1. Particle Filtering in Network Tomography
Mark Coates
McGill University

2. Network mapping
(opening it up can
disturb the system)
Brain mapping
(opening it up can
disturb the system)

3. Brain Tomography unknown object statistical model measurements Maximum
likelihood
estimate
maximize
physics
data
prior knowledge
counting &
projection
MRF model
Poisson

4. Link-level Network Tomography
unknown
object
statistical
model
routing &
counting
measurements
queuing behaviour
bi/multinomial
physics
Maximum
likelihood
estimate
maximize
likelihood
data

5. Likelihood Formulation
A = routing matrix (graph)
 = packet loss probabilities
or queuing delays
for each link
y = packet losses or delays
measured at the edge
= randomness inherent in
traffic measurements
Statistical
likelihood function

6. Classical Problem Interesting if A, , or  have special structures
Solve the linear system
Interesting if A, , or  have special structures
Maximize the likelihood function
or:

7. Network Tomography: The Basic Idea
sender
receivers

8. Network Tomography: The Basic Idea
sender
receivers

9. Loss Rate Network Tomography
Measure end-to-end losses of packets
‘0’ loss
‘1’ success
‘0’ loss
‘1’ success
Identifiability Problem: Cannot isolate where losses occur !

10. Multicast or Packet-Pair Measurement
cross-traffic
delay
measurement
packet pair
packet(1) and packet(2) experience (nearly) identical
losses and/or delays on shared links

11. Packets experience the
Loss Rate Estimation
Measure end-to-end losses of packet-pairs
Packets experience the
same fate on link 1
0 0
0 1
1 0
1 1
possible
outcomes
loss on link 2
loss on link 3

12. Modelling Time Variations
x-traffic
x-traffic
Nonstationary cross-traffic induces time-variation
Directly model the dynamics (but maybe not the traffic!)
Goal is to perform online tracking (and prediction) of network link characteristics

13. Non-stationary behaviour
Introduce time-dependence in parameters
Filtering exercise (track θt ):
(1) Describe dynamic behaviour of θt
(2) Form estimate:
(MMSE)

14. Particle Filtering Objective: Estimate expectations
with respect to a sequence of distributions
known up to a normalizing constant, i.e.
Monte Carlo: Obtain N weighted samples
where
such that

15. Sequential Monte Carlo Methods
With from in hand, goal is to obtain from
Sequential methods do not repeat work.
Combine importance sampling, resampling, MCMC.

16. Importance Sampling (1)
Cannot sample directly from
Introduce an importance function (pdf)
Ensure supports match:
where importance weight

17. Importance Sampling (2)
Sample
Then
where and

18. Sequential Importance Sampling (2)
Compute weights sequentially
At time t:

19. Optimal Filtering Evolution of parameters described by function
Observation described by function
We have
Importance weight update rule:

20. Optimal Filtering Algorithm
At time t: for i = 1,...,N,
Sample
Update the importance weights
Form an estimate:

21. Key Issues Choice of importance function: Make as close to as possible
Options: prior distribution, optimal distribution, locally optimal distributions, bridging techniques, etc.
Choice should attempt to ensure that particles focus on likely regions in the state space.
Mechanisms to avoid degeneracy (sample impoverishment)

22. Resampling As time goes by, some weights become dominant.
Many particles are wasted (sample impoverishment)
Number of effective particles Neff « N.
Estimate
Resampling : each particle spawns a number of children particles (copies)
Number of children C(i) related (proportional) to weight.
May introduce jitter in children to reduce clustering effects.

23. Delay Distribution Tracking
Time-varying delay distribution of window size R at time m
Delay unit
In each window, R probe measurements.
Form estimates of average delay and jitter over short time intervals
time
Delay units

24. Optimal Filtering Evolution of parameters described by dynamic model
Observations described by function
Interested in forming estimate of:
where
Estimate is:

25. Dynamic model Queue/traffic model:
reflected random walk on [0,max_del]
Probability
Delay units

26. Observations
Measurements:
Observe

27. Tracking Algorithm (Particle Filter)

28. Estimation of Delay Distributions
Sequential Monte Carlo Approximation to posterior mean estimate:
Message-passing algorithm
Particle weights
Estimate of time-varying delay distribution:

29. Analysis Convergence analysis of [Crisan, Doucet 01 ] applies.
Complexity: per measurement
Average Number of Unique Links
Number of Particles
Max. delay units per link
Convergence analysis of [Crisan, Doucet 01 ] applies.
The approximation to the posterior mean estimate converges to the true estimate as N ∞

30. Simulation Results
Delay Distributions
true
tracking
Mean Delay
time

31. Tracking shadow prices
Explicit congestion notification pricing mechanisms
Price variable maintained at each queue in the network.
Related to congestion, but not a specific performance measure (such as loss rate, queuing delay).
REM (random exponential marking)
Price p, marking probability m, total link traffic y, target queue length b* , measured queue length b

32. Tracking shadow prices (2)
Observations (relatively easy to collect !)
For one path:
nt : total traffic along a path defined by a row of routing matrix A during time period t.
xt : marked packets along same path.

33. Why Dynamic Models/Particle Filtering?
Summary
Why Dynamic Models/Particle Filtering?
Dynamic models allow us to account for non-stationarity
but it is difficult to generate and incorporate dynamic models derived from realistic traffic models
Particle filtering only appropriate when analytical techniques fail
non-Gaussian or non-linear dynamics or observations
Sequential structure allows on-line implementation
care must be taken to reduce computation at each step
Convergence, optimality results available
provided particle filters satisfy fairly mild constraints