Section: New Results

Dependability and extensions

Participants : Raymond Marie, Gerardo Rubino, Samira Saggadi, Bruno Tuffin.

We maintain a permanent research activity in different domains related to dependability, performability and vulnerability analysis of communication systems. Our focus is on evaluation techniques using both the Monte Carlo and the Quasi-Monte Carlo approaches. Monte Carlo (and Quasi-Monte Carlo) methods often represent the only available tool to solve complex problems in the area, and rare event simulation requires a special attention, in order to be able to efficiently analyze the model, that is, to be able to use good estimators having, in particular, a sufficiently small relative variance. Novel results in simulation can be decomposed into two subsets: results on rare event simulation, and those on Randomized Quasi-Monte Carlo methods.

The effectiveness of randomized quasi-Monte Carlo (RQMC) techniques is examined in [26] to estimate the integrals that express the discrete choice probabilities in a mixed logit model, for which no closed form formula is available. These models are used extensively in travel behavior research. We consider popular RQMC constructions, but our main emphasis is on randomly-shifted lattice rules, for which we study how to select the parameters as a function of the considered class of integrands. We compare the effectiveness of all these methods and of standard Monte Carlo (MC) to reduce both the variance and the bias when estimating the log-likelihood function at a given parameter value.

The main part of our activity in this simulation area in 2011 has been on rare event simulation though. The two major simulation families or rare event estimations are importance sampling and splitting. In [63] , we have provided a recent view of those methods, while in [64] we have overviewed how the zero-variance importance sampling can be approximated in classical reliability problems.

The problem of estimating the probability that a given set of nodes is connected in a graph (or network) where each link is failed with a given probability has received a lot of attention from us in 2011. We have proposed in [21] a new Monte Carlo method, based on dynamic Importance Sampling. The method generates the link states one by one, using a sampling strategy that approximates an ideal zero-variance importance sampling scheme. The approximation is based on minimal cuts in subgraphs. In an asymptotic rare-event regime where failure probability becomes very small, we prove that the relative error of our estimator remains bounded, and even converges to 0 under additional conditions, when the unreliability of individual links converges to 0. The empirical performance of the new sampling scheme is illustrated by examples. The method is even sped up in [50] by applying series-parallel reductions at each step of the algorithm.

The same problem is also analyzed in [15] by novel method that exploits a generalized splitting (GS) algorithm. We show that the proposed GS algorithm can accurately estimate extremely small unreliabilities and we exhibit large examples where it performs much better than existing approaches. Remarkably, it is also flexible enough to dispense with the frequently made assumption of independent edge failures. In [17] , another splitting approach is explored for the same problem, with very good results. It consists of a standard splitting procedure applied to the so-called Creation Process that can be associated with the initial static model. The paper discusses both a method for splitting this process, and an experimental analysis of the covering of the resulting estimator, showing its good behavior on different classes of test problems. Last, in [16] , always for the same static reliability problem, we proposed a new procedure belonging to the RVR family (Recursive Variance Reduction) where a new estimator based both in computed minpaths and mincuts of the graph, together with series-parallel reductions, allows to obtain very good accuracy in many rare events situations.

Finally, a versatile Monte Carlo method for estimating multidimensional integrals, with applications to rare-event probability estimation is presented in [39] , [75] . The method uses two distinct and popular Monte Carlo simulation techniques, namely Markov chain Monte Carlo (MCMC) and Importance Sampling, combined into a single algorithm. We show that for some illustrative and applied numerical examples the proposed Markov Chain Importance Sampling algorithm performs better than methods based solely on Importance Sampling or solely on MCMC.

Concerning the risk on spares for life-time maintenance purposes which is due to uncertainties on the mean up time, an extended version of a presentation made in 2010 has been published in [24] .