Multiscale Models for Network Traffic

Vinay Ribeiro

Rolf Riedi, Matt Crouse, Rich Baraniuk

Dept. of Electrical EngineeringRice University(Houston, Texas)

Outline

• Multiscale nature of network traffic

• Wavelets

• Wavelet models for traffic

• Network inference applications

Time Scales

timeunit

2n

2

1

(discrete time)

Multiscale Nature of Network Traffic

• 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

6ms

time unit

600msInternet

bytes/time trace

(LBL’93)

i.i.d. time series

(lognormal)

Fractional Gaussian Noise (fGn)

• Stationary Gaussian process,

• Covariance (Hurst parameter: 0<H<1)

• Long-range dependence (LRD) if ½<H<1

• Second-order self-similarity

Variance-timeplot

fGn is a 1/f-Process

• Power spectral density decays in a 1/f fashion– Low frequency components long-term

correlations

frequency

power

Towards Generalizations of fGn

• Variance decay of traffic not always straight line like fGn

• Goal: develop LRD models – Generalize fGn– Parsimonious (few parameters)– Fast synthesis for simulations

Variance-timeplot

Auckland Univ.Traffic

time scale

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

The Haar Wavelet Basis

Computing the Haar Transform

• Wavelet Transform: fine to coarse (bottom to top)• Inverse Wavelet Transform: coarse to fine (top to bottom)

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

Wavelets “Decorrelate” 1/f Processes

• 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

frequency

frequency

1/f spectrum

time domain1/f

strong correlation

wavelet domainnot 1/f

weak correlation

power

power

Haar Wavelet “Additive” 1/f Model

• Choose Wj,k i.i.d. within scale j

• Set var(Wj,k) to obtain required decay of var(Vj,k)

• Fast O(N) synthesis

• log2(N) parameters

• Asymptotically Gaussian

Sample Realization

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

Multiplicative Cascade Model

• Replace additive innovations by multiplicative innovations

• Aj,k 2 [0,1], example -distribution

• Choose var(Aj,k) to get appropriate decay

of var(Vj,k)

• Fast O(N) synthesis

• log2(N) parameters

• Positive data

• Asymptotically lognormal at fine time scales

Sample Realization

• Data is positive

• Same var(Vj,k) as additive model

Additive vs. Multiplicative Models

• Multiplicative model marginals closer to real data than additive model

Additive model Multiplicative modelInternet data(Auckland Univ)

6ms

12ms

24ms

timeunit

Queuing Experiment

• Additive and multiplicative models same var(Vj,k)

• Multiplicative model outperforms additive model• High-order moments can influence queuing (open loop)

real traffic

multiplicativemodel

additive model

Kilo bytes

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)

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,

Network Inference Applications

Why Network Inference?

Each dot is one Internet Service Provider

• Different parts of Internet owned by different organizations

• Information sharing difficult– Commerical

interests/trade secrets

– Privacy– Sheer volume of

“network state”

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

Cross-Traffic Inference

• Simple network path – single queue

• Spread of packet pair gives cross-traffic over small time interval

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

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

Multiscale Cross-Traffic Model

• Choose N leaf 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

root

leaves

Sensor Networks Application

• Each sensor samples local value of process (pollution, temperature etc.)

• Sensors cost money!• Find best placement for N sensors to measure

global average

Global average

possiblesensor location

Independent Innovations Trees

• Each node is a linear combination of parent and an independent random innovation

• Optimal solution obtained by a water-filling procedure

• : arbitrary set of leaf nodes; : size of X• : leaves on left, : leaves on right• : linear min. mean sq. error of estimating root using X •

•

•

Water-Filling

0 1 2 3 40 1 2 3 4

fL(l) f

R(l)

N=01243

• 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

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

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

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

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.