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1 c AR AIC SINET bit/sec

2 Leland et al. 1994, Paxon and Floyd 1995 FARIMA Resnick, 1997 Willinger et al., 1997; Yang and Petropulu, Kitagawa and Gersch, 1984 Kitagawa and Gersch 1984 AR

3 Kitagawa and Gersch 1984, Kitagawa and Gersch 1984, 1985 Yada et al., , 1994 Kitagawa and Gersch 1984, 2003; Ikoma et al., 2003; Shimizu et al., 2003 SINET, kbit/sec hour k y(k) (2.1) x T (k)=f T (k)x T (k 1) + G T (k)v T (k)

4 v T (k) N(0,τ 2 T ) 2.1 (2.2) x T (k)=[ T 1(k) T 2(k) ] F T (k) k 2 k =0(mod 24) (2.3) F T (k)= " k 0(mod 24) F T (k) G T (k) k 2 k =0(mod 24) # (2.4) G T (k)=[ 1 0 ] k 0(mod 24) G T (k) F T (k) G T (k) k hour x t(k)=[ t 1(k) t 2(k) ] v t(k) N(0,τ 2 t ) F t G t (2.5) x D(k)=F D(k)x D(k 1) + G D(k)v D(k) v D(k) N(0,τ 2 D) 2.5 (2.6) x D(k)=[ D 1(k) D 2(k) D 6(k) ] 6 F D(k) k 2

5 429 k =0(mod 24) (2.7) F D(k)= k 0(mod 24) F D(k) G D(k) k 2 k =0(mod 24) (2.8) G D(k)=[ 1 0 0] k 0(mod 24) G D(k) (2.9) x s(k)=f sx s(k 1) + G sv s(k) v s(k) N(0,τ 2 s ) 2.9 (2.10) x s(k)=[ s 1(k) s 2(k) s 23(k) ] 23 F s (2.11) G s F s = (2.12) G s =[ 1 0 0] 2.4 AR p AR AR (2.13) x a(k)=f ax a(k 1) + G av a(k)

6 v a(k) N(0,τ 2 a ) 2.13 (2.14) x a(k)=[ c 1(k) c 2(k) c p(k) ] AR F a (2.15) F a = a 1 a 2 a p a 1,a 2,...,a p G a (2.16) G a =[ 1 0 0] 2.5 (2.17) y k = T 1(k)+D 1(k)+s 1(k)+c 1(k)+w k w k N(0,σ 2 ) 2.6 ( x k = F k x k 1 + G k v k (2.18) y k = hx k + w k x k x T (k) x D(k) x s(k) x a(k) v k v T (k) v D(k) v s(k) v a(k) F k F T (k) F D(k) F s F a G k G T (k) G D(k) G s G a h Y k =(y 1,y 2,...,y k ) Y k k 3

7 431 k 1 p(x k Y k 1 ) p(x k Y k ) L>0 p(x k Y k+l ) p(x 0) p(x 0) x 0 V 0 ( x 0 0 = x 0 (3.1) V 0 0 = V 0 k =1 1 1 (3.2) 1 ( x k k 1 = F k x k 1 k 1 V k k 1 = F k V k 1 k 1 F k + G k QG k Q v k 1 p(y k Y k 1 ) ȳ k k 1 Σ k k 1 ( ȳ k k 1 = hx k k 1 (3.3) Σ k k 1 = hv k k 1 h + σ 2 (3.4) ( x k k = x k k 1 + k k e k V k k =(I k k h)v k k 1 I e k 1 (3.5) e k = y k ȳ k k 1 k k (3.6) k k = V k k 1 h Σ k k 1 1

8 Y N =(y 1,y 2,...,y N) 1 ( x k N = x k k + A k xk+1 N x k+1 k (3.7) A k V k N = V k k + A k V k+1 N V k+1 k A k (3.8) A k = V k k F k+1v 1 k+1 k 3.4 σ 2 τ 2 = τ 2 T,τD,τ 2 s 2,τa 2 AR a =(a1,a 2,...,a p) θ θ (3.9) l(θ) log p(y N;θ) = NX k=1 log p(y k Y k 1 ;θ) NX = N 2 log 2π 1 log Σ k k 1 2 k=1 NX 1 yk ȳ k k k=1 Σ k k , τ 2 ˆσ 2 τ 2 =ˆσ 2 τ 2 AR a 1 1 ȳ k k 1 Σ k k 1 ȳ k k 1 = ȳ k k 1 Σ k k 1 = σ 2 Σk k 1 σ σ 2 2 y k ȳ k k 1 (3.10) ˆσ 2 = 1 N NX k=1 Σ k k (3.11) l( τ 2,a) = N 2 log 2πˆσ2

9 NX k=1 log Σ k k 1 N 2 ˆτ 2 â 3.5 AR AIC Akaike, 1974 (3.12) AIC = 2l(ˆθ)+2 (θ ) AIC AIC 4. T t D s AR c 1 2 AR 1 2 AR AR PARCOR PARCOR ( 1,1) 0.05 PARCOR x 0 V AR τ 2 s 0 τ 2 s SINET SINET

