Ant Colony Optimized Importance Sampling: Principles, Applications - PowerPoint PPT Presentation

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

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

Model example 0,W 0,W-1 1,W-1 0,... 1,W-2 2,W-2 0,3 1,... 2,... ...,... State with loss 0,2 1,2 2,2 ...,2 W-2,2 λ Packet arrival rate 1/µ Packet transmission time 0,1 1,1 2,1 ...,1 W-2,1 W-1,1 λ µ 0,0 1,0 2,0 3,0 ..,0 W-1,0 W,0 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

Simulation problem 0,W When λ ≪ µ then rare packet loss 0,W-1 1,W-1 0,... 1,W-2 2,W-2 0,3 1,... 2,... ...,... State with loss 0,2 1,2 2,2 ...,2 W-2,2 λ Packet arrival rate 1/µ Packet transmission time 0,1 1,1 2,1 ...,1 W-2,1 W-1,1 λ µ 0,0 1,0 2,0 3,0 ..,0 W-1,0 W,0 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

Simulation with importance sampling 0,W Simulate with λ * ≫ µ * ⇒ packet loss not rare 0,W-1 1,W-1 0,... 1,W-2 2,W-2 0,3 1,... 2,... ...,... State with loss 0,2 1,2 2,2 ...,2 W-2,2 λ * Packet arrival rate 1/µ * Packet transmission time 0,1 1,1 2,1 ...,1 W-2,1 W-1,1 λ * µ * 0,0 1,0 2,0 3,0 ..,0 W-1,0 W,0 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

The problem with importance sampling 0,W How to determine the λ * and µ * ? 0,W-1 1,W-1 - scale too little - no effect - scale too much - biased estimates 0,... 1,W-2 2,W-2 0,3 1,... 2,... ...,... State with loss 0,2 1,2 2,2 ...,2 W-2,2 λ * Packet arrival rate 1/µ * Packet transmission time 0,1 1,1 2,1 ...,1 W-2,1 W-1,1 λ * µ * 0,0 1,0 2,0 3,0 ..,0 W-1,0 W,0 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

Adaptive change of measure in IS 0,W The optimal λ * and µ * depend on the state 0,W-1 1,W-1 - Large Deviation Theory - Asymptotic optimality 0,... 1,W-2 2,W-2 - Should depend on importance of state 0,3 1,... 2,... ...,... State with loss 0,2 1,2 2,2 ...,2 W-2,2 λ * Packet arrival rate 1/µ * Packet transmission time 0,1 1,1 2,1 ...,1 W-2,1 W-1,1 λ * µ * 0,0 1,0 2,0 3,0 ..,0 W-1,0 W,0 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

Swarm intelligence (ex. ants) Bad Food source Very good Initial phase: do (guided) random walk Ant nest 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

Swarm intelligence (ex. ants) Food source, the set R Stable phase: Follow (randomly) pheromones Ant nest 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

How ants guide IS parameters 0,W Scale the λ * and µ * according 0,W-1 1,W-1 to pheromone values 0,... 1,W-2 2,W-2 0,3 1,... 2,... ...,... State with loss 0,2 1,2 2,2 ...,2 W-2,2 λ * Packet arrival rate 1/µ * Packet transmission time 0,1 1,1 2,1 ...,1 W-2,1 W-1,1 λ * µ * 0,0 1,0 2,0 3,0 ..,0 W-1,0 W,0 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

How ants guide IS parameters Scaling factor (“pheromones”) α ij = Σ ij Rare Σ i event ∑ ij : sum (or max) j of “path evaluations” ∑ i : sum (or max) of all “path evaluations” in the state i 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

How ants guide IS parameters • ACO-IS approach = λ ij + α ij ( µ ji − λ ij ) λ ∗ ij = µ ji + α ij ( λ ij − µ ji ) µ ∗ ji 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

Inner workings Scaling is higher but original value lower Decreases as target is approached Increases as target is approached Scaling state -dependent more along the boudaries 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France than in the interior

Simulation cases 0,W 0,W-1 1,W-1 0,... 1,W-2 2,W-2 Different combinations of: - Number of resource, N=10, 20 0,3 1,... 2,... ...,... - State (in)dependent rates - Rare event set (single/multi-state) State with loss 0,2 - Balanced/unbalanced 1,2 2,2 ...,2 W-2,2 λ * Packet arrival rate 1/µ * Packet transmission time 0,1 1,1 2,1 ...,1 W-2,1 W-1,1 λ * µ * 0,0 1,0 2,0 3,0 ..,0 W-1,0 W,0 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

Numerical results High accuracy Low relative error 0,W 0,W-1 1,W-1 0,... 1,W-2 2,W-2 0,3 1,... 2,... ...,... 0,2 1,2 2,2 ...,2 W-2,2 0,1 1,1 2,1 ...,1 W-2,1 W-1,1   0,0 1,0 2,0 3,0 ..,0 W-1,0 W,0 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

High accuracy and low relative error observed for all cases Other simulation cases • Optical Burst Switch networks – Multiple service classes – Multiple service classes and preemptive priorities – Node and link failures 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

Tandem queues λ µ 1 µ 2 V.F. Nicola, T.S. Zaburnenko: Importance Sampling Simulation of Population Overflow in Two-node Tandem Networks . QEST 2005: 220-229 ˜ µ 1 1.4 1.2 ˜ λ 1.0 0.8 0.6 0.4 0.2 ˜ µ 2 1 2 3 4 x 2 b 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

Tandem queues • ACO-IS approach λ ∗ = λ i + α i,i +1 (min( µ 1 ,i , µ 2 ,i ) − λ i ) i = µ 1 ,i + α ii (max( µ 1 ,i , µ 2 ,i ) − µ 1 ,i ) µ ∗ 1 ,i µ ∗ = µ 2 ,i + ( λ i + µ 1 ,i ) − ( λ ∗ i + µ ∗ 1 ,i ) 2 ,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 7th Int'l Workshop on Rare Event Simulation, Sept 24-26 2008, Rennes, France

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