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Abate et al [1] showed this problem to be complementary to the problem of optimizing the control policy of an SHS such that the reach probability of some prescribed unsafe set remains below some given maximum level, and that the same dynamic programming-based computation of maximal safe sets can be used The dynamic programming approach becomes computationally intractable when the SHS considered is of large-scale type Prandini and Hu [39] developed a Markov chain approximation based method for the computation of reach probabilities for a continuous-time SHS This way the dynamic programming challenge s avoided, but the computational load of their method prohibits its application to a large-scale SHS Prajna et al [38] developed an approach which obtains an upper bound of the reach probability, but this cannot handle large-scale SHS either In theory, reach probability estimation can be done by simulating many trajectories of the process considered, and counting the fraction of cases where the simulated trajectory reaches the unsafe set within some given period T When the reach probability value is very small then the number of straightforward Monte Carlo (MC) simulations needed is impractically large The rare event estimation literature forms a potentially rich source of information for speeding up MC simulation, for example by combining methods from large-deviation and importance sampling theories [11, 29, 31] An early successful development in this area is sequential MC simulation for the estimation of the intensity of radiation that penetrates a shield of absorbing material in nuclear physics (see [10]) More recently this approach has also found application in non-nominal delay time and loss estimation in telecommunication networks [3] L Ecuyer et al [36] provide a very good recent overview of these sequential MC simulation developments In order to exploit rare event estimation theory within probabilistic reachability analysis of controlled SHS, we need to establish a theoretically unambiguous connection between the two concepts Implicitly, this connection has recently been elaborated by Del Moral and co-workers [16 18, 20, 21] They embedded theoretical physics equations, which supported the development of advanced MC simulations, within the stochastic analysis setting that is typically used for probabilistic reachability analysis They subsequently showed that this embedding provides a powerful background for the development and analysis of sequential MC simulation for rare event simulation In 3 of the present volume this novel development is well explained in the broader context of splitting techniques in rare event simulation The aim of this chapter is to present a part of the framework developed by Del Moral et al [16 18, 20, 21] in a probabilistic reachability setting, to further develop this for a large-scale SHS, and to demonstrate its practical use for safety veri cation of an advanced air traf c operation In [8, 9], the practical use of the approach of Del Moral [16 18, 20, 21] for safety veri cation of an advanced air traf c operation has already been demonstrated for some speci c scenarios In these scenarios, the main contributions to the reach probability value came from diffusion behavior It also became clear that the same sequential MC simulation approach failed to work for scenarios of the same air traf c. 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To scale a drawing onto a piece of paper in a pleasing manner requires a rectangular shape that somewhat resembles the proportions of standard paper sizes.Related: operation where the reach probability is determined by rare switching between modes This chapter aims to tackle such more demanding rare event estimation problems for large-scale controlled SHSs Essentially the approach is to introduce an aggregation of the discrete mode process, and to develop importance switching and Rao Blackwellization relative to these aggregated modes The chapter is organized as follows Section 92 develops a factorization of the reach probability Section 93 explains the approach of [16, 17, 20, 21] Section 94 presents an extension of this approach to hybrid systems Section 95 develops the aggregation mode process and characterizes key relations with the controlled SHS Section 96 develops a novel sequential MC simulation approach for estimating reach probabilities Section 97 brie y describes the free ight air traf c example considered Section 98 applies the novel approach to estimate reach probabilities for this air traf c example Section 99 presents oncluding remarks An early version of this chapter is [5]. 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