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MTTF H (Ci )) N (0, 1), / n
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where N (0, 1) is the standard normal distribution, when the number n of cycles tends to in nity In other words, just by dividing both the numerator and denom inator by Hn = n 1 n H (Ci ), i=1 n(MTTF MTTF) N (0, 1), /Hn when the number n of cycles tends to in nity A Monte Carlo standard estimation of the MTTF will be inef cient because the denominator is the probability of a rare event (it will be the numerator if we deal with the unavailability) IS is a relevant way to cope with that problem The rst class of IS strategies is called dynamic importance sampling (DIS) We will not rede ne IS here; for a more precise description, see 2 Basically, we replace the transition matrix P by another one P (with corresponding probability measure P and expectation E) If the likelihood ratio over a cycle is L(x0 , , xT ) = P{(X0 , , XT ) = (x0 , , xn )} = P{(X0 , , XT ) = (x0 , , xn )}
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T 1 i=0 P(xi , xi+1 ) , T 1 i=0 P(xi , xi+1 )
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provided T has nite expectation under P, then for any random variable Z de ned over paths, E[Z] = E[ZL] A new estimator of the MTTF is then MTTF =
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n i=1 G(Ci )Li n i=1 H (Ci )Li
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where Li is the likelihood associated with the ith cycle Another method, measure-speci c dynamic importance sampling (MSDIS), giving better results, was introduced in [11] This involves simulating independently the numerator and denominator of (63), using different IS measures P1 for the numerator and P2 for the denominator Indeed, the functions being different, reducing the variance for one does not necessary mean the same for the other Of the total of n cycles, n are used to estimate the numerator, and (1 )n for the denominator A new estimator is then MTTF =
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n (1) (1) i=1 G(Ci )Li /( n) (1 )n H (Ci(2) )L(2) /((1 )n) i i=1
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where the Ci(1) and L(1) (Ci(2) and L(2) ) are the cycles and likelihood ratios i i corresponding to IS measure P1 (P2 )
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We then have the following result, using independent cycles (thus, the covariance term does not exist anymore): if H(1 )n = 1 (1 )n
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(1 )n
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H (Ci(2) )L(2) i
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is an estimator of EP2 (H L(2) ), and if 2 2 2 = 1 (GL(1) ) + (MTTF)2 2 (H L(2) ) where i2 ( ) is for the variance using Pi as the underlying probability measure, then n(MTTF MTTF) N (0, 1) /H(1 )n
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632 Importance sampling simulation schemes and robustness properties
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Many IS simulation schemes have been proposed in the literature The basic principle is to increase the occurrence of failures We review such schemes here, dividing them into three categories: rst the basic schemes rst; then those using some topological information; and nally those directly trying to approach the zero-variance change of measure In each case, we will discuss the robustness properties as 0 The properties the literature has looked at are bounded relative error (BRE) and bounded normal approximation (BNA) Recall that BRE means that the relative variance remains bounded as 0, so that the relative precision of the con dence interval is insensitive to the rarity of the event, and BNA is a suf cient condition to assert that the coverage of the con dence interval will remain valid whatever the rarity For more precise de nitions, see 4 devoted to robustness properties Those properties have been discussed at great length for highly reliable Markovian systems [27, 39, 41, 42] Looking at all sample paths, necessary and suf cient conditions have been obtained Basically, it is not suf cient that the most likely paths to failure are not rare (ie, their probability is (1)) under the IS measure; other paths should not be too rare either (but not necessarily (1)) A string of properties has also been shown in [41, 42]: BNA implies that paths contributing the most to the variance are (1) under IS measure, meaning that the variance is asymptotically properly estimated ( 4 illustrates the problems that could occur otherwise), implying BRE, implying in turn that most likely paths to failure are (1) under IS measure For all those implications, the reverse assertion is not true in general; counterexamples have been highlighted in [42] In what follows, since in our model transitions are either failures or repairs, we denote by F the set of failures and by R the set of repairs If x = 0, we also denote Fx = {y : (x, y) F} and Rx = {y : (x, y) R}, and let fx =
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