10 l(ˆθ) AIC 2. T 1 +D+s+c τ 2 s 0 =10 l(ˆθ) AIC 1 AR 1 AIC AR AIC

11 AR

12 T 1 AR 2 AR 1 T 1 +D+s+c τ 2 T AR 8 3

13 437 AR σ = σ 7 7 Q = (10 1) 2= Q T D AR 8 AR 9 Box-Cox Box and Tiao, SINET 11 SINET 5 bit/sec SINET T+D+s+c τ 2 s 0 =10 2 τ 2 s 0 =10 AIC 3 AIC AR AIC 1 AIC AR 2 T 1 +D+s+c 2 AIC T 1 +D+s+c 2 2

14 SINET SINET SINET 14. SINET 1

15 SINET l(ˆθ) AIC 4. SINET T 1 +D+s+c SINET 4 T 1 +D+s+c

16 SINET 17. SINET 18. SINET AR 19. SINET

17 SINET 17 AR σ = ±4σ Q = Kitagawa and Gersch SINET AIC AIC 1 1

18 Doucet et al., Box-Cox A, J68-A 5, Akaike, H A new look at the statistical model identification, IEEE Transactions on Automatic Control, AC-19, Box, G. E. P. and Tiao, G. C Bayesian Inference in Statistical Analysis, Addison-Wesley. Doucet, A., de Freitas, N. and Gordon, N. J. eds Sequential Monte Carlo Methods in Practice, Springer, New York.

19 443 Ikoma, N., Shimizu, T., Imazu, K. and Yana, K Decomposition of network traffic of dialup access into trend and periodic components based on nonstationary Bayesian model, Proceedings of SICE Annual Conference 2003, Fukui, Japan, AR SINET FORTRAN77, , Kitagawa, G. and Gersch, W A smoothness prior state space approach to the modeling of time series with trend and seasonality, Journal of the American Statistical Association, , Leland, W. E., Taqqu, M. S., Willinger, W. and Wilson, D. V On the self-similar nature of Ethernet traffic, IEEE/ACM Transactions on Networking, 2, Paxon, V. and Floyd, S Wide-area traffic: The failure of Poisson modeling, IEEE/ACM Transactions on Networking, 3, Resnick, S. I Heavy tail modeling and teletrafic data, Annals of Statistics, 25 5, , A 4 47, Shimizu, T., Imazu, K., Ikoma, N. and Yana, K Nonstationary Bayesian time series analysis for network traffic by decomposition model into trend, periodic and AR components, Proceedings of Science of Modeling The 30th Anniversary of the Information Criterion AIC, Yokohama, Japan, Dec , , Willinger, W., Taqqu, M. S., Scherman, R. and Wilson, D. V Self-similarity through highvariability: Statistical analysis of Ethernets LAN traffic at the source level, IEEE/ACM Transactions on Networking, 5, , Yada, T., Kitawaga, G. and Inoue, M Prediction of telephone revenue by using a Kalman filter, Proceedings of the 14th International Teletraffic Congress ITC, 14, Yang, X. and Petropulu, A. P The extended alternating fractal renewal process for modeling traffic in high-speed communication networks, IEEE Transactions on Signal Processing, 49,

20 444 Proceedings of the Institute of Statistical Mathematics Vol. 54, No. 2, (2006) Quantitative Analysis Method of Network Traffic by Component Decomposition Using Bayesian Time Series Model Norikazu Ikoma 1 and Kazuo Yana 2 1 Faculty of Engineering, Kyushu Institute of Technology 2 Faculty of Engineering, Hosei University The ever-increasing expansion of the Internet constantly increases the importance of analyzing network traffic and making use of those results. To analyze network traffic in this study, we used a Bayesian time series model, which decomposes network traffic into trends, weekly and daily cycles, and colored noise components. The model is represented in a linear-gaussian state space model that consists of system equations for each component and an observation equation, in which sums of the components yields the observed network traffic. State estimation by Kalman filter is used to obtain each component. Variance and AR parameters in the model are obtained to maximize the likelihood. Then a comparison among candidate models and selection of the components proceed based on the Akaike Information Criterion (AIC). To provide an example of an analysis using this proposed method, we report on the decomposition results of traffic data of dialup access at Koganei Campus and those at the top network domain of Hosei University, Tokyo which connects to the SINET. Key words: Network traffic, decomposition, Bayesian time series model, state space model, Kalman filter.

θ 1 = φ n -n 2 2 n AR n φ i = 0 1 = a t - θ θ m a t-m 3 3 m MA m 1. 2 ρ k = R k /R 0 5 Akaike ρ k 1 AIC = n ln δ 2

θ 1 = φ n -n 2 2 n AR n φ i = 0 1 = a t - θ θ m a t-m 3 3 m MA m 1. 2 ρ k = R k /R 0 5 Akaike ρ k 1 AIC = n ln δ 2 35 2 2012 2 GEOMATICS & SPATIAL INFORMATION TECHNOLOGY Vol. 35 No. 2 Feb. 2012 1 2 3 4 1. 450008 2. 450005 3. 450008 4. 572000 20 J 101 20 ARMA TU196 B 1672-5867 2012 02-0213 - 04 Application of Time Series