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Modeling end-to-end internet delays using mixtures of Weibull distributions Iain W. Phillips and Jos e A. Hern andez Computer Science, Loughborough University July 2004 Iain W. Phillips and Jos e A. Hern andez Weibull Mixtures

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Modeling end-to-end internet delays using mixtures of Weibull distributions

Iain W. Phillips and Jos´ e A. Hern´ andez

Computer Science, Loughborough University

July 2004

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Introduction

History of work at Loughborough Other Measurement projects Visualisation of Measurements Mathematical Modelling Applications

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Ancient History

In 1994 JANET → SuperJanet, contract won by BT Built over SMDS—Switched Multi-megabit Data Service, and ATM networks University Research Initiative—Managing Multiservice Networks

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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MMN—Loughborough

Performance Monitoring and Measurement Researched and built a delay measurement tool Active Sender Used GPS for synchronisation Accurate to about 10µs

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Performance Monitoring

Ethernet MegaStream

R R R R R F M A M M R C SMDS

Manchester Birmingham Bristol Edinburgh London Loughborough

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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What causes performance problems?

Routing misconfiguration Link or Node failure Aggressive Applications

Peer-to-peer, video streaming, online gaming etc

Denial of Service attacks

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Visualisation

Tools to reduce working load of network operators FDV—Figurable Deformity Visualisation TMT—Trunk Monitoring Tool

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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FDV

2,3,4 small Degree of freedom 4 2 3 1 Reference 2,3,4 large 2 large 1 small 1 large Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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TMT

Trunk Monitoring Tool Uses SNMP to query trunk information from SMDS switches Presents this in a “single-look” view to operators. Deployed April 200

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Exceptions

Interesting Network Events, detected by: Manual Rule-based Neural networks All based on simple statistics, max in day, min in day, mean, max - min, variance etc

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Other Monitoring Projects

RIPE-NCC—Monitoring (mostly) European Delays SPRINT (US)—Monitoring for Traffic Engineering NLANR—Traceroute/ping delays Waikato (NZ) DAG hardware traffic capture Cambridge/Loughborough (EE) passive monitoring new UKLIGHTmas(t)

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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What to do next . . .

Can statistics/mathematics improve such displays? Can we predict Internet performance like the weather? How do we model?

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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The rest of this talk

Motivation Traffic modelling review Mixing Weibull distributions Expectation Maximisation algorithm Experiments and results Applications and discussion

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Motivation

The need to model network performance: Metrics to define network performance Low-level quantities: delay and loss End-to-end network performance status Packet probes such as ping or one-way delay UDP packets

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Previous work:

Traffic modelling and delay distributions: Network traffic shows self-similarity and long-range dependency. Current traffic strategies search for models compliant to these empirical properties: fBm, fARIMA, FSD, etc. When inputting such traffics into routers, the queue distribution exhibit heavy-tail distributions. Such distribution can be approximated to Weibull for the particular case of fBm. Such result has been previously validated in a single hop scenario.

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Previous work:

Traffic modelling and delay distributions: Our aim is to model multiple-hop (or end-to-end) delays with a combination of several Weibull distributions.

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Mixing Weibull distributions:

The Weibull distribution p(x|r, s) = sxs−1

rs

exp(− x

r

s)

1 2 3 4 5 6 7 8 9 10 0.002 0.004 0.006 0.008 0.01 x p(x|r,s) The Weibull distribution r=4 fixed 1 2 3 4 5 6 7 8 9 10 0.005 0.01 0.015 0.02 x p(x|r,s) s=4 fixed

r is concerned with the mode location. s is related to tail behaviour.

