ELSEVIER 0951-8320(95)00092-5 Reliability Engineering and System St~fety 52 (1996) 65-75 © 1996 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0951-8320]96/$15.IX) A Monte Carlo estimation of the marginal distributions in a problem of probabilistic dynamics P. E. Labeau* Service de Mdtrologie NuelOaire, Universit~ Libre de Bruxelles, Avenue F.D. Roosevelt, 50 1050 Bruxelles, Belgium (Received 2 May 1995; revised 3 July 1995; accepted 30 August 1995) Modelling the effect of the dynamic behaviour of a system on its PSA study leads, in a Markovian framework, to a development at first order of the Chapman-Kolmogorov equation, whose solutions are the probability densities of the problem. Because of its size, there is no hope of solving directly these equations in realistic circumstances. We present in this paper a biased simulation giving the marginals and compare different ways of speeding up the integration of the equations of the dynamics. © 1996 Elsevier Science Limited. 1 INTRODUCTION Recently, researchers in reliability have stressed the need to incorporate in a usual safety study the dynamic behaviour of a system and its influence on the transitions likely to occur in an accidental transient. I-4 Indeed, each reachable component state corresponds to a given dynamics, and, in turn, the transitions between states depend on the transition rates which are influenced by the evolution of the physical variables describing the system. This feedback is quantified by the probability densities of being in a given state at a given time, with given values of the physical variables. Estimating these distributions is far from being an easy computational task, since we are faced with high-dimensional problems in realistic circumstances. We shall first remind readers how this dynamic concept has been modelled in the Markovian assumption, as well as mentioning some previous attempts to deal with it. We shall then present a general way of simulating the behaviour of the system. We will then show how to bias this algorithm and apply it on a benchmark. Different techniques for accelerating the integration of the equations of the dynamics will be given and compared on the same benchmark. A more complicated application on a nuclear reactor transient will then be solved. Finally, we will give some concluding remarks. *Research Assistant (National Fund for Scientific Research, Belgium. 2 PROBABILISTIC DYNAMICS Let us consider how a system starting from state i behaves. The physical variables will evolve deter- ministically, according to the dynamics in state i d.f -x =f(-) (1) from time t = 0 to t = tj, where a stochastic transition occurs. The system then changes its state to j, in which a similar deterministic evolution will take place until the next transition. To quantify this deterministic-stochastic process, we introduce lr(i, £, t), density probability that the system is in state i at time t with physical variables £, for given initial conditions. It has been shown 5 that rc(i,Y,t) obeys a development at first order of the Chapman-Kolmogorov equation, if we assume a Markovian framework O~( i,£,t ) - - + div,~f~(Y);c(i,£,t)) + Ai(Y)Ir(i,Y,t) Ot 65 = ~p(j--->il£)rc(j,Y,t ) (2) j~i where Ai(£)andp(j---~il£ ) are the rate of transition out of state i and the transition rate from state j to state i, respectively. To solve system (2), the usual numerical schemes fail to work for realistic problems, since a great number of states have to be considered and enough variables have to be taken in order to model the