Multiscale Models for Network Traffic

1. Multiscale Models for Network Traffic Vinay Ribeiro Rolf Riedi, Matt Crouse, Rich Baraniuk Dept. of Electrical Engineering Rice University (Houston, Texas)

2. Outline • Multiscale nature of network traffic • Wavelets • Wavelet models for traffic • Network inference applications

3. (discrete time) Time Scales time unit 2n 2 1

4. Multiscale Nature of Network Traffic time unit Internet bytes/time trace (LBL’93) 600ms • Network traffic (local area networks, wide area networks, video traffic etc.) - variance decays slowly with aggregation • i.i.d. data – variance decays faster with aggregation 60ms i.i.d. time series (lognormal) 6ms

5. Stationary Gaussian process, Covariance (Hurst parameter: 0<H<1) Long-range dependence (LRD) if ½<H<1 Second-order self-similarity Fractional Gaussian Noise (fGn) Variance-time plot

6. fGn is a 1/f-Process power • Power spectral density decays in a 1/f fashion • Low frequency components  long-term correlations frequency

7. Towards Generalizations of fGn Variance-time plot Auckland Univ. Traffic • Variance decay of traffic not always straight line like fGn • Goal: develop LRD models • Generalize fGn • Parsimonious (few parameters) • Fast synthesis for simulations time scale

8. Wavelets • Consider only orthonormal wavelet basis in L2(R) • Prototype functions approximation function-wavelet function- • Basis formed by scaled and shifted versions of prototype functions • Approximation and wavelet coefficients

9. The Haar Wavelet Basis

10. Computing the Haar Transform • Wavelet Transform: fine to coarse (bottom to top) • Inverse Wavelet Transform: coarse to fine (top to bottom)

11. Wavelets and Filtering • Wavelet coefficients at any scale j is the output of a bandpass filter • Coarse scales  low frequency band • Fine scales  high frequency band • Width of bandpass filters increase exponentially frequency

12. Wavelets “Decorrelate” 1/f Processes 1/f spectrum power time domain 1/f strong correlation • Analysis of 1/f data • sample means converge faster in wavelet domain • estimate H in wavelet domain • Synthesis of 1/f data • Exploit weak correlation in wavelet domain • Generate independent wavelet coefficients with appropriate variance • Invert wavelet transform wavelet domain not 1/f weak correlation frequency power frequency

13. Haar Wavelet “Additive” 1/f Model • ChooseWj,ki.i.d. within scale j • Set var(Wj,k) to obtain required decay of var(Vj,k) • FastO(N) synthesis • log2(N) parameters • Asymptotically Gaussian

14. Sample Realization • Realization is Gaussian and can take negative values • Network traffic may be non-Gaussian and is always positive

15. Multiplicative Cascade Model • Replace additive innovations by multiplicative innovations • Aj,k2 [0,1],example -distribution • Choose var(Aj,k) to get appropriate decay of var(Vj,k) • FastO(N) synthesis • log2(N) parameters • Positive data • Asymptotically lognormal at fine time scales

16. Sample Realization • Data is positive • Same var(Vj,k) as additive model

17. Additive vs. Multiplicative Models time unit 24ms • Multiplicative model marginals closer to real data than additive model 12ms 6ms Internet data (Auckland Univ) Additive model Multiplicative model

18. Queuing Experiment • Additive and multiplicative models same var(Vj,k) • Multiplicative model outperforms additive model • High-order moments can influence queuing (open loop) multiplicative model real traffic additive model Kilo bytes

19. Shortcomings of Multiscale Models • Open-loop • Do not capture closed-loop nature of network protocols and user behavior • Physical intuition • Cascades model “redistribution” of traffic (multiplexing at queues, TCP)? • Stationarity: first order stationary but not second-order stationary • Time averaged correlation structure is close to fGn • Queuing of additive model close to stationary Gaussian data (simulations and theory)

