs f H rs 1 H rs (2) necessary for solving during construction of a multiservice n f f f c rs λ rs rs network, there is a determination problem of telecommunication's channels throughput capacities. Existing methods of this problem decision are constructed on the where c rs throughput of a telecommunication channel, bit per assumption about Poisson character of transmitted flows [1]. f second; λfrs and n rs - flows intensity, packets/s and an However research of statistical characteristics of streams average length of a packet, bit, in a telecommunication transmitted in a network has shown that this model not channel (r, s) Λ - the full traffic incoming on network's completely corresponds to real streams and that more N N λ ij . adequate models are models of a self-similar flow [2]. input Λ II. PROBLEM STATEMENT i 1 j1 Let has set A a i of a network subscribers - sources of This problem concerns problems of convex programming. For its decision it is offered to use a method of the quickest an information load of a various classes. Let's designate: descent. d ij λ ij , n ij , H ij - a vector parameters of network’s For determination of flows parameters in incoming information flows in assembly a i and transmitted telecommunication channels the following rule is used in assembly a j , where λ ij - intensity of network’s incoming λ ij n ij f i , j,r ,s M ij n rs , (3) messages, packets/s; n ij - an average length of the message, λ ij bit; Hij - Hurst's parameter; i , j,r ,s M ij B b km - matrix describing network topology: λfrs λ ij , (4) 1, if a k is adjacent to a m ; i, j,r ,s Mij b km Hij , . (1) 0, if not. Hf rs max (5) i , j,( r ,s )Mij Let's enter value w km which defines expenses necessary where Mij - the shortest way between final assemblies i and j; for the communication channel organization between n ij , λij, Hij - parameters of information flows between final assemblies a k and a m with preset throughput. assemblies. It is required to define flows characteristics that transmitted on networks telecommunication channels and throughput IV. ALGORITHM RESEARCH METHODOLOGY capacities of communication channels at which transmission of preset information flows is provided with network's minimum At the first stage the random array of a source data for the average delay Тср. Expenses for the organization of network dimensions in a range from 10 to 100 assemblies was generated. Then the network topology was synthesized, the flows distribution algorithm and choosing of communication Ageyev Dmytro - Kharkov National University of Radioelectronics, channels transmission capacities was applied and the analysis Lenin Avenue, 14, Kharkov, 61166, UKRAINE, of its work was carried out. E-mail: dm_ageyev@ukrpost.net The offered algorithm is iterative and on each iteration is Evlash Dmytro - Kharkov National University of Radioelectronics, Lenin Avenue, 14, Kharkov, 61166, UKRAINE, applied as problem decision composite algorithm by a E-mail: evlashdv@mail.ru dichotomy method which also is iterative. Thus, algorithm convergence is influenced by algorithm halt conditions as at a stage of dichotomy method application, so algorithm as a whole. In this work algorithm halt conditions researching has been carried out at its various stages on convergence and stability of its work. This algorithm was applied for researching of problem decision accuracy to network's configurations for which Hurst's parameter was equal 0,5, that corresponds to Poisson flow. The received results were compared to the results received with classical method application of "square root" which were received as the exact problem decision. Thus, during the experiment, on each iteration it was additional fixed accuracy of criterion function minimum determination. (a) V. THE ANALYSIS OF EFFICIENCY OF OFFERED ALGORITHM The analysis of criterion function value variation at various accuracy of minimum determination at a dichotomy method application stage has shown, that at use of smaller accuracy of point determination of a local minimum, the algorithm reached values near optimal faster. After that the search point started to fluctuate, gating through its optimum position. It is revealed, that if in the algorithm work process the relative change of objective function has received negative value the algorithm did not reach an optimum. Thus, it is possible to draw a conclusion, that fractional variation acceptance of objective function of negative value is a sufficient algorithm (b) halt conditions. Other decision in this case is the magnifying Fig.1 Researching results of offered method efficiency. accuracy of local minimum determination if demanded The dependence analysis of accuracy criterion function accuracy of the problem decision has not been reached. minimum determination from the received value of a halt The second stage of experiment is algorithm convergence condition (fig.2) has shown that for arrival of demanded rate researching and connection between accuracy of optimum calculation accuracy the halt parameter needs to be sampled value's determination of objective function and the accepted on 1..1,5 the smaller degree. value of halt parameter. There are two diagrams on (fig.1). On (fig.1,a) is presented fractional variation of objective