INVESTIGATION OF MATHEMATICAL MODEL OF COMMUNICATION NETWORK WITH UNSTEADY FLOW OF REQUESTS

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1 Trnsport nd Telecommuniction Vol No 4 9 Trnsport nd Telecommuniction 9 Volume No Trnsport nd Telecommuniction Institute Lomonosov Rig LV-9 Ltvi INVESTIGATION OF MATHEMATICAL MODEL OF COMMUNICATION NETWORK WITH UNSTEADY FLOW OF REQUESTS AN Tuenbev AA Nzrov Kliningrd Stte Technicl University Sovietsy prospect Kliningrd 36 Russi e-mil: tuenbev_iy@milru Toms Stte University Lenin prospect 36 Toms 6345 Russi e-mil: nzrov@sibmilcom When choosing structures nd chrcteristics of computer networs in prctice frequently priority is given to the clssicl techniques nd decisions However closer reserch nd disclosure of potentil opportunities of networs of dt trnsmission is promoted by crrying out mthemticl modeling In this wor mthemticl models of computer networs of csul ccess in the form of mss service systems with source of repeted clls nd the notifiction bout the conflict re constructed A modified method of symptotic nlysis to investigte mthemticl models of csul ccess networ ws developed The prcticl vlue of the wor consists in the results of scientific reserch to promote crrying out complex nlysis n optimiztion nd disclosing of potentil opportunities of telecommuniction systems of dt trnsmission Keywords: rndom ccess unstedy flow symptotic nlysis Introduction The improvement of dt processing is one of the most chllenging problems for computer networ cretors nd developers [ ] Therefore the reserch in the field of computer networs functioning is importnt nowdys More precisel problems relted to unstble functioning repeted trnsmission collision nd chrcteristics of input nd output dt flows in those networs re being investigted [3] Mthemticl modeling gives the possibility to predict the behvior of the networ under certin conditions without the physicl reliztion of the networ itself This llows you to sve money both t the stge of moderniztion or estblishment of networ connection nd t its further exploittion The study of informtion processes occurring in rel networs of rndom ccess should be crried out through rndom processes due to the presence of rndom effects Therefore to investigtion such systems the most efficient tool of nlyticl modeling is mss service theory [4 5] Among ll the possible methods of the gretest interest re methods of reserch which llow obtining nlyticl expressions for the probbilities of the sttes of simulted networ In this pper we use mthemticl modeling of multiple ccess protocols for defining the bsic probbilistic chrcteristics of such networs The Mthemticl Model of Rndom Access Networs with Unstedy Flow of Requests As mthemticl model of computer networs let us consider the mss service system (MSS) with the repet requests source (RRS) [6] There is one service device which simultes the overll chnnel the service time of which hs n exponentil distribution with prmeter μ = The unstedy flow of requests rriving t the entrnce of the system is n unstedy process with prmeter ρ ( Ech rrived request is immeditely strted to receive the service if the device is free On the other hnd if the device is bus the ccess conflict ( collision) occurs nd both requests go to RRS the stying time in which hs n exponentil distribution with prmeterγ A simultion progrm ws relized when creting model of rndom ccess networ The progrm helps to show tht the ssumption bout n exponentil distribution in RRS does not contrdict the hypothesis of the invrince of the distribution of the stte probbilities of the device to type of distribution of the length of the dely in RRS 8

2 Trnsport nd Telecommuniction Vol No 4 9 Since the conflict occurs in the chnnel the conflict notifiction stge is strted to be trnsmitted the durtion of which hs n exponentil distribution with prmeter μ = / We define the following vector ( i ( ( ) where i( is the request number in RRS nd ( the stte of the device The stte of the device cn te 3 vlues: the device is free ( = the device is bus the device hs the conflict stte The stte of the networ is defined s two-dimensionl vector ( i ( ( ) nd the chnges in the sttes ( i ( ( ) re Mrovin Becuse the networ tht we study is controlled by sttic rndom multiple ccess protocol for ny set of prmeter vlues ρ ( t ) γ for this MMS there is no sttionry regime Consider the non-sttionry probbilities { ( = i( = i} = P = P i These probbilities stisfy the following differentil system of finite-difference equtions with vrible coefficients: P + ( ρ ( + iγ ) P = P + P t P + ( + ρ ( + iγ ) P = ρ( P + ( i + ) γp ( i + () t P + ρ ( + P = ρ( P ( i + t + ( i ) γ P ( i + ρ( P ( i The solution of the system () sufficiently defines the opertion of mthemticl model of the networ connection but unfortuntel ccurte nlyticl methods for solving the system do not exist Therefore to study the system some uthors hve proposed modified method of symptotic nlysis of Mrovized systems [7] 3 Asymptotic Sstudy of Mthemticl Model of the Networ with Unstedy Flow of Requests Consider n unstedy flow the prmeter of which is slowly vrying function of time: ρ ( = ρ( γ ) t We introduce the following nottionγ = t = τ where is positive smll prmeter We define x = x( nd x(τ ) s follows: τ x ( = i nd x = lim x( In our first result we show tht under pproprite technicl conditions the process x ( exists nd is deterministic function with mening of the symptotic verge of normlized number of requests in RRS 9

