particle filtering in network tomography mark coates mcgill university
TRANSCRIPT
Particle Filtering in Network Tomography
Mark CoatesMcGill University
Brain mapping(opening it up candisturb the system)
Network mapping(opening it up candisturb the system)
Brain Tomography
unknown object
statistical model
measurements
Maximumlikelihood estimate
maximizelikelihood
physics
data
prior knowledge MRF model
counting &projection
Poisson
unknown object
statistical model
measurements
Maximumlikelihood estimate
maximizelikelihood
physics
data
Link-level Network Tomography
queuing behaviour
routing &counting
bi/multinomial
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
),|(),( AyfAl Statistical likelihood function
Ay
Likelihood Formulation
Classical Problem
Ay
Solve the linear system
Interesting if A, , or have special structures
)|()( Ayfl Maximize the likelihood function
)|(),( AyfAl or:
Network Tomography: The Basic Ideasender
receivers
sender
receivers
Network Tomography: The Basic Idea
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 !
Multicast or Packet-Pair Measurement
measurement packet pair
cross-traffic(2)packet (1)packet
(2)packet (1)packet
delay
packet(1) and packet(2) experience (nearly) identicallosses and/or delays on shared links
Loss Rate Estimation
Measure end-to-end losses of packet-pairs
0 00 11 01 1
possible outcomes loss on link 2
loss on link 3
Packets experience thesame fate on link 1
Modelling Time Variations
x-trafficx-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
Non-stationary behaviour
Introduce time-dependence in parameters
t t t ty A Filtering exercise (track θt ):
1:( | )ˆ [ ]
t tt p y t E
(1) Describe dynamic behaviour of θt
(2) Form estimate: (MMSE)
Particle Filtering
Objective: Estimate expectations 0: 0:( ) ( )t t th d with respect to a sequence of distributionsknown up to a normalizing constant, i.e.
Monte Carlo: Obtain N weighted samples
0t t
0: 0: 0:( ) ( )t t t t td d
( ) ( )0: 1, ,
,i it t i Nw
( ) ( )
1
0, 1N
i it t
i
w w
where such that
( ) ( )0: 0: 0:
1
Ni it t t t tN
i
w h h d
Sequential Monte Carlo Methods
• With from in hand, goal is to
obtain from .
• Sequential methods do not repeat work.
• Combine importance sampling, resampling, MCMC.
1t ( ) ( )0: 1 1 1, ,
,i it t i Nw
( ) ( )0: 1, ,
,i it t i Nw
t
Importance Sampling (1)
• Cannot sample directly from .
• Introduce an importance function (pdf)
• Ensure supports match:
where importance weight
0:t t
0:t tq
0: 0:0 0t t t tq
0: 0:0:
0: 0: 0:
t t t tt t
t t t t t
w q
w q d
0:0:
0:
t tt t
t t
wq
Importance Sampling (2)
• Sample .
• Then
where and
( )0: ~it tq
( )0:
( )0: 0:
1
( )0:
ˆ it
NN it t t t
i
it t t
d w d
w w
( )
1
1N
it
i
w
Sequential Importance Sampling (2)• Compute weights sequentially
• At time t:
0:0:
0:
0: 1 0: 11 0: 1
0: 1 0: 1
t tt t
t t
t t t tt t
t t t t
wq
qw
q
0: 1 0: 1 0: 1|t t t t t t tq q q
0:
0: 1 0: 1
1 0: 1 0: 1|t t
t t t t
t t t t t
w wq
Optimal Filtering
• Evolution of parameters described by function
• Observation described by function
• We have
• Importance weight update rule:
0: 0: 1:( | )t t t tp y
1
0: 1 0: 1
0: 1 1:
| |
| ,t t t t
t t t t
t t t t
f g yw w
q y
1( | )t tf
( | )t tg y
0:
0: 1 0: 1
1 0: 1 0:|t t
t t t t
t t t t t
w wq
Optimal Filtering Algorithm
• At time t: for i = 1,...,N,
• Sample
• Update the importance weights
• Form an estimate:
( ) ( )0: 1 1:~ | ,i i
t t tq y
( ) ( ) ( )1( ) ( )
1( ) ( )0: 1 1:
| |
| ,
i i it t t ti i
t ti it t t t
f g yw w
q y
( ) ( )1
1
ˆN
i it t t
i
w
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)
0:t t
0:t tq
0:t tq
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.
2( )1 ieff tN w
Delay Distribution Tracking• Time-varying delay distribution of window size R at time m
• In each window, R probe measurements.
• Form estimates of average delay and jitter over short time intervals
)(, kT Rm
time
Delay units
Delay unit
Optimal Filtering
• Evolution of parameters described by dynamic model
• Observations described by function
• Interested in forming estimate of:
where .
• Estimate is:
( | , )m m mg y ),|,( 11 mmmmf
, 1: , 1:( | ) ( | , ) ( | )j m m j m m m m mp x y p x y d y
( ) ( ) ( ), 1: ,
1
ˆ ( | ) ( | , , )N
i i ij m m j m m m m m
i
p x y p x y w
,m m m
Dynamic model• Queue/traffic model:
reflected random walk on [0,max_del]
),0(loglog 2,1, Nmjmj
mj ,
)exp()( ,,, mjmjmj kkp
Delay units
Probability
Observations
• Measurements:
Observe
)(~ ,, kpx mjmj
2,1),(,,
ixymjPathsmsmj
)(packet(1) m)(packet(2) m
)()2( my )()1( my
Limimi },{ ,,
Limimi },{ 1,1,
Tracking Algorithm (Particle Filter)
Estimation of Delay Distributions
• Sequential Monte Carlo Approximation to posterior mean estimate:
)()()(
1,, ),,|()(ˆ i
mi
mi
mm
N
imjmj wykxpkp
Message-passing algorithm
• Estimate of time-varying delay distribution:
Particle weights
, , 1:1
1ˆ ˆ( ) ( | )m
m R j l ll m R
T k p x yR
Analysis
• Complexity: per measurement)( 2NLKO
Average Number of Unique Links
Max. delay units per link
Number of Particles
• Convergence analysis of [Crisan, Doucet 01 ] applies.
• The approximation to the posterior mean estimate converges to the true estimate as N ∞
time
Mean Delay
Delay Distributions
Simulation Results
true
tracking
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
*1
1 t
t t t t
pt
p p b b y c
m
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.
~ ,
1 t
t t t
Apt
x Bi n q
q
SummaryWhy 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