Section: New Results

Monte Carlo

Participants : Gerardo Rubino, Bruno Tuffin, Pablo Sartor Del Giudice.

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, for two reasins: the need in accelerating the occurrence of those events and in getting unbiased estimators of them 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. Rare event simulation has been reviewed in [22] .

Multidimensional integrals. In [20] , we present a versatile Monte Carlo method for estimating multidimensional integrals, with applications to rare-event probability estimation. The method combines two distinct and popular Monte Carlo simulation techniques, 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.

Static models. Static reliability analysis has been the topic of an extensive activity in the group for years. Exact evaluation of static network reliability parameters belongs to the NP-hard family and Monte Carlo simulation is therefore a relevant tool to provide estimations for them.

In [67] , we first review a Recursive Variance Reduction (RVR) estimator which approaches the unreliability metric by recursively reducing the graph from the random choice of the first working link on selected cuts. We show that the method does not verify the bounded relative error (BRE) property as reliability of individual links goes to one, i.e., that 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 Importance Sampling technique. Two new estimators are presented: 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, tries to mimic the estimator with variance zero for this technique. We show that in both cases the BRE property is verified and, moreover, that a Vanishing Relative Error 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.

The same problem is also analyzed in [19] by a 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.

On the same type of model, we propose in [51] an adaptive parameterized method to approximate the zero-variance change of measure for the evaluation of static network reliability models, with links subject to failures. The method uses two rough approximations of the unreliability function, conditional on the states of any subset of links being fixed. One of these approximation, based on mincuts, under-estimates the true unknown unreliability, whereas the other one, based on minpaths, over-estimates it. Our proposed change of measure takes a convex linear combination of the two, estimates the optimal (graph-dependent) coefficient in this combination from pilot runs, and uses the resulting conditional unreliability approximation at each step of a dynamic Importance Sampling algorithm. This new scheme is more general and more flexible than a previously-proposed zero-variance approximation one, which is based on mincuts only and which was shown to be robust asymptotically when unreliabilities of individual links decrease toward zero. Our numerical examples show that the new scheme is often more efficient when the unreliabilities of the individual links are not so small but the overall unreliability is small because the system can fail in many ways. Part of these results are in the PhD [13] .

In [43] , we present a generalization of the above static models to cases for which the component failures are not independent. To model the dependence and also to develop effective simulation methods that estimate the system unreliability, we extend the static model into an auxiliary dynamic one where the components fail at random times, according to a Marshall-Olkin multivariate exponential distribution. We examine and compare different versions of this model and develop efficient unreliability estimation methods based on conditional Monte Carlo and on a generalized splitting methodology.

In [28] , a different splitting algorithm is proposed for solving the same static problem, which is converted into a dynamic one by means of the Creation Process of Elperin, Gerbtsbakh and Lomonosov. The classic splitting technique is then applied, and the obtained results are explored through several numerical experiments. The relative error and the covering properties of the obtained estimator are particularly studied.

In [29] , a generalization of the basic model is studied using Monte Carlo. The idea is that the system (the network) works when the terminal nodes are connected by at least one path whose length is less than or equal to a given parameter $d$. This is called Diameter Constrained Reliability. If the parameter $d$ is greater than or equal to the longest path in the network (or between terminals), the problem is the classic one. The paper proposes a variance reduction technique for the estimation of the system's reliability in this setting. In [21] , we analyze the particular case of $d=2$ using exact techniques. These results are part of the thesis [14] .

Finally, in [34] and [36] we made general presentations on the rare event problem in general, and on some of the team's results concerning the design of efficient techniques to analyze them.