Ant Colony Optimized Importance Sampling: Principles ...€¦ · Ant Colony Optimized Importance Sampling: Principles, Applications and Challenges Poul E. Heegaard Department of Telematics

7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

Ant Colony Optimized Importance Sampling: Principles, Applications and

Challenges

Poul E. Heegaard Department of Telematics

Norwegian University of Science and Technology (NTNU)

Werner Sandmann Department Information Systems & Applied Computer Science

University of Bamberg


Simulation problem

•  Strict QoS requirements need to be validated –  Analytic models need (too) strict assumptions be be solved –  Numerical solutions may suffer from state space explosion –  Hard to simulate, e.g. loss ratio less than 10-7 takes forever –  Importance sampling might work if parameters can be found

•  This presentation –  Model type handled –  Speed-up simulation approach –  Swarm technique for adaptation of simulation

parameters –  Numerical results –  Inner workings of adaptive scheme


•  D-dimensional discrete-state models (examples are Markovian) •  Finite or infinite restrictions in each dimension •  Transition affects at most two dimensions •  In paper described by transition classes

where –  Source state space –  Destination state function –  Transition rate function

•  The model is applied in performance and dependability evaluation of Optical Packet Switched networks [other papers by the authors]

Model description


Model example

0,0 1,0 2,0 3,0 ..,0 W-1,0 W,0

0,1 1,1 2,1 ...,1 W-2,1 W-1,1

0,2 1,2 2,2 ...,2 W-2,2

0,3 1,... 2,... ...,...

0,... 1,W-2 2,W-2

0,W-1 1,W-1

0,W

State with loss

Packet arrival rate Packet transmission time

λ 1/µ

λ µ


Simulation problem

0,0 1,0 2,0 3,0 ..,0 W-1,0 W,0

0,1 1,1 2,1 ...,1 W-2,1 W-1,1

0,2 1,2 2,2 ...,2 W-2,2

0,3 1,... 2,... ...,...

0,... 1,W-2 2,W-2

0,W-1 1,W-1

0,W

When λ≪µ then rare packet loss

λ µ

State with loss


λ 1/µ


Simulation with importance sampling

0,0 1,0 2,0 3,0 ..,0 W-1,0 W,0

0,1 1,1 2,1 ...,1 W-2,1 W-1,1

0,2 1,2 2,2 ...,2 W-2,2

0,3 1,... 2,... ...,...

0,... 1,W-2 2,W-2

0,W-1 1,W-1

0,W

Simulate with λ*≫µ* ⇒ packet loss not rare

λ* µ*

State with loss


λ*

1/µ*


The problem with importance sampling

0,0 1,0 2,0 3,0 ..,0 W-1,0 W,0

0,1 1,1 2,1 ...,1 W-2,1 W-1,1

0,2 1,2 2,2 ...,2 W-2,2

0,3 1,... 2,... ...,...

0,... 1,W-2 2,W-2

0,W-1 1,W-1

0,W

How to determine the λ* and µ*? -  scale too little - no effect - scale too much - biased estimates

λ* µ*

State with loss


λ*

1/µ*


Adaptive change of measure in IS

0,0 1,0 2,0 3,0 ..,0 W-1,0 W,0

0,1 1,1 2,1 ...,1 W-2,1 W-1,1

0,2 1,2 2,2 ...,2 W-2,2

0,3 1,... 2,... ...,...

0,... 1,W-2 2,W-2

0,W-1 1,W-1

0,W

The optimal λ* and µ* depend on the state -  Large Deviation Theory -  Asymptotic optimality -  Should depend on importance of state

λ* µ*

State with loss


λ*

1/µ*


Swarm intelligence (ex. ants)

Ant nest

Food source

Very good

Bad

Initial phase: do (guided) random walk


Swarm intelligence (ex. ants)

Ant nest

Food source, the set R

Stable phase: Follow (randomly) pheromones


How ants guide IS parameters

0,0 1,0 2,0 3,0 ..,0 W-1,0 W,0

0,1 1,1 2,1 ...,1 W-2,1 W-1,1

0,2 1,2 2,2 ...,2 W-2,2

0,3 1,... 2,... ...,...

0,... 1,W-2 2,W-2

0,W-1 1,W-1

0,W

Scale the λ* and µ* according to pheromone values

λ* µ*

State with loss


λ*

1/µ*



i

Rare event

∑i: sum (or max) of all “path evaluations” in the state

∑ij: sum (or max) of “path evaluations” j

!ij =!ij

!i

Scaling factor (“pheromones”)


•  ACO-IS approach


!!ij = !ij + "ij(µji ! !ij)µ!

ji = µji + "ij(!ij ! µji)


Inner workings Scaling is higher but original value lower

Increases as target is approached

Decreases as target is approached

Scaling state-dependent more along the boudaries than in the interior

State with loss


λ*

1/µ*


Simulation cases

0,0 1,0 2,0 3,0 ..,0 W-1,0 W,0

0,1 1,1 2,1 ...,1 W-2,1 W-1,1

0,2 1,2 2,2 ...,2 W-2,2

0,3 1,... 2,... ...,...

0,... 1,W-2 2,W-2

0,W-1 1,W-1

0,W

Different combinations of: - Number of resource, N=10, 20 -  State (in)dependent rates -  Rare event set (single/multi-state) -  Balanced/unbalanced

λ* µ*


Numerical results

0,0 1,0 2,0 3,0 ..,0 W-1,0 W,0

0,1 1,1 2,1 ...,1 W-2,1 W-1,1

0,2 1,2 2,2 ...,2 W-2,2

0,3 1,... 2,... ...,...

0,... 1,W-2 2,W-2

0,W-1 1,W-1

0,W

Low relative error

High accuracy


Other simulation cases

•  Optical Burst Switch networks

–  Multiple service classes

–  Multiple service classes and preemptive priorities

–  Node and link failures

High accuracy and low relative error observed for all cases

Tandem queues


1 2 3 4

0.2

0.4

0.6

0.8

1.0

1.2

1.4

! µ1 µ2

µ̃2

µ̃1

!̃

bx2

V.F. Nicola, T.S. Zaburnenko: Importance Sampling Simulation of Population Overflow in Two-node Tandem Networks. QEST 2005: 220-229

Tandem queues

•  ACO-IS approach


!!i = !i + "i,i+1(min(µ1,i, µ2,i)! !i)µ!

1,i = µ1,i + "ii(max(µ1,i, µ2,i)! µ1,i)µ!

2,i = µ2,i + (!i + µ1,i)! (!!i + µ!1,i)


Concluding remarks and further work

•  Speed-up simulations –  Importance sampling –  Adaptive parameters by ACO meta heuristics –  No a priori system knowledge required

•  Promising results •  Simulated rare packet loss in OPN

–  Buffer-less –  Multiple service classes

•  Further work –  Asymptotic behaviour –  Non-exponential distribution –  More complex system models –  Detailed studies of the inner working of the Ants+IS methods

Challenges

•  Initial phase: what are the consequences of biased sampling in initial phase?

•  Inner workings of the ACO-IS? What is the result of ACO-IS biasing?

•  Does it work for non-exponential distributions? Phase type distribution is the first to be checked?

•  Other models structures? Tandem queue example is the next to be checked


Documents

Ant Colony Optimized Importance Sampling: Principles ...€¦ · Ant Colony Optimized Importance Sampling: Principles, Applications and Challenges Poul E. Heegaard Department of Telematics