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Mixing Weibull distributions:

Problem statement: Let us assume we are given a sample of N delay measurements x = [x1, .., xN], which are supposed to be drawn from M Weibull distributions: [p(x|θ1), .., p(x|θM)] The result is: p(x|model) = M

j=1 αjp(x|θj)

αj = weight of the j-th component of the mixture. Obviously,

j αj = 1

θj = [rj, sj] shape and scale parameters of the j-th Weibull distribution Finding α and θ appropriate to best fit delay histograms represented by the measurements sample x

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Mixing Weibull distributions:

Expectation Maximisation To proceed, second random variable y, referred to as labels, is necessary to complete the problem formulation. p(yi = j|xi, Θ) = the probability of data xi being drawn from the j-th component of the mixture. Obviously,

p(xi|yi = j, Θ) = p(xi|θj), and p(yi = j|Θ) = αj

With this formulation EM defines an iterative procedure to obtain the maximum likelihood estimates, based on two steps:

E-step: Q(Θ, Θ(t)) = E[log L(Θ|x, y)|x, Θ(t)] M-step: Θ(t+1) = arg maxΘ Q(Θ, Θ(t))

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Mixing Weibull distributions:

Computing EM Expanding E-step: Q(Θ, Θ(t)) = M

j=1

N

i=1

  • log p(xi|θj)
  • p(yi = j|xi, Θ(t))

+ M

j=1

N

i=1

  • log αj
  • p(yi = j|xi, Θ(t))

Maximising: ∂Q(Θ, Θ(t)) ∂αj = 0 ∂Q(Θ, Θ(t)) ∂θj = 0

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Mixing Weibull distributions:

EM applied to mixtures of Weibull distributions

1 Computing parameters:

αj = 1

N

N

i=1 p(yi = j|xi, Θ)

rj = PN

i=1 x sj i p(yi=j|xi,Θ)

PN

i=1 p(yi=j|xi,Θ)

1/sj sj =

PN

i=1 p(yi=j|xi,Θ)

PN

i=1

x

sj i r sj j

−1

  • log
  • xi

rj

  • p(yi=j|xi,Θ)

2 Updating hidden probs:

p(yi = j|xi, Θ) =

αjp(x|θj) PM

k=1 αkp(xi|θk) Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Mixing Weibull distributions

Convergence speed - Initialisation

5 10 15 20 21 22 23 24 25 delay (ms) Delay over time 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF Histogram and its modelling Hist. Model 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF delay (ms) Mixture and its components Total model Single comps. 21 22 23 24 25 21 22 23 24 25 Percentiles of the data sample Quantiles of the model Q−Q plot

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Mixing Weibull distributions

Convergence speed - After 1 iteration

5 10 15 20 21 22 23 24 25 delay (ms) Delay over time 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF Histogram and its modelling 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF delay (ms) Mixture and its components 21 22 23 24 25 21 22 23 24 25 Percentiles of the data sample Quantiles of the model Q−Q plot Hist. Model Total model Single comps.

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Mixing Weibull distributions

Convergence speed - After 2 iterations

5 10 15 20 21 22 23 24 25 delay (ms) Delay over time 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF Histogram and its modelling 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF delay (ms) Mixture and its components 21 22 23 24 25 21 22 23 24 25 Percentiles of the data sample Quantiles of the model Q−Q plot Hist. Model Total model Single comps.

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Mixing Weibull distributions

Convergence speed - After 3 iterations

5 10 15 20 21 22 23 24 25 delay (ms) Delay over time 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF Histogram and its modelling 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF delay (ms) Mixture and its components 21 22 23 24 25 21 22 23 24 25 Percentiles of the data sample Quantiles of the model Q−Q plot Hist. Model Total model Single comps.

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Mixing Weibull distributions

Convergence speed - After 4 iterations

5 10 15 20 21 22 23 24 25 delay (ms) Delay over time 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF Histogram and its modelling 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF delay (ms) Mixture and its components 21 22 23 24 25 21 22 23 24 25 Percentiles of the data sample Quantiles of the model Q−Q plot Hist. Model Total model Single comps.

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Mixing Weibull distributions

Convergence speed - After 5 iterations

5 10 15 20 21 22 23 24 25 delay (ms) Delay over time 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF Histogram and its modelling 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF delay (ms) Mixture and its components 21 22 23 24 25 21 22 23 24 25 Percentiles of the data sample Quantiles of the model Q−Q plot Hist. Model Total model Single comps.

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Mixing Weibull distributions

Convergence speed - After 10 iterations

5 10 15 20 21 22 23 24 25 delay (ms) Delay over time 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF Histogram and its modelling 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF delay (ms) Mixture and its components 21 22 23 24 25 21 22 23 24 25 Percentiles of the data sample Quantiles of the model Q−Q plot Hist. Model Total model Single comps.

