Particle Filtering in Network Tomography Mark Coates McGill University

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

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

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

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

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:

Network Tomography: The Basic Idea sender receivers

Network Tomography: The Basic Idea sender receivers

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 cross-traffic delay measurement packet pair packet(1) and packet(2) experience (nearly) identical losses and/or delays on shared links

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

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

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

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

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

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

Importance Sampling (2) Sample . Then where and

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

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

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

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)

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.

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

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

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

Observations Measurements: Observe

Tracking Algorithm (Particle Filter)

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

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 ∞

Simulation Results Delay Distributions true tracking Mean Delay time

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

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.

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