Результати параметрів моделі оптимізації для нестаціонарного руху в телекомунікаційних та комп'ютерних мереж наведено. Для оптимального моделювання динаміки руху метрик Евкліда і просторів Гілберта використовуються. Техніка та алгоритм оптимізації моделей розроблені. Теоретичні положення ілюструються чисельними прикладами і схемами з використанням
ofMathCADРезультаты параметров модели оптимизации для нестационарного движения в телекоммуникационных и компьютерных сетей приведены . Для оптимального моделирования динамики движения метрик Евклида и пространств Гилберта используются . Техника и алгоритм оптимизации моделей разработаны . Теоретические положения иллюстрируются численными примерами и схемами с использованием
ofMathCADResults of models parameters optimization for the nonstationary traffic in telecommunication and computer networks are given. For optimum modeling of the traffic dynamics metrics of Euclid’s and Gilbert’s spaces are used. The technique and algorithm of models optimization are developed. Theoretical positions are illustrated by numerical examples and diagrams with use ofMathCA

Ignatov, V. O.

ZIJUAN, WU

English

Results of models parameters optimization for the nonstationary traffic in telecommunication and computer networks are given. For optimum modeling of the traffic dynamics metrics of Euclid’s and Gilbert’s spaces are used. The technique and algorithm of models optimization are developed. Theoretical positions are illustrated by numerical examples and diagrams with use ofMathCADРезультаты параметров модели оптимизации для нестационарного движения в телекоммуникационных и компьютерных сетей приведены . Для оптимального моделирования динамики движения метрик Евклида и пространств Гилберта используются . Техника и алгоритм оптимизации моделей разработаны . Теоретические положения иллюстрируются численными примерами и схемами с использованием
ofMathCADРезультати параметрів моделі оптимізації для нестаціонарного руху в телекомунікаційних та комп'ютерних мереж наведено. Для оптимального моделювання динаміки руху метрик Евкліда і просторів Гілберта використовуються. Техніка та алгоритм оптимізації моделей розроблені. Теоретичні положення ілюструються чисельними прикладами і схемами з використанням
ofMathCA