3 Trnsport nd Telecommuniction Vol No 4 9 In the system () we set τ x = x( = i( ) = i nd P = π For further use we lso set π = = π ( ) x τ Then we obtin system () below: π ) ( ρ( + x) π ) = π ) + π ) + π + ( + ρ + x) π τ )= π ( + ( x + ) π ( τ ) = ρ x x + () π + ρ + π = ρ π ( x τ )+ ( x ) π ( x ) + ρ( π ( ) + x Let us ssume tht the following limits exist: ( x ± j = π = ; j = lim π In the system () we let The result is system of liner lgebric equtions which solution (becuse of its homogeneity) cn be written s = R π = π with R + R + R R ( = The explicit form of the probbility distribution of the sttes of the device R = is given by: = G + G + G G + G + R = (3) G + R = G G + G where G = ρ( + x( + Now we hve to determine π ( x To this end in () we expnd the functions π ( x ± = into series Summing ll the equtions of the system we get the following equlity: π π = x + x {[ xr + ρ R + ( x + ρ ) R ] π ( } ο ( ) Dividing both sides by nd then ting the limit we get the eqution = x {[ ρ + R R )] π ( } x 3

4 Trnsport nd Telecommuniction Vol No 4 9 This eqution coincides with the degenerte Foer-Plnc eqution concerning the symptotic density of the distribution π of diffusion process x ( with diffusion coefficient equl to zero nd trnsfer coefficient A of the type + R R ( )) A = ρ τ Since the diffusion coefficient is equl zero the rndom process degenertes to deterministic function x = x( which stisfies n ordinry differentil eqution ( ) = ρ + R R ) x τ or using (3) = ρ ρ + x ( ρ() τ + x() + ( ρ + x) + x (4) with n initil condition Thus we hve shown tht the process x(τ ) is deterministic function defined by the ordinry differentil eqution (4) Depending on the considered system the differentil eqution tht determines the sort of function with the mening of the symptotic verge of normlized number of requests in RRS my hve one or more limiting points The studies hve shown tht in networs with rndom multiple ccess there my occur phenomenon of bi-stbilit which is chrcterized by the fct tht two sttes of the mny sttes of the networ re determined - the points of stbiliztion [8] In this cse the networ opertes s follows The system is fluctuting in neighborhood of point of stbiliztion nd then fter rndom move is going to nother point of stbiliztion nd then gin returns to the first point of stbiliztion etc The probbilistic-time chrcteristics of the networ in the neighborhood of one point of stbiliztion re cceptble while in the neighborhood of other point they my deteriorte mny times nd the networ is functioning very poorly Let us define τ y = y( = i( ) x( nd y = lim y( We show tht y ( is diffusion process of utoregression which hs the mening of devitions of the number of requests in RRS from the symptotic verge tht is locl pproximtion of the process of sttes' chnges of the MMS The distribution of probbilities of sttes of the device R = when is discrete Mrov process independent of the process y ( Agin we consider () nd s before we te ρ ( = ρ( γ ) Now we choose t = τ We lso set P = H Then we obtin the following system (5) of equtions: ) ) x ( x ) + ) [ + ( y + )] H ( + τ ) t ( ρ( + x( + y) H ) = H ) + H ) + ( + ρ + x + y) H ) = ρ H )+ γ = nd + x y (5) 3