function value diagram. On (fig.1,b) is presented accuracy of a objective function minimum presence vs. iteration number diagram. On diagrams results are presented for a network composed of 25 assemblies. For other network dimensions results are similar. Various curves on diagrams correspond to various network configurations. From the analysis of the received results it is possible to draw a conclusion, that algorithm convergence rate varies during its work. Also it is possible to notice, that algorithm convergence depends on initial conditions, i.e. it depend both Fig.2 Accuracy of the problem decision vs. threshold value on network topology, and from set of flows transmitted in a The problem brought above is simplified. In it unrecorded network. The additional analysis for a case of the least the kinds of traffic concerning various load classes also the convergence rate (a curve marked by daggers on the schedule) assumption about the infinite buffers dimensions in has shown, that to this case there corresponded a network's assemblies is made. Further the problem decision eliminating configuration which possessed high factor of gravitation the given lacks will be considered. between the assemblies which having a considerable quantity In this case set of flows information we will of transit assemblies in an information communication route between them. designate D(k) d ij (k) ; d ij (k ) λ ij (k ), n ij (k ), H ij (k ) - a vector of flows information parametres k-th class. Let's enter value w iK w K λ - cost of the communication centre organization, where λ - the total intensity of the flows serviced by this assembly. Value w km in this case depends on distance between This problem, taking into account additional gated in assemblies, from throughput of a communication channel and limiting, to similarly previous case, can be solved with use of from the buffer dimension of this communication channel, i.e. a quickest descent method. w km wl km , c km , x km , where x km - the buffer dimension. VI. CONCLUSION We will be limited additional, the probability of packet loss The problem of communication channels throughput Pkm between assemblies a k and a m should not exceed preset capacities determination providing flows transmission at the value Pп.max. minimum value network's average delay message and limiting Taking into account new parameters the mathematical on total value of expenses for the communication channels model of communication channels transmission capacities organization is solved, for a case of statistically network's determination will have the following appearance: self-similar flows and researching efficiency of application of min Tср D(k ), B , k 1..K ; (6) the offered method is carried out. Comparison of calculation results received with application w iK wl km , c km , x km b km Wm ax ; (7) of the offered method with results received by means of a i k m classical method of "a square root" for a case Poisson flows in 1 (1 Prs ) Pп.max , i, j, ai ,a j A ; (8) communication channels has shown their convergence. Sufficient condition halt algorithm is acceptances of ( r ,s )Mij f km c km , a k , a m A, b km 0 . (9) relative change parameter of criterion function value of negative value. For provision of demanded accuracy of the In work [3] probability dependence of packet loss in single- problem decision it is necessary to sample value of halt channel system with a self-similar incoming flow from parameter on 1…1,5 the smaller degree. average intensity of requests service has been received, At statistically self-similar information flows transmitted in average intensity of receipt of requests and Hurst's Н a network communication channels with large throughput in parameter. Using a given result for our case it is possible to comparison with a case Poisson flows are required at identical record intensity. c rs ( λ rs ) 2H rs REFERENCES Prs exp n rs x rs 2 2 H rs , (10) [1] W. Leland, M. Taqqu, W. Willinger, and D. Wilson. On the 2k (H rs ) 2 aλ rs Self-Similar Nature of Ethernet Traffic (Extended Version), IEEE/ACM Transactions on Networking, 2(1), where λ rs - average intensity of receipt of requests; a - February 1994, pp. 1-15. [2] Norros I. A Storage Model with Self-Similar Input. factor of disagreements; k (H) H H (1 H)1H . Queueing Systems, Volume 16, 1994. The average network's time delay of the message can be defined from Eq. (2).

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In this work is offered the determination method of communication channels throughput for network’s incoming self-similarity flow. Research of efficiency of the offered algorithm and the analysis of parameters influence of algorithm and dimension of a solved problem on algorithm convergence is carried out.

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My name is Dmitriy Evlash and I am an associate professor of Telecommunication systems department in Kharkiv National University of Radio and Electronics from Kharkiv, Ukraine. I graduated in 2006 with a master degree in Telecommunications from the Kharkiv National University of Radio and Electronics. I have finished my postgraduate education in 2011 with a Doctor of Philosophy degree.

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