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

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Optimal estimator for estimating E[h(Y )] = ∫ h(y)L(y)d˜P(y) Optimal change of measure: ˜P = |h(Y )| E[|h(Y )|] dP. Proof: for any alternative IS measure P ′ , leading to the likelihood ratio L ′ and expectation E ′ , Ẽ[(h(Y )L(Y )) 2 ] = (E[|h(Y )|]) 2 = (E ′ [|h(Y )|L ′ (Y )]) 2 ≤ E ′ [(h(Y )L ′ (Y )) 2 ]. If h ≥ 0, Ẽ[(h(Y )L(Y )) 2 ] = (E[h(Y )]) 2 , i.e., ˜σ 2 (h(Y )L(Y )) = 0. That is, IS provides a zero-variance estimator. Implementing it requires knowing E[|h(Y )|], i.e. what we want to compute; if so, no need to simulation! But provides a hint on the general form of a “good” IS measure. G. Rubino (INRIA) Monte Carlo DEPEND 2010 40 / 72
IS for a discrete-time Markov chain (DTMC) {Y j , j ≥ 0} X = h(Y 0 ,...,Y τ ) function of the sample path with ◮ P = (P(y, z) transition matrix, π0 (y) = P[Y 0 = y], initial probabilities ◮ up to a stopping time τ, first time it hits a set ∆. ◮ µ(y) = Ey [X]. IS replaces the probabilities of paths (y 0 ,...,y n ), n−1 ∏ P[(Y 0 ,...,Y τ ) = (y 0 ,... ,y n )] = π 0 (y 0 ) P(y j−1 ,y j ), by ˜P[(Y 0 ,...,Y τ ) = (y 0 ,...,y n )] st Ẽ[τ] < ∞. For convenience, the IS measure remains a DTMC, replacing P(y,z) by ˜P(y,z) and π 0 (y) by ˜π 0 (y). Then L(Y 0 ,...,Y τ ) = π τ−1 0(Y 0 ) ∏ P(Y j−1 ,Y j ) ˜π 0 (Y 0 ) ˜P(Y j=1 j−1 ,Y j ) . j=1 G. Rubino (INRIA) Monte Carlo DEPEND 2010 41 / 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 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 37 and 38: Estimator and goal of IS Take (Y i
Page 39: If A = [T, ∞), i.e., µ = P[Y ≥
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
Page 88 and 89: ε 1 ε 2 a 1 x a 2 before ε 4 ε
Page 90 and 91:
• The zero variance idea in IS le
Page 93 and 94:
• A variance-reduction procedure
Page 95 and 96:
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
Page 101 and 102:
• 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.
Page 117 and 118:
• Again, with P be a path and P-u
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- if the network is r 1 r 2 r 3 - t
Page 121 and 122:
- then if the network is - then Z =
Page 123 and 124:
• We partition Ω in the followin
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Ω = {L 1 L 2 , L 1 , L 1 L 2 } Z =
Page 127 and 128:
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
Page 148 and 149:
• A time-reduction procedure for
Page 150 and 151:
• Internal loop: sampling a graph
Page 152 and 153:
• We consider the case of case r
Page 154 and 155:
• For a table with M rows, - each
Page 156 and 157:
• Dividing the mean cost of the s
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• The procedure can be improved f
Page 161:
• Reference: G. Rubino, B. Tuffin
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Evaluating dependability metrics of critical systems: Monte ... - iaria

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