MODELING AND ANAL YSIS OF TRAFFIC IN HIGH SPEED NETW ORKS A CTS A TM In ternet w ork Pro ject Information and T elecomm unication T ec hnology Cen ter Univ ersit y of Kansas.
Mo deling and Analysis of T ra�c in High Sp eed Net w orks Organization � Motiv ation F ailure of Classical Mo dels. � In tro duction. Long Range Dep endence. � Analytical Mo del. Macro-dynamics Micro-dynamics � P erformance Analysis Metho dology . Mean Cell dela y . Cell loss probabilit y . � Data collection pro cess and the AAI Net w ork. A CTS A TM In ternet w ork Pro ject 1
Mo deling and Analysis of T ra�c in High Sp eed Net w orks Organization (con td.) � Sim ulation Mo del. Mo del Description. V alidation. � Exp erimen tal Ev aluation. Mean Cell dela y and Cell Loss Probabilit y results. T ra�c Micro-dynamics. Sensitivit y of n um b er of phases. Second-order statistics. � Conclusions. � F uture W ork. A CTS A TM In ternet w ork Pro ject 2
Mo deling and Analysis of T ra�c in High Sp eed Net w orks Motiv ation � T ra�c studies. � Queueing p erformance of a pro cess with in�nite v ariance. � Switc h bu�ers get �lled up faster than those predicted b y con v en tional mo dels � Example: Mean cell dela y: Mean Delay (B) (A) 0 Load 1 Figure 1: T ypical Dela y curv es for long-range dep enden t mo del (B) and con v en tional mo del (A). A CTS A TM In ternet w ork Pro ject 3
Mo deling and Analysis of T ra�c in High Sp eed Net w orks Motiv ation (con td.) � Example: Probabilit y of Cell Loss: log( P(Q>x) ) (B) (A) Buffer Size , x Figure 2: T ypical loss curv es for long-range dep enden t mo del (B) and con v en tional mo del (A). � T ra�c mo deling and p erformance prediction required for e�cien t op erational algorithms. A CTS A TM In ternet w ork Pro ject 4
Mo deling and Analysis of T ra�c in High Sp eed Net w orks In tro duction � Self similar tra�c mo dels: FBM, ARIMA, Mo deling with Chaotic Maps. � De�nition of self similarit y: Let = ( X , t = 0, 1, 2...) b e a co v ariance stationary pro cess X t 1 m � 1 ( m ) X = X X (1) k k m � i m i =0 Exactly: ( m ) � r ( k ) = r ( k ) ; k 0 (2) Asymptotically: ( m ) ! ! 1 ( k ) ( k ) ; (3) r r m A CTS A TM In ternet w ork Pro ject 5
Mo deling and Analysis of T ra�c in High Sp eed Net w orks In tro duction (con td.) � Rami�cations Non-summable auto correlation function. Div ergen t P o w er-Sp ectrum at the origin. � Implication Burstiness of Aggregate tra�c. � Origin ON-OFF source frame-w ork with ON and OFF p erio ds that follo w a distribution with in�nite v ariance. A CTS A TM In ternet w ork Pro ject 6
Mo deling and Analysis of T ra�c in High Sp eed Net w orks T ra�c Mo del � De�nition: Rate Pro cess: A random pro cess R ( t ), whic h is the short-term time a v erage of the arriv al pro cess A ( t ). 9 x 10 2.5 45 40 2 35 30 1.5 Number of Cells Rate(Mb/s) 25 20 1 15 10 0.5 5 0 0 0 5 10 15 0 5 10 15 Time (Hours) Time(Hours) Cell coun t pro cess Rate pro cess. A CTS A TM In ternet w ork Pro ject 7
Mo deling and Analysis of T ra�c in High Sp eed Net w orks T ra�c Mo del (con td.) � Mo del the Macro-dynamics and Micro-dynamics of the Rate Pro cess. � Rate pro cess is mo deled as a non-Mark o vian , stationary and ergo dic phase-pro cess S = f x g . ha ving a �nite state space ; x ; :::x 1 2 N Rate x N x k x 1 x 1 Time A CTS A TM In ternet w ork Pro ject 8
Mo deling and Analysis of T ra�c in High Sp eed Net w orks T ra�c Mo del (con td.) � The rate v ector � = [ � ] represen ts the b oundary rates for the states and � ; � ; :::; � 1 2 +1 N � � � � � :::: � . 1 1 +1 N Rate γ N+1 γ N . γ k . . γ 2 γ 1 Time A CTS A TM In ternet w ork Pro ject 9
