## Section: New Results

### Monte Carlo

Participants : Bruno Tuffin, Gerardo Rubino, Pablo Sartor.

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 in dealing with the rare event situation.

In [72] we have overviewed how the zero-variance importance sampling can be approximated in classical reliability problems. In general, we look for estimators such that the relative accuracy of the output is “controlled” when the rarity is getting more and more critical. Different robustness properties of estimators have been defined in the literature. However, these properties are not adapted to estimators coming from a parametric family for which the optimal parameter is random due to a learning algorithm. These estimators have random accuracy. For this reason, we motivate in [65] the need to define probabilistic robustness properties. We especially focus on the so-called probabilistic bounded relative error property. We additionally provide sufficient conditions, both in general and in Markov settings, to satisfy such a property, and hope that it will foster discussions and new works in the area.

In [43] and [18] we present results concerning the evaluation using Monte Carlo techniques, of a specific reliability metric for communication networks, based not only on connectivity properties, as in the classical network reliability measure, but also in the lengths of the paths. In [43] , we propose bounds of the metric that can be used to derive a variance reduction technique. In [18] , we describe techniques to analyze what could be called performability aspects of networks also based on the number of hops between sources and terminals. Let us also mention here our publication [16] , where we discuss the exact computation of these new types of metrics, and [29] , where other related combinatorial problems are discussed (here, optimization problems also based on connectivity properties, from the design point of view). In [17] , we propose a new version of the RVR principle, leading to a variance reduction technique for the classic network reliability problem. Paper [28] proposes a splitting algorithm for the same problem. The approach is quite straightforward, after the static problem is transformed into a dynamic one using the well known Creation Process. In [42] we explore a very general conditioning-based approach, including as a particular case the family of splitting procedures. We explore this idea through the analysis of dependability properties of complex systems using Markov models.

When looking specifically at static network reliability models, as described in the previous paragraph, it is often typically assumed that the failures of their components are independent. This assumption allows for the design of efficient Monte Carlo algorithms that can estimate the network reliability in settings where it is a rare-event probability. Despite this computational benefit, independent component failures is frequently not a realistic modeling assumption for real-life networks. In [39] we show how the splitting methods for rare-event simulation can be used to estimate the reliability of a network model that incorporates a realistic dependence structure via the Marshal-Olkin copula.

In [15] , we present a versatile Monte Carlo method for estimating multidimensional integrals, with applications to rare-event probability estimation. The method fuses two distinct and popular Monte Carlo simulation methods, Markov chain Monte Carlo and importance sampling, into a single algorithm. We show that for some applied numerical examples the proposed Markov Chain importance sampling algorithm performs better than methods based solely on importance sampling or MCMC.

Finally, in two presentations [67] and [32] we discuss the main problems when analyzing rare events using Monte Carlo methods, focusing on robustness properties of the corresponding estimators.