Evaluating dependability metrics of critical systems: Monte ... - iaria

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Importance Sampling (IS) Let X = h(Y ) for some function h where Y obeys some probability law P. IS replaces P by another probability measure ˜P, using ∫ ∫ E[X] = h(y)dP(y) = h(y) dP(y) d˜P(y) d˜P(y) = Ẽ[h(Y )L(Y )] ◮ L = dP/d˜P likelihood ratio, ◮ Ẽ is the expectation associated with probability law ˜P. Required condition: d˜P(y) ≠ 0 when h(y)dP(y) ≠ 0. If P and ˜P continuous laws, L ratio of density functions f (y)/˜f (y). ∫ ∫ E[X] = h(y)f (y)dy = h(y) f (y) ˜f (y)dy = Ẽ[h(Y )L(Y )] . ˜f (y) If P and ˜P are discrete laws, L ratio of indiv. prob p(y i )/˜p(y i ) E[X] = ∑ i h(y i )p(y i ) = ∑ i h(y i ) p(y i) ˜p(y i )˜p(y i) = Ẽ[h(Y )L(Y )] . G. Rubino (INRIA) Monte Carlo DEPEND 2010 36 / 72
Estimator and goal of IS Take (Y i , 1 ≤ i ≤ n) i.i.d; copies of Y , according to ˜P. The estimator is 1 n∑ h(Y i )L(Y i ). n i=1 The estimator is unbiased: [ ] 1 n∑ E h(Y i )L(Y i ) = 1 n n i=1 n∑ E [h(Y i )L(Y i )] = µ. i=1 Goal: select probability law ˜P such that ˜σ 2 [h(Y )L(Y )] = Ẽ[(h(Y )L(Y ))2 ] − µ 2 < σ 2 [h(Y )]. It means changing the probability distribution such tat the 2nd moment is smaller. G. Rubino (INRIA) Monte Carlo DEPEND 2010 37 / 72
Page 1 and 2: Evaluating dependability metrics of
Page 3 and 4: Outline 1 Introduction 2 Monte Carl
Page 5: Rare events Rare events occur when
Page 8 and 9: Outline 1 Introduction 2 Monte Carl
Page 10 and 11: Accuracy: how accurate is ¯X n ? W
Page 12 and 13: A fundamental example: evaluating i
Page 14 and 15: Other examples Reliability at t:
Page 16 and 17: From the accuracy point of view, th
Page 18 and 19: Variance reduction: antithetic vari
Page 20 and 21: Variance reduction: common variable
Page 22 and 23: Variance reduction: control variabl
Page 24: Monte Carlo drawbacks So, is there
Page 27 and 28: What is crude simulation? Assume we
Page 29 and 30: Inefficiency of crude Monte Carlo:
Page 31 and 32: Robustness properties: Bounded rela
Page 33 and 34: Relation between BRE and AO AO weak
Page 35: Outline 1 Introduction 2 Monte Carl
Page 39 and 40: If A = [T, ∞), i.e., µ = P[Y ≥
Page 41 and 42: IS for a discrete-time Markov chain
Page 43 and 44: Zero-variance IS estimator for Mark
Page 45 and 46: Zero-variance approximation Use a h
Page 47 and 48: Drawbacks of the learning technique
Page 49 and 50: Other procedure: optimization withi
Page 51 and 52: Adaptive learning in Cross-Entropy
Page 53 and 54: Splitting: general principle Splitt
Page 55 and 56: Splitting and Markov chain {Y j ; j
Page 57 and 58: The different implementations Fixed
Page 59 and 60: Issues to be solved How to define t
Page 61 and 62: Optimal values In a general setting
Page 63 and 64: Simplified setting: fixed splitting
Page 65 and 66: Illustration, fixed effort: a tande
Page 67 and 68: Confidence interval issues Robustne
Page 69 and 70: IS probability used Failure Biasing
Page 71 and 72: Asymptotic explanation When ε smal
Page 73 and 74: Tutorial on Monte Carlo techniques,
Page 76 and 77: • Very used in dependability anal
Page 78 and 79: • Usual situation: at some time s
Page 80 and 81: first order (in ε) expressions bef
Page 82 and 83: efore after a 1 a 2 x ε 2 ε 1 a 1
Page 84 and 85: symbolically, for an operational st
Page 86 and 87:
• Always the same idea: take adva
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ε 1 ε 2 a 1 x a 2 before ε 4 ε
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• The zero variance idea in IS le
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• A variance-reduction procedure
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s t r 1 r 2 r 3 r 4 r 5 R st = r 1
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A COMMUNICATION NETWORK backbone, o
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A COMMUNICATION NETWORK • nodes a
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• Ω : set of all partial sub-gra
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• #failed = 0 • for m = 1, 2,
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• the correct answer given by the
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R = 1 R = r × R = r |K | = 1 |K |
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1 R r 2 = R r 1 r 2 R r 1 r 2 r 3 r
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1 R r 2 = R r 1 r 2 R r 1 r 5 etc.
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• Again, with P be a path and P-u
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- if the network is r 1 r 2 r 3 - t
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- then if the network is - then Z =
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• We partition Ω in the followin
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Ω = {L 1 L 2 , L 1 , L 1 L 2 } Z =
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Ω = {L 1 L 2 , L 1 , L 1 L 2 } Z =
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• It consists of generalizing thi
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Ω L 1 L’ 2 L’ 3 L 1 L 1 L’ 2
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L 1 L’ 2 L’ 3 L 1 L 1 L’ 2 L
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L 1 L’ 2 L’ 3 L 1 L 1 L’ 2 L
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L 1 L’ 2 L’ 3 L 1 L 1 L’ 2 L
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• sum = 0.0 • for m = 1, 2, …
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|V | = 9, |K | = 4, |E | = 12 M = 1
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• We varied K (|K | = 2, |K | = 5
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• We also varied r i = r from 0.9
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• A time-reduction procedure for
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• Internal loop: sampling a graph
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• We consider the case of case r
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• For a table with M rows, - each
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• Dividing the mean cost of the s
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• The procedure can be improved f
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• Reference: G. Rubino, B. Tuffin
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