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

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Robustness properties: Asymptotic Optimality (AO) BRE has often been found difficult to verify in practice (ex: queueing systems). Weaker property: asymptotic optimality (or logarithmic efficiency) if ln(E[X 2 (ǫ)]) lim = 2. ǫ→0 ln(µ(ǫ)) Equivalent to say that lim ǫ→0 ln(σ 2 [X(ǫ)])/ln(µ(ǫ)) = 2. Property also called logarithmic efficiency or weak efficiency. Quantity under limit is always positive and less than or equal to 2: σ 2 [X(ǫ)] ≥ 0, so E[X 2 (ǫ)] ≥ (µ(ǫ)) 2 and then ln E[X 2 (ǫ)] ≥ 2ln µ(ǫ), i.e., ln E[X 2 (ǫ)] ≤ 2. lnµ(ǫ) Interpretation: the second moment and the square of the mean go to zero at the same exponential rate. G. Rubino (INRIA) Monte Carlo DEPEND 2010 32 / 72
Relation between BRE and AO AO weaker property: if we have BRE, ∃κ > 0 such that E[X 2 (ǫ)] ≤ κ 2 µ 2 (ǫ), i.e., ln E[X 2 (ǫ)] ≤ lnκ 2 + 2lnµ(ǫ), leading to lim ǫ→0 ln E[X 2 (ǫ)]/ln µ(ǫ) ≥ 2. Since this ratio is always less than 2, we get the limit 2. Not an equivalence. Some counter-examples: ◮ an estimator for which γ = e −η/ε with η > 0, but for which the variance is Q(1/ε)e −2η/ε with Q a polynomial; ◮ exponential tilting in queueing networks. Other robustness measures exist (based on higher degree moments, on the Normal approximation, on simulation time...) G. Rubino (INRIA) Monte Carlo DEPEND 2010 33 / 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: Robustness properties: Bounded rela
Page 37 and 38: Estimator and goal of IS Take (Y i
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
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efore after a 1 a 2 x ε 2 ε 1 a 1
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symbolically, for an operational st
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• 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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Evaluating dependability metrics of critical systems: Monte ... - iaria

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