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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) of all “path evaluations” in the state
∑ij: sum (or max) of “path evaluations” j
!ij =!ij
!i
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
!!ij = !ij + "ij(µji ! !ij)µ!
ji = µji + "ij(!ij ! µji)
7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France
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
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
! 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
7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France
!!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)
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