Наукові журнали Національного Авіаційного Університету

11Проблеми інсЬорматизаиії та управління. 1(23)’2008 UDC 004.738 (043.3)Ignatov V.O., Doctor of Engineering, Prof.WU ZIJUAN, Post-graduateMETHOD OPTIMIZATION OF MODELS PARAMETERS FOR THE NONSTATIONARY TRAFFIC IN TELECOMMUNICATION NET­WORKInstitute of information and diagnostic NAUResults of models parameters optimization for the nonstationary traffic in telecommunication and computer networks are given. For optimum modeling o f the traffic dynamics metrics o f  Euclid’s and Gilbert’s spaces are used. The technique and algorithm o f models optimization are developed. Theoretical positions are illustrated by numerical examples and diagrams with useofMathCADIntroductionProblems of modeling of the traffic in telecommunication and computer networks become more and more actual, in process of development of these networks, use of principles of construction of open information systems and OSI ISO standards, growth of a supply and demand in the market of optimum service models of the traffic at a transport level [1-3]. The successful decision of these problems restrains complexity of the problems decision of modeling for nonstationary and nonlinear cases. In the known literature there are no techniques and results of optimization and the comparative analysis of models of the nonstationary traffic. In many respects it is caused by weak development of mathematical methods of stochastic approximation of casual nonstationary processes, difficulties of the proved choice of the traffic standards. The given work is logic continuation of the work[4].The purpose of workThe purpose of work is optimization and comparison of models of the nonstatio­nary traffic in telecommunication and com­puter networks.For achievement of the purpose the fol­lowing tasks are put and solved: use of a technique and algorithm of the comparative analysis of optimum models of the nonsta­tionary traffic in telecommunication and computer networks [4], a choice of the spac­es metrics of models comparison and criteria of their optimality, substantiation and a choice of optimality criteria of models of the nonstationary traffic, a substantiation of atask statement and construction of optimum models of the nonstationary traffic, the com­parative analysis of adequacy of optimum models of the nonstationary traffic on the basis of the chosen adequacy parameters, the decision of numerical examples and the for­mulation of conclusions by results of models optimization of the nonstationary traffic.Statement of problemAs well as in the work [4], in a role of the initial data the initial polynomial repre­sentation of the reference nonstationary traf­fic with known Gaussian m - measured dis­tribution of decomposition independent fac­tors and the Markov’s non-uniform models of the nonstationary traffic are chosen [4]. The basic methods of construction of optimi­zation models of the nonstationary traffic are chosen a method of the traffic quantization and Markov’s approximation of random process of the traffic conditions change, the Ritz’s method and the method of the maxim­al plausibility. The results of the problem decision are optimization models parameters of the nonstationary traffic constructed on the basis of metrics of Euclid’s and Gilbert’s spaces, the comparative analysis and also conclusions on results of the analysis.The problem decisionWe shall begin with the optimization of the traffic quantization values with the purpose of achievement of the best approxi­mation of the modeling and reference mo­ments of the traffic on the normalized inter­val of comparison time. We shall use the technique and the algorithm of the compara­tive analysis of the work [4].nWe shall use the metrics of the Euc­lid’s space then the square of distance be­tween the first initial moments of the refer­ence and model of the nonstationary traffic can be submitted as [4]4Z„Z2)2 = j j t k -(2-tk)-[Z2 -(Z 2 -Z,)./>0.e ^ ‘] f , (1 )  1,=0where Z/ and Z2 - the traffic quantization values,tk (2-tk) -  normalized on an interval of com­parison the standard of the traffic, r] -  intensity of change of the nonstationary traffic,tk -  the k— th moment of the traffic supervi­sion time,P 10 -  probability of the traffic presence in the first condition at the initial moment of time.When we use the metrics of the Gil­bert’s space then the square of distance be­tween the first initial moments of the refer­ence and model of the nonstationary traffic can be submitted as [4]f (Z„Z2)2 = jfc (2 - /J -[Z 2 -(Z2 - z, ) . ] M  -(2) 0From formulas (1), (2) follows, that with a choice of quantized values Z/ and Z2 it is possible to provide the best approximation of the initial moments of the reference traf­fic. Similar expression can be written down concerning the second central moments of the reference and model of the nonstationary traffic. For a case of the Gaussian multiva­riate distribution the optimum choice of the first two moments of the traffic model the method