Section: New Results

Monte Carlo

Participants : Pierre L'Ecuyer, Gerardo Rubino, Bruno Tuffin.

We maintain a research activity in different areas related to dependability, performability and vulnerability analysis of communication systems, using both the Monte Carlo and the Quasi-Monte Carlo approaches to evaluate the relevant metrics. Monte Carlo (and Quasi-Monte Carlo) methods often represent the only tool able to solve complex problems of these types. However, when the events of interest are rare, simulation requires a special attention, to accelerate the occurrence of the event and get unbiased estimators of the event of interest with a sufficiently small relative variance. This is the main problem in the area. Dionysos' work focuses then on dealing with the rare event situation. For example, [39] presents an exponential tilting method for exact simulation from the truncated multivariate student-t distribution in high dimensions as an alternative to approximate Markov Chain Monte Carlo sampling.

A non-negligible part of our activity on the application of rare event simulation was about the evaluation of static network reliability models. Our paper [16] focuses on a technique known as Recursive Variance Reduction (RVR) which approaches the unreliability by recursively reducing the graph from the random choice of the first working link on selected cuts. This previously known method is shown to not verify the bounded relative error (BRE) property as reliability of individual links goes to one, i.e., the estimator is not robust in general to high reliability of links. We then propose to use the decomposition ideas of the RVR estimator in conjunction with the IS technique. Two new estimators are presented in the paper: the first one, called Balanced Recursive Decomposition estimator, chooses the first working link on cuts uniformly, while the second, called Zero-Variance Approximation Recursive Decomposition estimator, combines RVR and our zero-variance IS approximation. We show that in both cases BRE property is verified and, moreover, that a vanishing relative error (VRE) property can be obtained for the Zero-Variance Approximation RVR under specific sufficient conditions. A numerical illustration of the power of the methods is provided on several benchmark networks. In [54] , we explore the use of the same powerful RVR idea, but applied in a very general context, where the system is model by a monotone structure function. In the paper, we illustrate the approach with a very widely used model, a series of $k$-out-of-$m$ modules.

In a static network reliability model one typically assumes that the failures of the components of the network are independent. This simplifying assumption makes it possible to estimate the network reliability efficiently via specialized Monte Carlo algorithms. Hence, a natural question to consider is whether this independence assumption can be relaxed, while still attaining an elegant and tractable model that permits an efficient Monte Carlo algorithm for unreliability estimation. In [14] we provide one possible answer by considering a static network reliability model with dependent link failures, based on a Marshall-Olkin copula, which models the dependence via shocks that take down subsets of components at exponential times, and propose a collection of adapted versions of permutation Monte Carlo (PMC, a conditional Monte Carlo method), its refinement called the turnip method, and generalized splitting (GS) methods, to estimate very small unreliabilities accurately under this model. The PMC and turnip estimators have bounded relative error when the network topology is fixed while the link failure probabilities converge to 0, whereas GS does not have this property. But when the size of the network (or the number of shocks) increases, PMC and turnip eventually fail, whereas GS works nicely (empirically) for very large networks, with over 5000 shocks in our examples. In [41] we focus on a method proposed by Fishman making use of bounds on the structure function describing in terms of configurations of (independent) link states if the considered nodes are connected. The bounds are based on the computation of (independent) mincuts disconnecting the set of nodes and (independent) minpaths ensuring that they are connected. We analyze here the robustness of the method when the unreliability of links goes to zero. We show that the conditions provided by Fishman are based on a bound and are therefore only sufficient, and provide more insight and examples on the behavior of the method.

PMC is an effective way of estimating the unreliability of a static network when this unreliability is very small and the network is not too large. We generalize the method in [31] to cover a wider range of applications, in which an estimation problem can be reframed in terms of the hitting time of a given set of states by a continuous-time Markov chain. The estimator is then defined as a function of the sample path of the underlying discrete time chain only, via Conditional Monte Carlo.We prove that the method gives bounded relative error for rare event probability estimation in certain settings. We show how it can be used to estimate the cumulative distribution function, or the density, or some moment of the hitting time. We provide examples for which the method can be applied and we give numerical illustrations.

Another family of models of interest in the group are the highly reliable Markovian systems, where a Markov chain models the evolution of a multicomponent system with failures and repairs of its components. In [27] we explore a new approach in the context of these models, and in the rare event case, called Conditional Monte Carlo with Intermediate Estimations (CMIE). The target are models with complex structures, where it is hard to design a good importance function dealing to good Importance Sampling schemes. The paper shows that the method belongs to the variance reduction family, and some examples illustrate its performances. It can be seen as a generalization of the class of splitting simulation procedures.

Finally, in Quasi-Monte Carlo (QMC), we reviewed in [64] the recent development on array-RQMC, a randomized quasi-Monte Carlo method for we had developed estimating the state distribution at each step of a Markov chain with totally ordered (discrete or continuous) state space. It can be used in particular to obtain a low-variance unbiased estimator of the expected total cost up to some random stopping time, when state-dependent costs are paid at each step. In [21] , a combination of sequential MC with RQMC to accelerate convergence proposed by Gerber and Chopin is compared with our array-RQMC.

But simulation requires the use of pseudo-random generators. In [45] we provide a review of the state of the art on the design and implementation of random number generators (RNGs) for simulation, on both sequential and parallel computing environments. A general review of pseudo-random and quasi-random number generation is also provided in [73] . A tool for the generation of rank-1 lattice rules is described in [22] .