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Chapters 9

Chapters 9. Self-Similar Traffic. Introduction- Motivation. Validity of the queuing models we have studied depends on the Poisson nature of data traffic Recent studies show that Internet traffic patterns are often bursty and self-similar , not Poisson (exponential)

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Chapters 9

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  1. Chapters 9 Self-Similar Traffic

  2. Introduction- Motivation • Validity of the queuing models we have studied depends on the Poisson nature of data traffic • Recent studies show that Internet traffic patterns are often bursty and self-similar, not Poisson (exponential) • Patterns appear through time • Understanding of the characteristics of self-similar traffic is essential to enable analysis of modern networks Chapter 9 – Self-Similar Traffic

  3. d dx f(x) = F(x) = e -x Introduction- Motivation Exponential Distribution Exponential Density E[X] = X = 1/ F(x) = Pr[Xx] = 1 – e-x Chapter 9 – Self-Similar Traffic

  4. Order Sierpinski triangle …can be found …in chaos. Chapter 9 – Self-Similar Traffic

  5. 72 144 72 328 144 72 72 ??? 0 72 216 288 648 720 864 936 ??? 432 216 216 0 216 648 864 Self-Similar Time Series Example 0 8 24 32 72 80 96 104 216 224 240 248 288 296 312 320 648 656 672 680 720 728 744 752 864 872 888 896 936 944 960 968 Chapter 9 – Self-Similar Traffic

  6. Relevance in Networking • Clustering (a.k.a. burstiness) is common in network traffic patterns • patterns are typically persistent through time • the clusters may themselves be clustered • Poisson traffic demonstrates clustering in short term, but smoothes over long term (known as the “memoryless” property) • During bursts due to clustering, queue sizes in switches/routers may build up more than the classical M/M/1 model predicts • impact on buffer sizes? Chapter 9 – Self-Similar Traffic

  7. Relevance in Networking Bottom line - The M/M/1 queuing model assumes that arrivals and service times are exponential and, therefore, “memoryless” … real network traffic patterns might not be memoryless, therefore the M/M/1 model might be “optimistic” Chapter 9 – Self-Similar Traffic

  8. E[x(at)] aH Var[x(at)] a2H Rx(at, as) a2H Continuous-Time Self-Similar Stochastic Processes • Continuous time: 0 t  • A stochastic process x(t) is statistically self-similar with parameter H (0.5  H  1), if for a  0: a -H x(at) has the same statistical properties as x(t) • In other words: • E[x(t)] = mean • Var[x(t)] = variance • Rx(t, s) = autocorrelation Chapter 9 – Self-Similar Traffic

  9. Hurst Parameter (H) • Measure of the persistence of a statistical phenomenon….. the measure of the long-range dependence of a stochastic process • H = 0.5 indicates absence of long-term dependence • As H approaches 1, the greater the degree of long-term dependence • Note: Brownian motion process B(t) is self-similar with H = 0.5 SEE Appendix 9A Chapter 9 – Self-Similar Traffic

  10. It is possible to define self-similar stochastic processes with heavy-tailed distributions • Leads to manageable distribution models • Can be used to characterize probability densities for traffic processes, e.g., packet interarrival times and burst lengths • Distribution of random variable X is heavy-tailed if: 1 – F(x) = Pr[X  x] ~ , as x  , 0   1 x  Heavy-Tailed Distributions Chapter 9 – Self-Similar Traffic

  11. k  x k +1 x  -1  k Pareto Heavy-Tailed Distribution • Characteristics: f(x) = F(x) = 0 (x  k) F(x) = 1 - (x  k,   0) f(x) = (x  k,   0) E[X] = k (  1) • Note that for k = 1, •  2, infinite variance •  1, infinite mean and variance Chapter 9 – Self-Similar Traffic

  12. +1  k kx f (x) = Pareto and Exponential Examples Density Functions Compared Chapter 9 – Self-Similar Traffic

  13. Examples of Self-Similar Data Traffic • LAN (Ethernet): • self-similar, H = 0.9 • Pareto fit for  = 1.2 • World-Wide Web: • browser traffic nicely fits Pareto distribution for 1.16    1.5 • file sizes on WWW seem to fit this distribution as well • TCP - FTP, TELNET: • session arrivals approximate Poisson • traffic patterns are bursty… heavy-tailed Chapter 9 – Self-Similar Traffic

  14. Mean Waiting Time (Delay) – Ethernet/ISDN Study Chapter 9 – Self-Similar Traffic

  15. Findings/Implications? • Higher loads lead to higher degrees of self-similarity • performance issues most relevant at high loads • Traditional Poisson modeling of traffic proven inadequate • leads to inaccurate queuing analysis results • increased delays, buffer size requirements • applicable to ATM, frame relay, 100BaseT switches, WAN routers, etc. • note excessive cell loss in first generation ATM switches • Pareto modeling yields better (more conservative) results Chapter 9 – Self-Similar Traffic

  16. Self-Similar Modeling (Norros) • Attempts to develop reliable analytical model of self-similar behavior • uses Fractional Brownian Motion (FBM) process (sect. 9.2) as basis • Buffer size can often be estimated using: q = 1/[2(1-H)] / (1-)H/(1-H) (note that for H = 0.5, this simplifies to /(1-), the M/M/1 model) Chapter 9 – Self-Similar Traffic

  17. Self-Similar Storage Model (Norros) Chapter 9 – Self-Similar Traffic

  18. Findings/Implications? • Buffer requirements much higher at lower levels of utilization for higher degrees of self-similarity (higher H) THEREFORE… • If higher levels of utilization are required, much larger buffers are needed for self-similar traffic than would be predicted using classical modeling So, what does all this really mean? Chapter 9 – Self-Similar Traffic

  19. Modeling and Estimating Self-Similarity • Task: determine if time series of data is actually is self-similar, and estimate the value of H • Variance Time Plot: Var[x(m)] vs. m • R/S Plot: log[R/S] vs. N • Periodogram: spectral density estimate • Whittle’s Estimator: estimate H, assuming self-similarity Chapter 9 – Self-Similar Traffic

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