of optimization of quantized values and their dispersions allows to solve a prob­lem of the best approximation of the normal density of distribution of the reference non­stationary traffic and the normal density dis­tribution of the model of the nonstationary traffic.Let's illustrate the specified opportuni­ty with the problem decision of the best ap­proximation of a population mean of the ref­erence nonstationary traffic by a population mean of the model of the traffic. Considering (1), (2) as optimization criteria of quantized values in a nonlinear bidimentional problem of optimization without restrictions, we use a classical method of search of the optimal quantized values.This problem we shall solve in the fol­lowing statement: expression of optimization criterion quantized values Zj, Z2 as function­al (1) or (2) is known, the classical method of search of a minimum (1) or (2) requires to find Ziopt, Z2opu delivering a minimum func­tional (1) (a discrete case) or functional (2) (a continuous case):min s 2( Zr Z 2) = s ^ ( Z lopl, Z 2opi) (3)Z\.Z2Differentiating functionals (1), (2) on parameters Z/, Z2, we shall receive two equa­tions of optimization concerning values Z/, Z2.1. For a discrete case:B M 2 - g - [ Z 2- ( Z 2- Z , ) P l0 •*-'" ]]• (4)■ (Pn - e * ' )  = oB fe  • (2 - t t ) -IZ2 - (Z2 - Z , ) P W- e""‘ ]]• (5)<* =o(l-/>0 ^'*)=02. For a continuous case:jfe • (2-tk)-[z2 -(Z 2 -z.) • j>0 • ]]• (Pm-e'" )dtk =0 (6)0j  fc • (2-<t)-[Z 2 -  (Z2 -  Z,) ■ P]0 ■ e'"“ ]] ■ (1 -  /?„• e+'> )dtk = 0 '(7 )  0Let's result these equations in an initial form of system from two linear algebraic eq­uations, we shall receivea \ l^ l  a i2^2 ~  1^ • (8)#21^ 1 + a22Z2 — b2. (9)1. For a discrete case:an = ± (P l0. e ^ ytk=0 9a2l = £ P l 0 e-™-(l-Pl0-e ^ ‘)<4=0 ) \ /a\2 = ^ ( l - P ]0- e ^ ) P l 0 e~^'*= 0 9a22 = ^ ( l -P l0 e-^ )2 n n/,=0 * V 'bl =  (2- t k)] Pl0 - e */*=09b2 = '£[tt V - t k)\-Q-Pn - e * ) '  ( 1 2 )2. For a continuous case:13I7po6/ieMu inAopMammauiï ma ynpaeninm. 1(23)’20080alx=\{Pw - e ^ ) - { \ - P w -e~^)dtk , (13) 0an =\ [ ( \ -Pw -e^) - {Pia-e^) ]d tk ,0a22= ) { \ - P X0-e -^fd t r  (14)0bl = ][tk(2-tt)HPl0-e-«')dtk,00Using Kramer’s rule, we shall receive optimum quantized values of controlled va­riableswhere |A|, |A1|, |A2| - determinants of the equations system (8), (9).Substituting these values in functional (1) or (2), we shall find its minimal value at optimum quantized values (16).Example 1. We shall show feature of definition optimum quantized values and mi­nimal root-mean-square deviations of the moments of the first order nonstationary pa­rabolic traffic at the following initial data: P,o = 0.95, P20 = 0.05, /// = 3.414.Using formulas (10) - (12), we shall find parameters of system of the algebraic equations for a discrete casea ll  = 0.903, al2 = 0.078, 61 = 0.031,a21 = 0.078, a l l  = 0.941, b l  = 0.969 .Using formulas (13) - (15), we shall find parameters of system of the algebraic equations for a continuous casean =0.132, an =0.137, 6, =0.108, a2] = 0.137 , a22 = 0.594, b2 = 0.559 .|A2| float,7 -> .8727696 = 0.873Substituting these values in formulas for determinants, we shall find their values for a continuous case\A\ float, 7—» 5.961473 • 10“2 = 0.06 |zll| float, 7 —» —1.267710• 1 O' 2 = -0.013 \A1\ float, 7 —» 5.905071 • 10~2 = 0.059We use these values of determinants and we shall find under formulas (16) re­quired optimum quantized values. In system MathCAD the decision looks like:1. For a discrete caseZ10 = -0 .213 ,Z20 = 0.990538. (17)2. For a continuous caseZ10 =-0.054, Z20 =1.034. (18)Substituting optimum quantized values Z10, Z20 in the formula (3), we shall find the minimal values target functional and root- mean-square errors of modeling of a popula­tion mean of the reference traffic:1. For a discrete case£ram(.Z\,Zl) -  1.624<10CT„,„= ^ .«4x/(T 7 (]9)2. For a continuous case£ mîn(Z11Z 2)  =  2,587x10 3a min= J 2,587*1O'3 = 0,05086 =  5,086% ( 2 Q )Let's define the minimal value of aver­age factor of a variation of an error on an interval nonstationary:1. For a discrete caseV min&  minmo0.04% ~ 0.667 = 0.06% , ( 21)2. For a continuous caseSubstituting these values in formulas for determinants, we shall find their values for a discrete case|A| float,7 —» .8440187 = 0.844|Al| float,7 -» -,4593524e-l =-0.046F  min —(7  min mo5,0860,667 = 7,6252% , ( 22)where average value (a constant com­ponent) normalized traffic on an interval nonstationary is defined under the formula14_mO = \[tk {2 -tk)]dtk . (23)0On fig. 1 results of modeling of a pop­ulation mean of the reference nonstationary traffic m(tk) by model M(Zi , Z2 , t/J opti­mized in Gilbert space with parameters (18), and a parameter of accuracy of the modeling (22) optimized in Euclid space by model Ml(tk), and also model M2(ty) in which in a role quantized values average values of quantums are used are shown. Comparing curves, it is uneasy to notice, that model M(Zi , Z2 , tk) optimized in Gilbert space with parameters (18), and a parameter of ac­curacy of modeling (22) yields the best re­sults of modeling of the reference traffic. Fig. 1 evidently illustrates that the model op­timized in Gilbert space approximates the reference traffic much better.F5(ZItrilopl)min = 2.993 10 s > 2.587 I ff3. Hence, selection r\l the optimum decision to improve is impossible.Research of influence of other un­guided parameters of models - initial proba­bilities P jo, P 11 is similarly carried out. It is interesting to note, that concerning these pa­rameters of the equation of optimization are linear algebraic, therefore values of probabil­ities are optimum.piFig. 