5 Trnsport nd Telecommuniction Vol No 4 9 x + ρ + H = ρ H ( y + [ x + ( y )] H ( y + ρ H ( τ ) + y lim H We ssume tht the following limits exist: ( y ± j = H = ; j = We lso ssume tht the functions H = re twice differentible functions of y In the system of equtions (5) we te limits s The solution of the new system of liner lgebric equtions in the functions H = we find s R H H = where H = H( + H + H 3( nd where R = re determined by (3) It remins to determine H In (5) we expnd the functions H ( y ± j = ; j = in series by increments of the rgument y to within ο( ) ccurcy The solutions = of system (5) re of the form: H = R H + h + ο( ) H (6) To determine the functions h = we get system of liner lgebric equtions Its solution cn be written s h ( y G = + G h R R G G { x R + x R } yh h = h + R yh ) + (7) ( τ + { x R ρ( R (ρ + x( ) R } ( In (5) we expnd the functions ( y ± = in series by increments of the rgument y to within ο ) ccurcy Summing of the system of equtions we get the following equlity ( x H = { ( x( + y) H ρ( H ( ( x + y) H ) ρ( H )} + { x( H ) + ρ( H ) + ( + 4ρ( ) H ( τ } + ο( ) + x y ) Substituting (6) nd (7) into this equlit fter simple trnsformtions we get the following eqution for H : 3

6 Trnsport nd Telecommuniction Vol No 4 9 ( = H( { A( yh( } + B ( (8) where A ( nd B ( re given by: R A( = R (9) B ( = ( ρ( ) R + ρ( R + (3ρ( + + G ) R + R { ρ( R + ( ρ( + ) R ( ) ( τ ( ρ( R ) R ( R ) } + {( ρ( R ) R + ( ρ( ) R } () Eqution (8) is the Foer-Plnc eqution for the probbility density H of the diffusion uto regression process y ( where the coefficients of trnsfer nd diffusion re determined respectively by formuls (9) nd () Let us write the stochstic differentil eqution for the process y ( : α y dτ β dw dy = + () where w ( is stndrd Wiener process nd the coefficients α nd (9) nd () with given symptotic verge of x ie A ( ρ + x α = B ( ρ + x β = y Explicit expression for the process y(τ ) hs the form τ () τ = exp α() s ds y() + β () s exp α() u τ s du ds β re defined by equlities As usul y() is given the initil vlue solutions of differentil equtions () stochsticlly independent of the vlues of Wiener process w ( 4 Conclusion Crrying out reserch s presented in this rticle llows us to understnd the nture of informtion processes tht te plce in rel communiction networs The im of the study is to find the probbility distributions of different sttes of the investigted system Knowledge of the probbility distribution provides the most complete in probbilistic sense description of the functioning of the model nd llows clculting different chrcteristics of the system These chrcteristics cn be further used for development tss design prmeter optimiztion of networs with rndom multiple ccess protocol Thus nowledge of the distribution of sttes of the investigted networ gives us the possibility to predict nd control the rndom processes tht te plce in networs Using the obtined informtion bout the investigted systems we cn control their opertion in the future 33

7 Trnsport nd Telecommuniction Vol No 4 9 References Olifer VG Olifer NA Computer networs Principles technologies protocols SPb: Piter p Ege-Lopez E Mrtinez-Sl A Vles-Alonso J Grci-Hro J Mlgos-Snhuj J Wireless communictions deployment in industry: review of issues options nd technologies Computers in Industr V 56 Issue 5 pp Queues: Flows Systems Networs Proceedings of the Interntionl Conference Mthemticl Methods for Anlysis nd Optimiztion of Informtion Telecommuniction Networs Mins Jnury p 4 Artlejo JR Accessible Bibliogrphy on Retril Queues Mthemticl nd Computer Modelling V 3 Issue pp -6 5 Artlejo JR Gómez-Corrl A Retril Queueing Systems: A Computtionl Approch Berlin Heidelberg: Springer 8 38 p 6 Seisenbeov BE Tuenbev AN Investigtion of the mthemticl model of networ of csul ccess with recurrent strem rrives Computtionl technologies V 3 Specil issue 5 8 pp 6-7 Tuenbev AN Mthemticl modelling of computer networs operted by reports of csul multiple ccesses Doctorl disserttion 7 p 8 Tuenbev AN Computer networs with rndom ccess Astn: ENU 6 5 p 34

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response