20. Selected References • Self-similar traffic and networks (upto 1996) • W. Willinger, M. Taqqu, A. Erramilli, “A bibliographical guide to self-similar traffic and performance modeling for modern high-speed networks”, Stochastic Networks: Theory and Applications, vol. 4, Oxford Univ. Press, 1996. • Wavelets • S. Burrus and R. Gopinath, “Introduction to Wavelets and Wavelet Transforms”, Prentice Hall, 1998. • I. Daubechies, “Ten lectures on wavelets”, SIAM, New York, 1992. • Additive model • S. Ma and C. Ji, “Modeling heterogeneous network traffic in wavelet domain”, IEEE Trans. Networking, vol. 9, no. 5, Oct 2001. • Multiplicative model • R. Riedi, M. Crouse, V. Ribeiro, R. Baraniuk, “A multifractal wavelet model with application to network traffic”, IEEE Trans. Info. Theory, vol. 45, no. 3, April 1999. • A. Feldmann, A. C. Gilbert, W. Willinger, “Data networks as cascasdes: investigating the multifractal nature of Internet WAN traffic”, ACM SIGCOMM, pp. 42-55, 1998. • P. Mannersalo and I. Norros, “Multifractal analysis of real ATM traffic: A first look”, Technical report, VTT Information Technology, 1997, COST257TD(97)19,

21. Network Inference Applications

22. Why Network Inference? • Different parts of Internet owned by different organizations • Information sharing difficult • Commerical interests/trade secrets • Privacy • Sheer volume of “network state” Each dot is one Internet Service Provider

23. Edge-based Probing • Inject probe packets into network • Infer internal properties from packet delay/loss • Current tools infer • Topology • Link bandwidths • End-to-end available bandwidth • Congestion locations

24. Cross-Traffic Inference • Simple network path – single queue • Spread of packet pair gives cross-traffic over small time interval 

25. Inferring cross-traffic over large time interval [0,T] • Probing uncertainty principle • Dense sampling: accurate inference, affect cross-traffic • Sparse sampling: less accurate inference, less influence on cross-traffic

26. Problem Statement Given N probe pairs, how must we space them over time interval [0,T] to optimally estimate the total cross-traffic in [0,T] • Answer depends on • cross-traffic • optimality criterion

27. Multiscale Cross-Traffic Model root • Choose Nleaf nodes to give best linear estimate (in terms of mean squared error) of root node • Take a guess! • Bunch probes together • Exponentially space probes pairs • Uniformly space probes over interval • Your favorite solution leaves

28. Sensor Networks Application Global average • Each sensor samples local value of process (pollution, temperature etc.) • Sensors cost money! • Find best placement for N sensors to measure global average possible sensor location

29. Independent Innovations Trees • Each node is a linear combination of parent and an independent random innovation • Optimal solution obtained by a water-filling procedure

30. Water-Filling • : arbitrary set of leaf nodes; : size of X • : leaves on left, : leaves on right • : linear min. mean sq. error of estimating root using X 0 1 2 4 3 N= • Repeat at next lower scale with N • replaced by l*N(left) and (N-l*N) (right) • Result: If innovations identically • distributed within each scale then • uniformly distribute leaves, l*N=b N/2 c fL(l) fR(l) 0 1 2 3 4 0 1 2 3 4

31. Covariance Trees • Distance : Two leaf nodes have distance j if their lowest common ancestor is at scale j • Covariance tree : Covariance between leaf nodes with distance j is cj(only a function of distance), covariance between root and any leaf node is constant,  • Positively correlated tree : cj>cj+1 • Negatively correlated tree : cj<cj+1

32. Covariance Tree Result • Result:For a positively correlated tree choosing leaf nodes uniformly in the tree is optimal. However, for negatively correlated trees this same uniform choice is the worst case! • Optimality proof:Simply construct an independent innovations tree with similar correlation structure • Worst case proof: The uniform choice maximizes sum of elements of SX Using eigen analysis show that this implies that uniform choice minimizes sum of elements of S-1X

33. Future Directions • Sampling • More general tree structures • Non-linear estimates • Non-tree stochastic processes • Traffic estimation • More complex networks • Sensor networks • jointly optimize with other constraints like power transmission

34. References • Estimation on multiscale trees • A. Willsky, “Multiresolution Markov models for signal and image processing”, Proc. of the IEEE 90(8), August 2002. • Optimal sampling on trees • V. Ribeiro, R. Riedi, and R. Baraniuk, “Optimal sampling strategies for multiscale models and their application to computer networks”, preprint.