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

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Modelling analysis of robustness: parameterisation of rarity In rare-event simulation models, we often parameterize with a rarity parameter ǫ > 0 such that µ = E[X(ǫ)] → 0 as ǫ → 0. Typical example ◮ For a direct Bernoulli r.v. X = 1A , ǫ = µ = E[1 A ]. ◮ When simulating a system involving failures and repairs, ǫ can be the rate or probability of individual failures. ◮ For a queue or a network of queues, when estimating the overflow probability, ǫ = 1/C inverse of the capaciy of the considered queue. The question is then: how behaves an estimator as ǫ → 0, i.e., the event becomes rarer? G. Rubino (INRIA) Monte Carlo DEPEND 2010 30 / 72
Robustness properties: Bounded relative error (BRE) An estimator X(ǫ) is said to have bounded relative variance (or bounded relative error) if σ 2 (X(ǫ))/µ 2 (ǫ) is bounded uniformly in ǫ. Equivalent to saying that σ(X(ǫ))/µ(ǫ) is bounded uniformly in ǫ. Interpretation: estimating µ(ǫ) with a given relative accuracy can be achieved with a bounded number of replications even if ǫ → 0. When the confidence interval comes from the central limit theorem, it means that the relative half width c α σ(X(ǫ)) √ n remains bounded as ǫ → 0. G. Rubino (INRIA) Monte Carlo DEPEND 2010 31 / 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: Inefficiency of crude Monte Carlo:
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 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
Page 88 and 89:
ε 1 ε 2 a 1 x a 2 before ε 4 ε
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• 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
Page 119 and 120:
- if the network is r 1 r 2 r 3 - t
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- then if the network is - then Z =
Page 123 and 124:
• We partition Ω in the followin
Page 125 and 126:
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
Page 131 and 132:
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
Page 145 and 146:
• 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
Page 158:
• 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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