1. 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

2. 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

3. 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 Model description

4. 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France Model example 0,01,02,03,0..,0W-1,0W,0 0,11,12,1...,1W-2,1W-1,1 0,21,22,2...,2W-2,2 0,31,...2,......,... 0,...1,W-22,W-2 0,W-11,W-1 0,W State with loss Packet arrival rate Packet transmission time    

5. 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France Simulation problem 0,01,02,03,0..,0W-1,0W,0 0,11,12,1...,1W-2,1W-1,1 0,21,22,2...,2W-2,2 0,31,...2,......,... 0,...1,W-22,W-2 0,W-11,W-1 0,W When  then rare packet loss   State with loss Packet arrival rate Packet transmission time  

6. 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France Simulation with importance sampling 0,01,02,03,0..,0W-1,0W,0 0,11,12,1...,1W-2,1W-1,1 0,21,22,2...,2W-2,2 0,31,...2,......,... 0,...1,W-22,W-2 0,W-11,W-1 0,W Simulate with  *  * ⇒ packet loss not rare   State with loss Packet arrival rate Packet transmission time   

7. 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France The problem with importance sampling 0,01,02,03,0..,0W-1,0W,0 0,11,12,1...,1W-2,1W-1,1 0,21,22,2...,2W-2,2 0,31,...2,......,... 0,...1,W-22,W-2 0,W-11,W-1 0,W How to determine the  *  and  * ? - scale too little - no effect - scale too much - biased estimates   State with loss Packet arrival rate Packet transmission time   

8. 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France Adaptive change of measure in IS 0,01,02,03,0..,0W-1,0W,0 0,11,12,1...,1W-2,1W-1,1 0,21,22,2...,2W-2,2 0,31,...2,......,... 0,...1,W-22,W-2 0,W-11,W-1 0,W The optimal  *  and  * depend on the state - Large Deviation Theory - Asymptotic optimality - Should depend on importance of state   State with loss Packet arrival rate Packet transmission time   

9. 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

10. 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

11. 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France How ants guide IS parameters 0,01,02,03,0..,0W-1,0W,0 0,11,12,1...,1W-2,1W-1,1 0,21,22,2...,2W-2,2 0,31,...2,......,... 0,...1,W-22,W-2 0,W-11,W-1 0,W Scale the  *  and  * according to pheromone values   State with loss Packet arrival rate Packet transmission time   

12. 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 Scaling factor (“pheromones”)

13. How ants guide IS parameters ACO-IS approach 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

14. 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

15. State with loss Packet arrival rate Packet transmission time    7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France Simulation cases 0,01,02,03,0..,0W-1,0W,0 0,11,12,1...,1W-2,1W-1,1 0,21,22,2...,2W-2,2 0,31,...2,......,... 0,...1,W-22,W-2 0,W-11,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  

16. 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France Numerical results 0,01,02,03,0..,0W-1,0W,0 0,11,12,1...,1W-2,1W-1,1 0,21,22,2...,2W-2,2 0,31,...2,......,... 0,...1,W-22,W-2 0,W-11,W-1 0,W  Low relative error High accuracy

17. 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

18. Tandem queues; example 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France V.F. Nicola, T.S. Zaburnenko: Importance Sampling Simulation of Population Overflow in Two-node Tandem Networks. QEST 2005: 220-229

19. Tandem queues; example ACO-IS approach 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

20. 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 ACO-IS 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

21. 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