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

More documents

Recommendations

Info

Outline 1 Introduction 2 Monte Carlo: the basics 3 Inefficiency of crude Monte Carlo, and robustness issue 4 Importance Sampling 5 Splitting 6 Confidence interval issues G. Rubino (INRIA) Monte Carlo DEPEND 2010 8 / 72
Monte Carlo In all the above problems, the goal is to compute µ = E[X] for some random variable X (that is, it can be written in this form). Monte Carlo simulation (in its basic form) generates n independent copies of X, (X i , 1 ≤ i ≤ n). Then, ◮ ¯Xn = 1 n∑ X i is an approximation (an estimation) of µ; n ◮ i=1 ¯X n → µ with probability 1, as n → ∞ (Strong Law of Large Numbers). G. Rubino (INRIA) Monte Carlo DEPEND 2010 9 / 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 35 and 36: Outline 1 Introduction 2 Monte Carl
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
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
Page 97 and 98:
A COMMUNICATION NETWORK backbone, o
Page 99 and 100:
A COMMUNICATION NETWORK • nodes a
Page 101 and 102:
• Ω : set of all partial sub-gra
Page 103 and 104:
• #failed = 0 • for m = 1, 2,
Page 105 and 106:
• the correct answer given by the
Page 107 and 108:
R = 1 R = r × R = r |K | = 1 |K |
Page 109 and 110:
1 R r 2 = R r 1 r 2 R r 1 r 2 r 3 r
Page 111:
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
Page 121 and 122:
- 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 =
Page 127 and 128:
Ω = {L 1 L 2 , L 1 , L 1 L 2 } Z =
Page 129 and 130:
• It consists of generalizing thi
Page 131 and 132:
Ω L 1 L’ 2 L’ 3 L 1 L 1 L’ 2
Page 133 and 134:
L 1 L’ 2 L’ 3 L 1 L 1 L’ 2 L
Page 135 and 136:
L 1 L’ 2 L’ 3 L 1 L 1 L’ 2 L
Page 137 and 138:
L 1 L’ 2 L’ 3 L 1 L 1 L’ 2 L
Page 139 and 140:
• sum = 0.0 • for m = 1, 2, …
Page 141 and 142:
|V | = 9, |K | = 4, |E | = 12 M = 1
Page 143 and 144:
• 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
show all

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

Create successful ePaper yourself

Delete template?

Save as template?