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Self-Similarity in Network Traffic. Kevin Henkener 5/29/2002. What is Self-Similarity?. Self-similarity describes the phenomenon where a certain property of an object is preserved with respect to scaling in space and/or time.

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Self similarity in network traffic

Self-Similarity in Network Traffic

Kevin Henkener


What is self similarity
What is Self-Similarity?

  • Self-similarity describes the phenomenon where a certain property of an object is preserved with respect to scaling in space and/or time.

  • If an object is self-similar, its parts, when magnified, resemble the shape of the whole.

The famous data
The Famous Data

  • Leland and Wilson collected hundreds of millions of Ethernet packets without loss and with recorded time-stamps accurate to within 100µs.

  • Data collected from several Ethernet LAN’s at the Bellcore Morristown Research and Engineering Center at different times over the course of approximately 4 years.

Why is self similarity important
Why is Self-Similarity Important?

  • Recently, network packet traffic has been identified as being self-similar.

  • Current network traffic modeling using Poisson distributing (etc.) does not take into account the self-similar nature of traffic.

  • This leads to inaccurate modeling which, when applied to a huge network like the Internet, can lead to huge financial losses.

Problems with current models
Problems with Current Models

  • Current modeling shows that as the number of sources (Ethernet users) increases, the traffic becomes smoother and smoother

  • Analysis shows that the traffic tends to become less smooth and more bursty as the number of active sources increases

Problems with current models cont d
Problems with Current Models Cont.’d

  • Were traffic to follow a Poisson or Markovian arrival process, it would have a characteristic burst length which would tend to be smoothed by averaging over a long enough time scale. Rather, measurements of real traffic indicate that significant traffic variance (burstiness) is present on a wide range of time scales

Definitions and properties
Definitions and Properties

  • Long-range Dependence

    • covariance decays slowly

  • Hurst Parameter

    • Developed by Harold Hurst (1965)

    • H is a measure of “burstiness”

      • also considered a measure of self-similarity

    • 0 < H < 1

    • H increases as traffic increases

Definitions and properties cont d
Definitions and Properties Cont.’d

  • low, medium, and high traffic hours

  • as traffic increases, the Hurst parameter increases

    • i.e., traffic becomes more self-similar

Self similar measures
Self-Similar Measures

  • Background

    • Let time series: X = (Xt : t = 0, 1, 2, ….) be a covariance stationary stochastic process

    • autocorrelation function: r(k), k ≥ 0

    • assume r(k) ~ k-βL(t), as k∞where 0 < β < 1

      • limt∞ L(tx) / L(t) = 1, for all x > 0

Second order self similar
Second-order Self-Similar

  • Exactly

    • A process X is called (exactly) self-similar with self-similarity parameter H = 1 – β/2 if

    • for all m = 1, 2, …. var(X(m)) = σ2m-β

    • r(m)(k) = r(k), k ≥ 0

  • Asymptotically

    • r(m)(k) = r(k), as m∞

    • aggregated processes are the same

  • Current model shows aggregated processes tending to pure noise

Measuring self similarity
Measuring Self-Similarity

  • time-domain analysis based on R/S statistic

  • analysis of the variance of the aggregated processes X(m)

  • periodogram-based analysis in the frequency domain

Methods of modeling self similar traffic
Methods of Modeling Self-Similar Traffic

  • Two formal mathematical models that yield elegant representations of self-similarity

    • fractional Gaussian noise

    • fractional autoregressive integrated moving-average processes


  • Ethernet traffic is self-similar irrespective of time

  • Ethernet traffic is self-similar irrespective of where it is collected

  • The degree of self-similarity measured in terms of the Hurst parameter h is typically a function of the overall utilization of the Ethernet and can be used for measuring the “burstiness” of the traffic

  • Current traffic models are not capable of capturing the self-similarity property

Results cont d
Results Cont.’d

  • There exists the presence of concentrated periods of congestion at a wide range of time scales

  • This implies the existence of concentrated periods of light network load

  • These two features cannot be easily controlled by traffic control.

    • i.e., burstiness cannot be smoothed

Results cont d1
Results Cont.’d

  • These two implications make it difficult to allocated services such that QOS and network utilization are maximized.

  • Self-similar burstiness can lead to the amplification of packet loss.

Problems with packet loss
Problems with Packet Loss

  • Effects in TCP

    • TCP guarantees that packets will be delivered and will be delivered in order

    • When packets are lost in TCP, the lost packets must be retransmitted

    • This wastes valuable resources

  • Effects in UDP

    • UDP sends packets as quickly as possible with no promise of delivery

    • When packets are lost, they are not retransmitted

    • Repercussions for packet loss in UDP include “jitter” in streaming audio/video etc.

Possible methods for dealing with the self similar property of traffic
Possible Methods for Dealing with the Self-Similar Property of Traffic

  • Dynamic Control of Traffic Flow

  • Structural resource allocation

Dynamic control of traffic flow
Dynamic Control of Traffic Flow of Traffic

  • Predictive feedback control

    • identify the on-set of concentrated periods of either high or low traffic activity

    • adjust the mode of congestion control appropriately from conservative to aggressive

Dynamic control of traffic flow cont d
Dynamic Control of Traffic Flow Cont.’d of Traffic

  • Adaptive forward error correction

    • retransmission of lost information is not viable because of time-constraints (real-time)

    • adjust the degree of redundancy based on the network state

      • increase level of redundancy when traffic is high

        • could backfire as too much of an increase will only further aggrevate congestion

      • decrease level of redundancy when traffic is low

Structural resource allocation
Structural Resource Allocation of Traffic

  • Two types:

    • bandwidth

    • buffer size

  • Bandwidth

    • increase bandwidth to accommodate periods of “burstiness”

    • could be wasteful in times of low traffic intensity

Structural resource allocation cont d
Structural Resource Allocation Cont.’d of Traffic

  • buffer size

    • increase the buffer size in routers (et. al.) such that they can absorb periods of “burstiness”

    • still possible to fill a given router’s buffer and create a bottleneck

  • tradeoff

    • increase both until they complement each other and begin curtailing the effects of self-similarity