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y Fx P(x, y) and rx = from state x y Rx
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P(x, y) be the failure and repair probabilities
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Basic schemes The rst proposal, called failure biasing (FB), rst appeared in [23] It simply increases the probability of the failure transitions to a xed value (0, 1); typically, 05 09 Then the probability of getting a failure is no longer o(1) The transition probabilities are changed as follows: x U, x = 0, (x, y) F : x U, x = 0, (x, y) R : P(x, y) P(x, y) = ; fx P(x, y) = (1 ) P(x, y) rx
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The P(0, )s are not modi ed (since only a failure can happen from 0) Observe that the total probability of failure from x is now equal to From states in D, the probabilities are not changed Similarly, as soon as D has been reached, we switch back to P It was shown in [39] that, even if BRE is not satis ed in general by FB, it is the case for so-called balanced systems, that is, systems for which, from every state x, each failure transition has a probability of the same order of magnitude in terms of But some (important) paths are still too rare when using FB, because one of its failure transitions in a given state can still have probability o(1) due to a less rare failure under the initial law and which does not lead to interesting states Based on this, balanced failure biasing (BFB) was suggested [39]; here, for the subset of failure transitions, the conditional individual probabilities, taken proportionally to the initial ones in FB, are replaced by uniform ones Formally, x U, (x, y) F : P(x, y) = 1 ; Card(Fx ) P(x, y) rx
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x U, x = 0, (x, y) R :
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P(x, y) = (1 )
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From 0, we just use = 1 It is then shown in [39] that this scheme satis es BRE, and in [41] that BNA is also satis ed Inverse failure biasing (IFB) [36] is based on the ef cient simulation of the M/M/1 queue, involving switching arrival and service rates P is then chosen as follows: 1 Card(F0 ) rx x U, x = 0 y : (x, y) F, P(x, y) = , Card(Fx ) fx and if (x, y) R, P(x, y) = Card(Rx ) If x = 0, y : P(0, y) > 0, P(0, y) =
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The probability of having a repair is then o(1) Very ef cient when most likely paths to failure involve only failures, it performs poorly when those paths involve some repairs [5] (because those paths are o(1) under IFB) In [3], simple balanced likelihood ratio methods are introduced to increase the frequency of component failures, but keeping bounded at the same time the likelihood ratios associated with regenerative cycles The idea is to de ne stacks, initialized to empty sets, corresponding to failures with a given order of magnitude in terms of Throughout the simulation of a cycle, likelihood ratios for a component failure are put on top of the corresponding stack, and this value is taken back (and removed) from the stack when there is a component repair which has a failure with the same order of magnitude, in order to cancel the current likelihood ratio As a consequence, BRE is satis ed See [3] for more details and a complete description In the above IS estimators, the variance comes from the variations of the likelihood ratio (disregarding the fact that we either do or do not hit the rare set) In [21] it is highlighted that reducing the variance of that likelihood ratio can increase the ef ciency of the IS estimator if its does not signi cantly reduce the probability of the rare event, and if it does not require much more work Such a variance reduction can be obtained using weight windows Indeed, the likelihood ratio can be viewed as a weight that the simulated chain has accumulated so far For each state the current weight multiplied by the expected remaining likelihood ratio must be equal to the value of interest Given some estimation of the expected likelihood ratio from any state, we can decide at each step of the simulation to apply splitting to chains with excessive weights (therefore decreasing those weights), or to apply Russian roulette (increasing the weight if the chain is not killed), if the weight is not included in a window Another version simulates a xed number of chains in parallel The weight windows algorithm is then applied just to keep the number of chains constant, and each weight as close as possible to the expected value It has been noted in [21] that the savings can be large, but the estimator can also be very poor if the weight windows are wrongly selected The dif culty is to get a good approximation of the expected likelihood ratio from any state As a rough approximation, the most likely path, or direct paths, can be considered Some re nements were proposed in [17, 18] for the case where we have deferred (or grouped) repairs; that is, when there are states other than 0 for which only failures are possible This induces high probability cycles for which the above methods can lead to very large and even in nite variances In [17, 18] the probabilities along those cycles are not reduced too much How do these IS schemes perform in practice For space reasons, we limit ourselves to studying Example 1 with BFB and IFB For an extensive numerical study, depending on the topology, the reader may consult [5] Assume that failure rates of processors, controllers and disks are 5 10 6 , 2 10 6 , and 2 10 6 respectively, leading to = E 1(TD < T0 ) 555 10 6 , the dif cult component in the estimation of the MTTF This is not too rare, but allows us to compare with standard simulation (a rarer event would indeed yield a con dence
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