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