## Section: New Results

### Monte Carlo

Participants : Bruno Tuffin, Gerardo Rubino.

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.

**Rare event simulation of regenerative systems.**
Rare events occur by definition with a very small probability but are important to analyze because of potential catastrophic consequences. In [32], we focus on rare event for so-called regenerative processes, that are basically processes such that portions of them are statistically independent of each other. For many complex and/or large models, simulation is the only tool at hand but it requires specific implementations to get an accurate answer in a reasonable time. There are two main families of rare-event simulation techniques: Importance Sampling (IS) and Splitting.
In a first part, we briefly remind them and compare their respective advantages but later (somewhat arbitrarily) devote most of the work to IS.
We then focus on the estimation of the mean hitting time of a rarely visited set. A natural and direct estimator consists in averaging independent and identically distributed copies of simulated hitting times, but an alternative standard estimator uses the regenerative structure allowing to represent the mean as a ratio of quantities. We see that in the setting of crude simulation, the two estimators are actually asymptotically identical in a rare-event context, but inefficient for different, even if related, reasons: the direct estimator requires a large average computational time of a single run whereas the ratio estimator faces a small probability computation. We then explain that the ratio estimator is advised when using IS.
In the third part, we discuss the estimation of the distribution, not just the mean, of the hitting time to a rarely visited set of states. We exploit the property that the distribution of the hitting time divided by its expectation converges weakly to an exponential as the target set probability decreases to zero. The problem then reduces to the extensively studied estimation of the mean described previously. It leads to simple estimators of a quantile and conditional tail expectation of the hitting time. Some variants are presented and the accuracy of the estimators is illustrated on numerical examples.

In [46], we introduce and analyze a new regenerative estimator. A classical simulation estimator of this class is based on a ratio representation of the mean hitting time, using crude simulation to estimate the numerator and importance sampling to handle the denominator, which corresponds to a rare event. But the estimator of the numerator can be inefficient when paths to the set are very long. We thus introduce a new estimator that expresses the numerator as a sum of two terms to be estimated separately. We provide theoretical analysis of a simple example showing that the new estimator can have much better behavior than the classical estimator. Numerical results further illustrate this.

**Randomized Quasi-Monte Carlo for Quantile Estimation.**
Quantile estimation is a key issue in many application domains, but has been proved difficult to efficiently estimate. In [42], we compare two approaches for quantile estimation via randomized quasi-Monte Carlo (RQMC) in an asymptotic setting where the number of randomizations for RQMC grows large but the size of the low-discrepancy point set remains fixed. In the first method, for each randomization, we compute an estimator of the cumulative distribution function (CDF), which is inverted to obtain a quantile estimator, and the overall quantile estimator is the sample average of the quantile estimators across randomizations. The second approach instead computes a single quantile estimator by inverting one CDF estimator across all randomizations. Because quantile estimators are generally biased, the first method leads to an estimator that does not converge to the true quantile as the number of randomizations goes to infinity. In contrast, the second estimator does, and we establish a central limit theorem for it. Numerical results further illustrate these points.

**Reliability analysis with dependent components. **
In the reliability area, the Marshall-Olkin copula model has emerged as the standard tool for capturing dependence between components in failure analysis. In this model, shocks arise at exponential random times, affecting one or several components, thus inducing a natural correlation in the failure process. However, because the number of parameter of the model grows exponentially with the number of components, the tool suffers from the “curse of dimensionality.” These models are usually intended to be applied to design a network before its construction; therefore, it is natural to assume that only partial information about failure behavior can be gathered, mostly from similar existing networks. To construct them, we propose in [22] an optimization approach to define the shock's parameters in the copula, in order to match marginal failures probabilities and correlations between these failures. To deal with the exponential number of parameters of the problem, we use a column-generation technique. We also discuss additional criteria that can be incorporated to obtain a suitable model. Our computational experiments show that the resulting tool produces a close estimation of the network reliability, especially when the correlation between component failures is significant.

The Creation Process is an algorithm that transforms a static network model into a dynamic one. It is the basis of different variance reduction methods designed to make efficient reliability estimations on highly reliable networks in which links can only assume two possible values, operational or failed. In [18] the Creation Process is extended to let it operate on network models in which links can assume more than two values. The proposed algorithm, that we called Multi-Level Creation Process, is the basis of a method, also introduced here, to make efficient reliability estimations of highly reliable stochastic flow networks. The method proposed, which consists in an application of Splitting over the Multi-Level Creation Process, is empirically shown to be accurate, efficient, and robust. This work was the first step towards a way to implement an efficient estimation procedure for the problem of flow reliability analysis. Our first solution in that direction was presented in [54], where not only we could develop a procedure providing a significant variance reduction but that allows a direct extension to the final target, the solution to the same estimation problem in the more general case of models where the components are dependent. The idea is an original way of implementing a splitting procedure that leads simultaneously to these two properties.

**Rare events in risk analysis. **
One of the main tasks when dealing with critical systems (systems where specific classes of failures can deal to human losses, or to huge financial losses) is the ability to quantify the associated risks, which is the door that, when opened, leads to paths towards understanding what can happen and why, and towards capturing the relationships existing between the different parts of the system, with respect to those risks. This is also the necessary preliminary work allowing to evaluate the relative importance of different factors, always from the viewpoint of the considered risks, an important component of any disaster management system. Identifying the dominant ones is important to know which parts of the system we must reinforce. The keynote [29] described different tools available for these tasks, and how they can be used depending on the objectives to reach. The focus was on Monte Carlo techniques, the only available ones in general, because the only ones able to evaluate any kind of system, and how they deal with rare events. It also discussed the main related open research problems.
The tutorial [64] is closely related to previous talk, but the presentation explores more in general the estimation problem and the main families of techniques available for its solution (Importance Sampling, and the particular case of Zero-Variance methods, Splitting, Recursive Variance Reduction techniques, etc.).