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The applications are numerous Dependability analysis (which inherits from the classical theory an older concept of reliability analysis) is a critical issue in telecommunications, computer science, and manufacturing, among other elds, [13] For instance, catastrophic failures in transport systems or in nuclear power plants could lead to human losses Similarly, the failure of a computer or a network (telecommunication, electricity, etc) may lead to important monetary losses These systems are thus designed in such a way that these undesirable events with serious consequences happen rarely (ie, have very small probabilities) The ability to numerically evaluate the risks associated with their use is therefore a major concern Example 1 A typical example is that of a large computing system This type of model, originally from [11], has been used in most of the papers focusing on highly reliable Markovian systems Our example consists of two sets of processors, each with two sets of disk controllers and six clusters of disks with four disks per cluster Data are replicated in each cluster so that one disk can fail without affecting the system Figure 61 describes the system Failure propagations are possible, as will be seen later There are two failure modes for each component The system is considered as operational if all data are accessible from each processor type This translates as follows: at least one processor, one controller in each set, and three out of four disks in each cluster are operational

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Figure 61 Block diagram for Example 15 These nite Markovian models can in theory be analyzed by means of a rich set of ef cient numerical procedures Moreover, in some cases these techniques are basically insensitive to the rarity phenomenon In practice things are different: the power of these representations that can capture quite accurately the behavior of complex systems leads very often to huge state spaces, rendering numerical approaches impracticable Simulation is then the only possible evaluation tool, but the rarity problem becomes the bottleneck of the solution process The same happens obviously if the model is not Markovian, for instance, if it is a

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MARKOVIAN MODELS FOR DEPENDABILITY ANALYSIS

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semi-Markov process The theory is much less rich in this case, and simulation is almost always the method of choice We will brie y consider this case at the end of the chapter Another issue is the fact that some transitions, namely failures, are much rarer than repairs From a modeling assumption point of view, we will introduce a rarity parameter , such that transition rates are decreasing with , while repair rates do not depend on it Then the smaller is, the less likely the general system will fail As extensively explained in previous chapters, standard (naive) simulation is inef cient because the event of interest, the failure of the system, is rare, and special procedures have to be implemented Splitting and importance sampling (IS) are again the tools at hand Note that, here, splitting is not relevant when 0 Indeed, observe that for the type of model we are looking at, it is each individual failure which is rare, and the number of transitions required to reach a failed state is therefore small (otherwise, the probability of the event would be meaningless) Splitting can therefore hardly be ef ciently applied because it does not change the probabilities of individual transitions; it would be necessary to decompose each (rare) component failure in sub-events to make those events less rare themselves, which would be cumbersome We will then focus on IS The chapter reviews the main results and techniques obtained in the domain mainly for steady-state but also for transient analysis [3, 5, 7, 9 11, 14, 17, 18, 21, 23, 25, 27, 36, 39 43], in the case of Markovian modeling We also brie y deal with the non-Markovian case [15, 30, 32 35, 38,], and with sensitivity analysis [26, 29] The chapter is organized as follows Section 62 describes the mathematical model and its rare event parameterization Section 63 looks at the estimation of steady-state measures, unavailability and the MTTF;1 it shows how they can be estimated and describes the known IS schemes for obtaining an ef cient simulation Robustness properties, as described in 4 of this book, are also discussed Section 64 looks in the same way at transient measures, such as the reliability at a given time Section 65 gives a short introduction to and references on sensitivity analysis and non-Markovian models The following notation is used throughout the chapter For a function f : (0, ) R, we say that f ( ) = o( d ) if f ( )/ d 0 as 0; f ( ) = O( d ) if |f ( )| c1 d for some constant c1 > 0 for all suf ciently small; f ( ) = O( d ) if |f ( )| c2 d for some constant c2 > 0 for all suf ciently small; and f ( ) = ( d ) if f ( ) = O( d ) and f ( ) = O( d )

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