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
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
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
MMN—Loughborough Performance Monitoring and Measurement Researched and built a delay measurement tool Active Sender Used GPS for synchronisation Accurate to about 10 µ s Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures
Performance Monitoring SMDS R Edinburgh R Manchester Loughborough R Birmingham Bristol R MegaStream London R R C Ethernet M M M F A Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures
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
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
FDV 2 Degree of freedom 1 3 4 Reference 1 small 2 large 2,3,4 large 1 large 2,3,4 small Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures
Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures
Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures
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
Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures
Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures
Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures
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
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
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
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
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
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
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
Mixing Weibull distributions: � s ) The Weibull distribution p ( x | r , s ) = sx s − 1 � x exp( − r s r The Weibull distribution r is concerned with the 0.01 r=4 fixed 0.008 mode location. 0.006 p(x|r,s) 0.004 s is related to tail 0.002 0 behaviour. 0 1 2 3 4 5 6 7 8 9 10 x 0.02 s=4 fixed 0.015 p(x|r,s) 0.01 0.005 0 0 1 2 3 4 5 6 7 8 9 10 x Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures
Mixing Weibull distributions: Problem statement: Let us assume we are given a sample of N delay measurements x = [ x 1 , .., x N ], which are supposed to be drawn from M Weibull distributions: [ p ( x | θ 1 ) , .., p ( x | θ M )] The result is: p ( x | model ) = � M j =1 α j p ( x | θ j ) α j = weight of the j -th component of the mixture. Obviously, � j α j = 1 θ j = [ r j , s j ] 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
Mixing Weibull distributions: Expectation Maximisation To proceed, second random variable y , referred to as labels, is necessary to complete the problem formulation. p ( y i = j | x i , Θ) = the probability of data x i being drawn from the j -th component of the mixture. Obviously, p ( x i | y i = j , Θ) = p ( x i | θ j ), and p ( y i = j | Θ) = α j 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 max Θ Q (Θ , Θ ( t ) ) Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures
Mixing Weibull distributions: Computing EM Expanding E-step: � M � N Q (Θ , Θ ( t ) ) = � � p ( y i = j | x i , Θ ( t ) ) log p ( x i | θ j ) j =1 i =1 + � M � N � � p ( y i = j | x i , Θ ( t ) ) log α j j =1 i =1 Maximising: ∂ Q (Θ , Θ ( t ) ) = 0 ∂α j ∂ Q (Θ , Θ ( t ) ) = 0 ∂θ j Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures
Mixing Weibull distributions: EM applied to mixtures of Weibull distributions 1 Computing parameters: � N α j = 1 i =1 p ( y i = j | x i , Θ) N sj � 1 / s j � P N i p ( y i = j | x i , Θ) i =1 x r j = P N i =1 p ( y i = j | x i , Θ) P N i =1 p ( y i = j | x i , Θ) s j = sj � x � � � xi P N i − 1 log p ( y i = j | x i , Θ) sj i =1 rj r j 2 Updating hidden probs: α j p ( x | θ j ) p ( y i = j | x i , Θ) = P M k =1 α k p ( x i | θ k ) Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures
Mixing Weibull distributions Convergence speed - Initialisation Delay over time Histogram and its modelling 1.2 Hist. 25 Model 1 24 delay (ms) 0.8 PDF 23 0.6 0.4 22 0.2 21 0 0 5 10 15 20 21 22 23 24 25 Mixture and its components Q−Q plot 1.2 Total model 25 Single comps. Quantiles of the model 1 24 0.8 PDF 0.6 23 0.4 22 0.2 21 0 21 22 23 24 25 21 22 23 24 25 delay (ms) Percentiles of the data sample Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures
Mixing Weibull distributions Convergence speed - After 1 iteration Delay over time Histogram and its modelling 1.2 Hist. 25 Model 1 24 delay (ms) 0.8 PDF 23 0.6 0.4 22 0.2 21 0 0 5 10 15 20 21 22 23 24 25 Mixture and its components Q−Q plot 1.2 Total model 25 Single comps. Quantiles of the model 1 24 0.8 PDF 0.6 23 0.4 22 0.2 21 0 21 22 23 24 25 21 22 23 24 25 delay (ms) Percentiles of the data sample Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures
Mixing Weibull distributions Convergence speed - After 2 iterations Delay over time Histogram and its modelling 1.2 Hist. 25 Model 1 24 delay (ms) 0.8 PDF 23 0.6 0.4 22 0.2 21 0 0 5 10 15 20 21 22 23 24 25 Mixture and its components Q−Q plot 1.2 Total model 25 Single comps. Quantiles of the model 1 24 0.8 PDF 0.6 23 0.4 22 0.2 21 0 21 22 23 24 25 21 22 23 24 25 delay (ms) Percentiles of the data sample Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures
Mixing Weibull distributions Convergence speed - After 3 iterations Delay over time Histogram and its modelling 1.2 Hist. 25 Model 1 24 delay (ms) 0.8 PDF 23 0.6 0.4 22 0.2 21 0 0 5 10 15 20 21 22 23 24 25 Mixture and its components Q−Q plot 1.2 Total model 25 Single comps. Quantiles of the model 1 24 0.8 PDF 0.6 23 0.4 22 0.2 21 0 21 22 23 24 25 21 22 23 24 25 delay (ms) Percentiles of the data sample Iain W. Phillips and Jos´ e A. Hern´ andez Weibull Mixtures
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