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Mixing Weibull distributions

Convergence speed - After 15 iterations

5 10 15 20 21 22 23 24 25 delay (ms) Delay over time 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF Histogram and its modelling 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF delay (ms) Mixture and its components 21 22 23 24 25 21 22 23 24 25 Percentiles of the data sample Quantiles of the model Q−Q plot Hist. Model Total model Single comps.

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Mixing Weibull distributions

Convergence speed - After 25 iterations

5 10 15 20 21 22 23 24 25 delay (ms) Delay over time 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF Histogram and its modelling 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF delay (ms) Mixture and its components 21 22 23 24 25 21 22 23 24 25 Percentiles of the data sample Quantiles of the model Q−Q plot Hist. Model Total model Single comps.

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Mixing Weibull distributions

Convergence speed - After 50 iterations

5 10 15 20 21 22 23 24 25 delay (ms) Delay over time 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF Histogram and its modelling 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF delay (ms) Mixture and its components 21 22 23 24 25 21 22 23 24 25 Percentiles of the data sample Quantiles of the model Q−Q plot Hist. Model Total model Single comps.

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Mixing Weibull distributions

Convergence speed - After 100 iterations

5 10 15 20 21 22 23 24 25 delay (ms) Delay over time 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF Histogram and its modelling 21 22 23 24 25 0.2 0.4 0.6 0.8 1 1.2 PDF delay (ms) Mixture and its components 21 22 23 24 25 21 22 23 24 25 Percentiles of the data sample Quantiles of the model Q−Q plot Hist. Model Total model Single comps.

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Experiments and results

Measurement testbed The following delay measurements, provided by RIPE NCC1, have been utilised for this experiments. 245 24-hour exps. ≈ 3000 meas. per exp.

  • Total: ≈ 700, 000 measurements

GPS accuracy ≈ few hundred of nanoseconds error. Matching error = √P(hist-model)2

P hist

× 100%

1http://www.ripe.net

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Experiments and results

Full experiments model validation

5 10 15 20 25 30 35 10 20 30 40 50 60 % matching error Cases Results

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Experiments and results

Example of a five Weibull matching result

5 10 15 20 64 64.1 64.2 64.3 64.4 64.5 64.6 64.7 delay (ms) Delay over time 64 64.2 64.4 64.6 1 2 3 4 5 PDF Histogram and its modelling 64 64.2 64.4 64.6 1 2 3 4 5 PDF delay (ms) Mixture and its components 64 64.2 64.4 64.6 64 64.1 64.2 64.3 64.4 64.5 64.6 64.7 Percentiles of the data sample Quantiles of the model Q−Q plot Hist. Model Total model Single comps.

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Experiments and results

Example of Parameter Evolution

5 10 15 20 13 14 15 16 17 18 19 20 delay (ms) Delay over time 14 16 18 20 0.2 0.4 0.6 0.8 PDF Mixture and its components delay (ms) Histogram Total model Single comps. 5 10 15 20 0.2 0.4 0.6 0.8 Evolution of weight q1 Time of day r=0.56 s=2.1 5 10 15 20 0.2 0.4 0.6 0.8 Evolution of weight q2 Time of day r=2.2 s=1.3

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Discussion

Two main conclusions arise from this work: A combination of Weibull distributions look very suitable to match end-to-end delay histograms. The Weibull parameters impact the appearance of the Weibull distribution.

r is related to the location of the mode/maximum/peak for that particular Weibull component. s concerns tail behaviour: the smaller the slower the tail decays.

Expectation Maximisation is a suitable algorithm to find the parameters defining such model, both easily and

  • ptimally.

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Applications

Performance related applications: Traffic engineering. Fault tolerance and troubleshooting. Provisioning. Admission control. . . .

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures

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Thanks

Other researchers at Loughborough especially: David Parish, Omar Bashir, Mark Sandford, Antony Pagonis. Jose’s PhD is a Loughborough CS Scholarship. RIPE for 1 year’s (75GB) measurements.

Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures