PSI2 : Envelope Perfect Sampling of Non Monotone Systems c 1 , Bruno - - PowerPoint PPT Presentation

PSI2 : Envelope Perfect Sampling of Non Monotone Systems

Ana Buˇ si´ c1, Bruno Gaujal2,3 Ga¨ el Gorgo2,3 and Jean-Marc Vincent2,3

1INRIA/ENS Paris 2INRIA-Rhˆ

• ne-Alpes 3Laboratoire 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

Finite queuing networks (dynamic routing) Call centers Grid/cluster scheduling Rare event estimation Statistical verification of program

Models

Discrete vector state-space X Event based models Xn+1 = Φ(Xn, en+1) , en ∈ E Stochastic recurrence equation Independent events (iid)

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

All the trajectories Time −i −j −τ ∗ S t a t i

• n

a r y P r

• c

e s s X X X X Zi Zj Z−τ ∗ = {X0} Z0 = X collapse

Synchronizing pattern = ⇒ finite backward scheme τ ∗ < ∞

[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

−(n + 1) X X time −n m′ M m M′

same convergence condition complexity in O(Eτ ∗) ⇒ 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 −(n + 1) −n time

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

Exhaustive state simulation

−i −j −τ ∗ S t a t i

• n

a r y P r

• c

e s s X X X X

Splitting point Envelope algorithm

Time

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

λ Batch arrival B C = 200 µ = 1 B = 2 with probability 1

2 and B = 3 with probability 1 2

C µ C µ C µ C µ C µ

Linux 2.6.31 Intel Core2 Duo 2.8GHz Memory 3.9Go Sample size = 1000 batch of size 2 and 3 batch of size 1 30 35 40 45 50 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 sampling time (ms) equivalent arrival rate 20 15 10 5 25

⇒ Almost monotone systems

QEST 2010 J-M. Vincent et al. INRIA-LIG Envelope Perfect Sampling Williamsburg september 2010 7 / 10

Sampling efficiency

Coxian queues

1 − p p = 1

2

λ = 1 µ2 = 2 C = 1000 λ C µ1 µ1 p C µ2 Cox Server

Coxian service time 150 200 250 300 350 400 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Exponential service time Equivalent service rate Sampling time (ms) 50 100 Sample size 1000

⇒ 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) ...

Ψ2 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