Mo deling and Analysis of T ra�c in High Sp eed Net w orks T ra�c Mo del (con td.) � The probabilit y v ector � � = [ � ; � ; :::; � ] denotes the steady-state probabilit y v ector 1 2 N of the phase pro cess. � The steady state phase probabilities of the phase pro cess are assumed to follo w a distribution with a hea vy tail. � The rates in � � and the probabilit y v ector � � de�ne the macro-dynamics. A CTS A TM In ternet w ork Pro ject 10
Mo deling and Analysis of T ra�c in High Sp eed Net w orks T ra�c Mo del (con td.) � � � is obtained b y partitioning the range [ � , � ] where � and � represen t the min max min max minim um and maxim um rates resp ectiv ely . � The rate v ector � for phases is: � N = (4) � � 1 min � � � max min = + = 2 ; 3 ; 4 ; (5) � � ; i :::; N 1 i N � = � : (6) N +1 max A CTS A TM In ternet w ork Pro ject 11
Mo deling and Analysis of T ra�c in High Sp eed Net w orks T ra�c Mo del (con td.) � � is obtained from the CDF of ( t ). � R � The CDF of ( t ) can b e obtained from a theoretical in�nite v ariance distribution. R Here, a theoretical P areto Distribution w as used. A P areto Distribution can b e giv en as: � � � F ( x ) = 1 K x ; � > 1 ; x > K : (7) X The maxim um lik eliho o d estimate of the shap e parameter � , for a giv en set of data f d g samples D = ; d ; ::; d is giv en as 1 2 M 1 � = (8) 1 M l n ( d ) P i =1 i M A CTS A TM In ternet w ork Pro ject 12
Mo deling and Analysis of T ra�c in High Sp eed Net w orks T ra�c Mo del (con td.) � The CDF can also b e obtained from an empirical CDF of the rate pro cess. � The steady-state probabilit y � for state x , whic h is b ounded b y rates � and � is i +1 i i i computed as: � � � � � = P [ X � ] P [ X � ] ; i = 1 ; 2 ; 3 :::N 1 i +1 i i � � = ( � ) ( � ) ; = 1 ; 2 ; 3 :::; 1 (9) F F i N i +1 X X i 1 CDF 0 γ γ γ γ γ γ 1 2 3 4 Ν Ν+1 RATE Figure 3: Quan tizing the CDF of R ( t ). A CTS A TM In ternet w ork Pro ject 13
Mo deling and Analysis of T ra�c in High Sp eed Net w orks T ra�c Mo del (con td.) � Within eac h state x , with asso ciated probabilit y � , the arriv al pro cess is mo deled as a i i p oin t pro cess with a with �nite mean and �nite v ariance. � i +1 � The distribution function de�nes the tra�c micro-dynamics of that state. � Assuming � as the mean rate in state x is a conserv ativ e assumption as it is the i +1 i upp er b ound on the rates in eac h state. A CTS A TM In ternet w ork Pro ject 14
Mo deling and Analysis of T ra�c in High Sp eed Net w orks P erformance Analysis Metho dology � Let Z denote the random v ariable asso ciated with a p erformance parameter of a queue with R ( t ) as the input pro cess. R x 1 x 2 x i x x 4 3 x n Figure 4: Concept for p erformance analysis metho dology . � F ollo wing the linearit y prop ert y of exp ected v alue, the exp ected v alue of the random v ariable can b e written as Z j S [ Z ] = [ Z = ] (10) E X � E x i i i 2 S � In the ab o v e equation the tra�c macro-dynamics are describ ed b y the v alues of � and � j S the micro-dynamics are represen ted b y E [ Z = x ]. i A CTS A TM In ternet w ork Pro ject 15
Mo deling and Analysis of T ra�c in High Sp eed Net w orks P erformance Analysis Metho dology (con td.) � P erformance prediction in terms of Mean Cell Dela y and Cell loss Probabilit y . � Notation: Let Slotted system, in�nite bu�er. n denote the n um b er of cells in the system at a giv en time. i P ( j ) represen ts the probabilit y that there are n cells in the queue at the end of the n th j slot, giv en that the input pro cess is in state x . i i p denote the probabilit y that there are k arriv als to the system when the input k pro cess is in state . x i � Queue dynamics are describ ed b y: n +1 i i i P ( j + 1) = X P ( j ) p (11) + n � ( k � 1) n k =0 k + � f 0 ; x g . ( x ) represen ts the maxim um of A CTS A TM In ternet w ork Pro ject 16
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