Net Delay Tomography

1
Net Delay Tomography

• Earl Lawrence
• Joint with George Michailidis and Vijay Nair

2
General Framework

• Active probing.
• Measure end-to-end statistics on injected
  traffic.
• Tree topologies.
• Single sender with multiple receivers.
• Independence and stationarity.
• K-cast probes.
• Probes are sent from the sender to collections of
  receivers simultaneously.

3
General Framework

• Let Xk be the link delay accumulated on link k.
• We measure Yr the end-to-end delay at receiver
  r.
• We want to reconstruct the X processes the
  delay distributions.

4
Discrete Delay Problem

• Discretize the continuous delay based on a common
  bin width q.
• Xk?0,,bq where b is the maximum delay.
• Goal is to estimate ?k(i)PXkiq based on a
  collection of k-cast experiments.

5
K-Cast Experiments.

• A single k-cast experiment consists of a series
  of probes sent to a collection of k receivers.
• Special cases
• Multicast send probes to all receivers.
• Unicast send probes to one receiver.

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K-cast Example

• Some valid collections of k-cast experiments for
  this tree
• lt4,5gt, lt5,6gt, lt6,7gt, lt7,8gt,
• lt4,5,6gt, lt6,7gt, lt7,8gt,
• lt4,5gt, lt5,7gt, lt6gt, lt7,8gt,
• lt4,5,6,7,8gt.

7
K-cast Identifiability

• In order to identify the distribution of every
  link in the tree, a collection of k-cast
  experiments must meet two conditions.
• Every link in the tree much be covered by at
  least one experiment.
• Every internal node in the tree must be used as a
  splitting node for at least one experiment.

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K-cast ID Example

• Bad
• lt4,5,6gt, lt7,8gt
• lt4gt, lt5gt, lt6gt, lt7gt, lt8gt
• Good
• lt4,5gt, lt5,6gt, lt6,7gt, lt7,8gt
• lt4,5,6gt, lt6,7gt, lt7,8gt
• lt4,5gt, lt5,7gt, lt6gt, lt7,8gt
• lt4,5,6,7,8gt

9
Discrete EM

• EM is used to compute the MLE.
• The algorithm is straightforward, but can be
  somewhat intensive.
• Must compute the expected count for each possible
  link delay set.

10
Numerical Example

• Simulated data from a three-layer binary tree in
  ns-2.
• lt4,5gt, lt5,6gt, lt6,7gt, lt4,7gt
• Used a bin size of .01s.
• Estimation does well even under the violation of
  temporal dependence.

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Other Issues

• Fast Algorithms
• Distributed EM.
• Local estimation.
• Treat the data from each experiment as if it came
  from a two-layer tree.
• Combine results.
• Design
• Choosing the group of k-casts.
• Probe allocation among the experiments.

13
Continuous Delay Problem

• Parametric estimation.
• MLE is difficult, if not impossible, in most
  situations.
• Need to find efficient estimating equations.
• Similar to passive tomography of Vardi and Cao et
  al.

14
Continuous Parameter Estimation

• Consider modeling each link delay distribution as
  exponential with mean parameter ?k.
• Observe bicast probes lt2,3gt.

15
Continuous Delay Problem

• We are looking into methods of using similar
  ideas without a parametric assumption.
• Model the variance as some function of the mean.
• Just estimate means, variances, etc.
• Quasi-likelihood methods.

16
Conclusion

• Identifiability and algorithms for nonparametric
  discrete delay.
• Algorithms for continuous delay estimation.
• We hope to verify some of these techniques in the
  very near future using the Avaya testbed.