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

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A fundamental example: evaluating integrals ∫ Assume µ = f (x)dx < ∞, with I an interval in R d . I With an appropriate change of variable, we can assume that I = [0,1] d . There are many numerical methods available for approximating µ. Their quality is captured by their convergence speed as a function of the number of calls to f , which we denote by n. Some examples: ◮ Trapezium rule; convergence speed is in n −2/d , ◮ Simpson’s rule; convergence speed is in n −4/d , ◮ Gaussian quadrature method having m points; convergence speed is in n −(2m−1)/d . For all these methods, the speed decreases when d increases (and → 0 when d → ∞). G. Rubino (INRIA) Monte Carlo DEPEND 2010 12 / 72
The “independence of the dimension” Let now U be an uniform r.v. on the cube [0,1] d and X = f (U). We immediately have µ = E(X), which opens the path to the Monte Carlo technique for approximating µ statistically. We have that ◮ ¯Xn is an estimator of our integral, ◮ and that the convergence speed, as a function of n, is in n −1/2 , thus independent of the dimension d of the problem. This independence of the dimension of the problem in the computational cost is the main advantage of the Monte Carlo approach over quadrature techniques. In many cases, it means that quadrature techniques can not be applied, and that Monte Carlo works in reasonable time with good accuracy. G. Rubino (INRIA) Monte Carlo DEPEND 2010 13 / 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 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 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
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Simplified setting: fixed splitting
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Illustration, fixed effort: a tande
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Confidence interval issues Robustne
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IS probability used Failure Biasing
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Asymptotic explanation When ε smal
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Tutorial on Monte Carlo techniques,
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• Very used in dependability anal
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• Usual situation: at some time s
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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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