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Statistical Multiplexing

Statistical Multiplexing. 1 조 : 오 호 일 이 은 희 고 한 호. 차례. Introduction Superposition of ATM traffics - Fundamental limit Theorem - Superposition of Poisson, Renewal, Point process - Limit Theorem 에 관한 결론 Theoretical Apporach of Superposition - PDF

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Statistical Multiplexing

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  1. Statistical Multiplexing 1조 : 오 호 일 이 은 희 고 한 호

  2. 차례 • Introduction • Superposition of ATM traffics - Fundamental limit Theorem - Superposition of Poisson, Renewal, Point process - Limit Theorem에 관한 결론 • Theoretical Apporach of Superposition - PDF - Autocorrelation • Experimental Study of Superposition

  3. Introduction • The key characteristic of ATM networks Statistical Multiplexing(switching node) • switching node의 역할 1. Decomposition(demultiplexing) function 2. Switching function 3. Multiplexing function

  4. TV SOURCE 1 MULTIPLEXING QUEUE TV SOURCE 2 ATM TRUNK TV SOURCE n 사용된 모델 [그림 : Multiplexing n TV sources] Individual ATM traffics의 superposition 으로 묘사됨.

  5. TV SOURCE 1 TV SOURCE 2 TV SOURCE n SUPERPOSITION Superposition [그림 : Superposition of n different sources]

  6. Superposition approach • Theoretical approach • Poisson process -> fundamental limit position • Superposed traffic의 pdf와 auto-correlation 유도 • experimental approach • 여러 index를 정의 -> arrival process분류 기준 • Modeling approach • renewal process • MMPP(Markov Modulated Poisson Process) • semi-Markov process

  7. Fundamental limit Theorem of Superposition • 기본적 개념 : total cell arrival rate를 constant하게 유지 Poisson process가 됨. Superposition of Poisson process Superposition of Renewal process Superposition of Point process

  8. Superposition of Poisson Process • independent Poisson process의 superposition Poisson Process • 수학적인 유도에 관련된 reference : D. P. Heyman and M. J. Sobel “Stochastic Models in Operation Research” McGraw Hill, Vol. 1.

  9. Superposition of Renewal Process • Poisson process를 이끌어 내기 위한 2가지 가정 1. if and only if all the component processes are Poisson 2. By convergence in distribution

  10. Superposition of Point Process Non-homogeneous Poisson process (by Franken and Grigelionis)

  11. Limit Theorem에 대한 결론 1. Convergence이론에 의해 얻어진 Poisson process 는 correlated structure 와 무관하게 얻어진 것이다. 2. When superposing TV sources, the behavior of the pooled arrival process can be considered like a Poisson process only over short time intervals relatively to the contribution and the number of the different sources involved in the multiplexing. This effect will be direct application when queueing the pooled arrivals.

  12. Theoretical Approach of Superposition • PDF of Superposed Process - Repartition function - complementary function(survivor function) - mean inter-arrival time

  13. 5가지의 elementary configurations(1)

  14. 5가지의 elementary configurations(2)

  15. -i 번째 source가 시간 에서 존재하는 cell이 될 확률 5가지의 elementary configurations(3)

  16. SUPERPOSITON OF TV SOURCE : RESULTING PDF 0.6 0.5 0.4 0.3 0.2 0.1 0 N=16 N=10 N=5 N=2 0 5 10 15 20 25 30 35 40 INTERVAL TIME Resulting PDF [그림 : Superposing of actual TV sources(PDF after n superpositions)]

  17. C(t) t1 t2 t3 t4 t5 t6 t7 t N(0,t) 0 t Autocorrelation Function(1) 1. infinitesimal(differential) 2가지 접근법 2. integral

  18. 에 착안

  19. SUPERPOSITION OF TV SOURCES : RESULTING AUTOCORRELATION 1.2 1 0.8 0.6 0.4 0.2 0 N=15 N=10 AUTOCORRELATION N=5 N=2 0 5 10 15 20 25 30 35 40 INTERVAL TIME Autocorrelation Function(3) [그림 : Superposing actual TV sources(autocorrelation function of the counts after n superpositions)]

  20. Experimental Study of Superposition(1) • index를 정의 - Counting process를 위한 index Index of dispersion - DSPP의 경우 index I(n,t) 는 Poisson 과의 연관성 추정 근거

  21. Experimental Study of Superposition(2) • Inter-Arrival Time process를 위한 index • k-interval squared coefficient of variation Inter-arrival이 correlation 되었을 경우

  22. Index를 통한 Arrival process분류 기준 • For a Poisson process, • For a Renewal process, • For a point process,

  23. COUNTING PROCESS [n(0,t)]/INCREASING INTERVALS(T=10) 35 30 25 20 15 10 5 0 -5 I(1,t) I(1,t)(-.), skew(-),kurtosis(--) KURTOSIS SKEW 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Counting interval t x10 4 Supporting Figures(1) [그림 : Counting processes on one single TV source(index of dispersion I(t), skewness S and kurtosis K at increasing counting intervals)]

  24. COUNTING PROCESS [n(0,t)]/INCREASING INTERVALS(U=10) 60 50 40 30 20 10 0 I(7,t) I(4,t) I(2,t)(-),I(4,t)(-),I(7,t)(..) I(2,t) 0 20 40 60 80 100 120 140 160 Counting interval t Supporting Figures(2) [그림 : Index of dispersion of superposed TV traffics on short range counting intervals]

  25. SQUARED COEFFICIENT OF VARIATION FOR n=3,5,7 9 8 7 6 5 4 3 2 1 (ckn*ckn) : squared coef of variation 0 20 40 60 80 100 120 Number of consecutive intervals : k Supporting Figures(3) [그림 : k-interval squared coefficient of variation for superposed TV traffics]

  26. 20 16 12 8 4 0 SINGLE VOICE SOURCE n=1 18 16 14 12 10 8 6 4 2 0 n=2 I(t) SQUARED COEFFICIENT OF VARIATION n=10 n=60 POISSON n=inf. 0 .10 .20 .30 .40 .50 .60 .70 .80 .90 1.0 1 2 5 10 20 50 100 200 500 1000 2000 CONSECUTIVE INTERVALS (k) t(SECONDS) Supporting Figures(4) [그림 : Index of dispersion for number of arrivals in (0,t) for voice sources] [그림 : k-interval squared coefficient of Variation for n voice sources]

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