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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

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

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

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

• 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

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

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

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

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 Packet arrival rate Packet transmission time λ 1/µ

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

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 Packet arrival rate Packet transmission time λ* 1/µ*

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

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 Packet arrival rate Packet transmission time λ* 1/µ*

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

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 Packet arrival rate Packet transmission time λ* 1/µ*

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

Swarm intelligence (ex. ants)

Ant nest Food source

Very good Bad

Initial phase: do (guided) random walk

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

Swarm intelligence (ex. ants)

Ant nest Food source, the set R

Stable phase: Follow (randomly) pheromones

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

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 Packet arrival rate Packet transmission time λ* 1/µ*

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

How ants guide IS parameters

i Rare event

∑i: sum (or max)

• f all “path evaluations”

in the state ∑ij: sum (or max)

• f “path evaluations”

j

αij = Σij Σi

Scaling factor (“pheromones”)

How ants guide IS parameters

• ACO-IS approach

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

λ∗

ij

= λij + αij(µji − λij) µ∗

ji

= µji + αij(λij − µji)

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

Inner workings

Scaling is higher but

• riginal 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 Packet arrival rate Packet transmission time λ* 1/µ* 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

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*

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

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

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

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

7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France 1 2 3 4 0.2 0.4 0.6 0.8 1.0 1.2 1.4

λ µ1 µ2 ˜ µ2 ˜ µ1 ˜ λ b x2

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

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

λ∗

i

= λi + αi,i+1(min(µ1,i, µ2,i) − λi) µ∗

1,i

= µ1,i + αii(max(µ1,i, µ2,i) − µ1,i) µ∗

2,i

= µ2,i + (λi + µ1,i) − (λ∗

i + µ∗ 1,i)

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

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

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