PSI2 : Envelope Perfect Sampling of Non Monotone Systems c 1 , Bruno Gaujal 2 , 3 Ana Buˇ si´ el Gorgo 2 , 3 and Jean-Marc Vincent 2 , 3 Ga¨ 1 INRIA/ENS Paris 2 INRIA-Rhˆ one-Alpes 3 Laboratoire d’Informatique de Grenoble Outline 1 Motivations 2 Perfect Sampling 3 Sampling efficiency 4 Future Work and References Work partially supported by ANR Setin Checkbound QEST 2010 J-M. Vincent et al. INRIA-LIG Envelope Perfect Sampling Williamsburg september 2010 1 / 10
Motivations Perfect Sampling of Complex Large Scale Markov Chains Applications Models Discrete vector state-space X Finite queuing networks (dynamic routing) Event based models Call centers X n +1 = Φ( X n , e n +1 ) , e n ∈ E Grid/cluster scheduling Stochastic recurrence equation Rare event estimation Independent events (iid) Statistical verification of program Provide independent samples of stationary states. PSI2 : a Perfect Sampler Library of monotone events Simulation kernel Efficient simulator : polynomial in the model dimension = ⇒ Extension to non-monotone events QEST 2010 J-M. Vincent et al. INRIA-LIG Envelope Perfect Sampling Williamsburg september 2010 2 / 10
Perfect Sampling Perfect Sampling Principle X s s c e o r P X r y a n o t i a S t X Time X 0 All the trajectories collapse − i Z 0 = X − j Z i Z j Z − τ ∗ = { X 0 } − τ ∗ ⇒ finite backward scheme τ ∗ < ∞ Synchronizing pattern = [NSMC 2003, LAA 2004] QEST 2010 J-M. Vincent et al. INRIA-LIG Envelope Perfect Sampling Williamsburg september 2010 3 / 10
Perfect Sampling Monotone Perfect Sampling X M ′ X m ′ M m time − n − ( n + 1) same convergence condition complexity in O ( E τ ∗ ) ⇒ polynomial in model dimension [NSMC 2006,QEST 2008] QEST 2010 J-M. Vincent et al. INRIA-LIG Envelope Perfect Sampling Williamsburg september 2010 4 / 10
Perfect Sampling Envelopes Perfect Sampling X X time − n − ( n + 1) Synchronizing pattern for envelopes complexity unknown but practically efficient QEST 2010 J-M. Vincent et al. INRIA-LIG Envelope Perfect Sampling Williamsburg september 2010 5 / 10
Perfect Sampling Envelopes and Splitting Perfect Sampling X s e s c o r P y X r a n o t i a S t X Time X 0 Exhaustive state simulation − i Splitting point − j Envelope algorithm − τ ∗ Guarantees the convergence complexity unknown but practically more efficient [VALUETOOLS 2008] QEST 2010 J-M. Vincent et al. INRIA-LIG Envelope Perfect Sampling Williamsburg september 2010 6 / 10
Sampling efficiency Batch arrivals Batch arrival B C µ C µ C µ C µ C µ λ B = 2 with probability 1 2 and B = 3 with probability 1 C = 200 µ = 1 2 50 45 batch of size 2 and 3 40 sampling time (ms) 35 30 batch of size 1 25 20 Sample size = 1000 15 Intel Core2 Duo 2.8GHz 10 Memory 3.9Go 5 Linux 2.6.31 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 equivalent arrival rate ⇒ Almost monotone systems QEST 2010 J-M. Vincent et al. INRIA-LIG Envelope Perfect Sampling Williamsburg september 2010 7 / 10
Sampling efficiency Coxian queues Cox Server C µ 1 µ 1 C µ 2 λ p 1 − p p = 1 C = 1000 µ 2 = 2 λ = 1 2 400 Sample size 1000 350 300 Sampling time (ms) 250 Coxian service time 200 150 Exponential service time 100 50 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Equivalent service rate ⇒ pre-computation for phase-type distribution QEST 2010 J-M. Vincent et al. INRIA-LIG Envelope Perfect Sampling Williamsburg september 2010 8 / 10
Future Work and References Synthesis Non-monotone events Negative customers, join, ...(frontier in 0) Batch arrivals or services (large coupling time ⇒ splitting) Event triggering on non-monotone condition Coxian and phase-type distribution (transform the state-space) ... Ψ 2 Implementation adaptation of the kernel user defined envelopes process basic events in the library Some references - Methodological reference A. Buˇ si´ c, B. Gaujal, J-M Vincent. Perfect Simulation and Non-monotone Markovian Systems. Valuetools’08, Athens, 2008 - Non-monotone load-sharing policies G. Gorgo, J-M Vincent. Perfect Sampling of Load Sharing Policies in Large Scale Distributed Systems. ASMTA, LNCS, 6148 - Performance comparison in statistical model-checking D. Elrabih, G. Gorgo, N. Pekergin, J-M. Vincent. Steady-state Property Verification : a Comparison Study. VECoS, Paris, 2010. QEST 2010 J-M. Vincent et al. INRIA-LIG Envelope Perfect Sampling Williamsburg september 2010 9 / 10
Future Work and References Download : http://gforge.inria.fr/projects/psi QEST 2010 J-M. Vincent et al. INRIA-LIG Envelope Perfect Sampling Williamsburg september 2010 10 / 10
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