1. Comparison of models of the nonstatio­nary trafficIntensity and initial probabilities in problems of optimization act, as a rule, in a role of unguided parameters. Despite of it is not superfluous to check up their influence on the optimum decision, to estimate an op­portunity of optimization of their values for the further improvement of the decision. We shall show feature and principles of such check on an example of optimization rjl.Example 2. We shall present target functional as function of intensity rjl:For a continuous caseFig. 2. Find the optimal value piIn a role of criteria of adequacy we shall choose parameters concordance model­ing and reference values of the traffic on an interval nonstationary. We shall enter a pa­rameter concordance modeling and reference values of the traffic in points of supervision th k =1, m,P(^10,^20»^ k)2 ‘m(tk)’M(Z10 Z20,tk) m(tk)2 +M(Z10 Z20,tk)2Example 3. We shall calculate values of a parameter (25) for the models optimized in spaces Euclid and Gilbert at m = 10. Re­sults of calculation we shall reduce in tab. 1 and we shall display on fig. 3..rçl)=j[f* • (2-ft) -[Z20 -(Z20 -Z,0) ■ /|0 • e~nUk ]f dtk -(24) 0Also we shall find a graphic method (fig. 2) a minimum value of functional (24), on rjl at optimum quantized value of the traf­fic. We shall receive r\lopt = 2.81,a. A discrete caseПроблеми інсЬорматизаиії та управління. 1(23)’2008 иlO j  t j j  J jb. A continuous caseFig. 3. Diagrams of dependences of parameters concordance models from time Tab. 1. Calculation results p(Z10 Z20,tk)4 0.0 0.1 0.2 0.3 0.4Pki 0.0 0.905 0.941 0.967 0.983Pk2 0.0 0.998 0.991 0.992 0.996tk 0.5 0.6 0.7 0.8 0.9 1.0Pkl 0.993 0,998 1.0 1.0 1.0 1.0Ж . 0.999 1.0 1.0 0.999 0.999 0.999Analyzing given tab. 1 and fig. 3 it is possible to conclude, that the optimized models precisely enough display the refer­ence traffic, is especial in second half of in­terval nonstationary. In a continuous case more exact coordination of model and the standard is observed.Application of various criteria of adequacy allows all-round check of adequacy of models, enables to execute also an estimation of influence of unguided variables on optimum decisions. Especially it is necessary to pay attention to the nonlinear effects connected to application of decompo­sition in a number on indicative functions and use intensities of change of the traffic as parameters an exhibitor.Conclusions1. Application of quantization and Markov approximations allows building of adequate models of the nonstationary traffic submitted initial polynomial by decomposi­tion with casual factors. With growth of di­mension of optimum models accuracy of modeling nonstationary the traffic will grow also. The significant interest represents re­search of this law.2. Offered the system of parameters and a technique of verification of models nonstationary traffic with various number of conditions of the traffic allow to carry out verification of models and to use quantitative estimations of a degree of adequacy of mod­els to the real nonstationary traffic.References1. Игнатов В.А., Жуков И.А., ГузийH.H. Данилина Г.В. Моделювання перехідних режимів трафікукомп’ютерної мережі. Информационные технологии и безопасность. -  Сборник научных трудов. -  К.: Национальная ака­демия наук Украины, Институт проблем регистрации информации, 2006. -  Вып. 9. -С . 84 -87 .2. Гузій М.М., Даниліна Г.В., Ігнатов В. О., Милокум Я.В. Методи і ал­горитми оптимального управління трафіком в обчислювальних мережах. Проблеми інформатизації та управління- К.: НАУ, 2006. -  Вип. 17. -  С. 32 -  37.3. Гузій М.М., Даниліна Г.В., Ігнатов В.О. Мачалін І.О. Марковські рандомізовані моделі. Вісник Жито­мирського державного технологічного університету. Житомир 2006. -  Вип. IV (39).-С . 179- 184.4. Гузій М.М., У Цзы Цзюанъ,Ігнатов В. О. Сравнительный анализ адек­ватности моделирования нестационарного трафика в телекоммуникационных и ком­пьютерных сетях. Проблемиінформатизації та управління. -  К.: НАУ, 2007.-В ип. 21.-С . 127- 134.

Метод оптимізації параметрів моделей для нестаціонарного TRAFFIC в телекомунікаційних мережі

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Method optimization of models parameters for the nonstationary traffic in telecommunication network

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Наукові журнали Національного Авіаційного Університету