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[33] VF Nicola, MK Nakayama, P Heidelberger, and A Goyal Fast simulation of dependability models with general failure, repair and maintenance processes In Proceedings of the 20th International Symposium on Fault-Tolerant Computing, pp 491 498 IEEE Computer Society Press, 1990 [34] VF Nicola, MK Nakayama, P Heidelberger, and A Goyal Fast simulation of highly dependable systems with general failure and repair processes IEEE Transactions on Computers, 42(12): 1440 1452, December 1993 [35] VF Nicola, P Shahabuddin, P Heidelberger, and PW Glynn Fast simulation of steady-state availability in non-Markovian highly dependable systems In Proceedings of the 23rd International Symposium on Fault-Tolerant Computing, pp 38 47 IEEE Computer Society Press, 1993 [36] C Papadopoulos A new technique for MTTF estimation in highly reliable Markovian systems Monte Carlo Methods and Applications, 4(2): 95 112, 1998 [37] A Ridder Importance sampling simulations of Markovian reliability systems using cross-entropy Annals of Operations Research, 143: 119 136, 2005 [38] P Shahabuddin Fast transient simulation of Markovian models of highly dependable systems Performance Evaluation, 20: 267 286, 1994 [39] P Shahabuddin Importance sampling for the simulation of highly reliable Markovian systems Management Science, 40(3): 333 352, 1994 [40] P Shahabuddin, VF Nicola, P Heidelberger, A Goyal, and PW Glynn Variance reduction in mean time to failure simulations In Proceedings of the 1988 Winter Simulation Conference, pp 491 499 IEEE Press, 1988 [41] B Tuf n Bounded normal approximation in simulations of highly reliable Markovian systems Journal of Applied Probability, 36(4): 974 986, 1999 [42] B Tuf n On numerical problems in simulations of highly reliable Markovian systems In Proceedings of the 1st International Conference on Quantitative Evaluation of Systems (QEST), pp 156 164 Los Alamitos, CA: IEEE Computer Society Press, 2004 [43] B Tuf n, W Sandmann, and P L Ecuyer Robustness properties in simulations of highly reliable systems In Proceedings of RESIM 2006 , University of Bamberg, Germany, October 2006
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Rare event analysis by Monte Carlo techniques in static models
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H ctor Cancela, Mohamed El Khadiri e and Gerardo Rubino
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This chapter discusses Monte Carlo techniques for rare event simulation in the case of static models, that is, models in which time is not an explicit variable The main example and the one that will be used in the chapter is the network reliability analysis problem, where the models are graphs with probabilities associated with their components (with arcs or edges, and/or with nodes) Other typical names in this domain are fault trees, block diagrams, etc All these models are in general solved using combinatorial techniques, but only for quite small sizes, because their analysis is extremely costly in terms of computational resources The only methods able to deal with models having arbitrary size are Monte Carlo techniques, but there the main dif culty is with the rare event case, the focus of this chapter In many areas (eg, telecommunications, transportation systems, energy productions plants), either the components are very reliable or redundancy schemes are adopted, resulting in extremely reliable systems This means that a system s failure is (or should be) a rare event
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Rare Event Simulation using Monte Carlo Methods Edited by Gerardo Rubino and Bruno Tuffin 2009 John Wiley & Sons, Ltd ISBN: 978-0-470-77269-0
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Introduction
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The most commonly discussed example in the area of static models in dependability analysis is the network reliability problem This concerns the evaluation of reliability metrics of large classes of multicomponent systems We will denote by E the set of components in the system (which will shortly be represented by the set of edges of the undirected graph modeling the system) In general, the structure of such a system is represented by a binary function of |E| binary variables The usual convention for the state of a component or for the whole system is that 1 represents the operational state (the device, component or system is operational or up) and 0 represents the failed or down state A state vector or system con guration is a vector x = (x1 , , x|E| ) where xi is a possible state, 0 or 1, of the ith component (ie, x is an element of [0, 1]|E| ) With this notation, (x) = 1 if the system is up when the con guration is x, and 0 otherwise We may have different structure functions associated with the same system, each addressing a speci c aspect of interest that must be evaluated (see below) Frequently (but not always) structure functions are coherent, corresponding to systems satisfying the following properties: (i) when all the components are down (up), the system is down (up); (ii) if the system is up (down) and we change the state of a component from 0 to 1 (from 1 to 0), the system remains up (down); (iii) all the components are relevant (a component i is irrelevant if the state of the system does not depend on the state of i) Formally, let us denote by 0 (by 1) a state vector having all its entries equal to 0 (equal to 1) We also denote by x y the relation xi yi for all i, by x < y the fact that x y with, for some j , xj < yj , and by (x, 0i ) (by (x, 1i )) the state vector constructed from x by setting xi to 0 (to 1) Then, is coherent if and only if (i) (0) = 0, (1) = 1; (ii) if x < y then (x) (y); and (iii) for each component i there exists some state vector x such that (x, 0i ) = (x, 1i ) (and thus, due to (ii), (x, 0i ) = 0 and (x, 1i ) = 1) After specifying the function , which de nes how the system provides the service for which it was designed, a probabilistic structure must be added to take the failure processes into account The usual framework is to assume that the state of the ith component is a random binary (Bernoulli) variable Xi with expectation E(Xi ) = ri , and that the |E| random variables X1 , , X|E| are independent The numbers ri = P(Xi = 1) (called the elementary reliabilities) are input data Sometimes we will also use the notation qi for the unreliability of link i, that is, qi = 1 ri The output parameter is the reliability R of the system, de ned by R = P( (X) = 1) = E( (X)) (71)
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where X = (X1 , , X|E| ), or its unreliability Q = 1 R Observe that this is a static problem, that is, time is not explicitly used in the analysis When time relations are considered, the context changes and the general framework in which the analysis is usually done is the theory of stochastic processes and, in particular,
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