Mohamed El Khadiri

A new simulation method based on the RVR principle for the rare event network reliability problem

In this paper we consider the evaluation of the well known -network unreliability parameter by means of a new RVR Monte-Carlo method. This method is based on series-parallel reductions and a partitioning procedure using pathsets and cutsets for recursively changing the original problem into similar ones on smaller networks. By means of several experimental results, we show that the proposed method has good performances in rare event cases and offers significant gains over other state-of-the-art variance reduction techniques.

Balanced and Approximate Zero-Variance Recursive Estimators for the Network Reliability Problem

Exact evaluation of static network reliability parameters belongs to the NP-hard family, and Monte Carlo simulation is therefore a relevant tool to provide their estimations. The first goal of this work is to review a Recursive Variance Reduction (RVR) estimator, which approaches the unreliability 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—that is, 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—the Balanced Recursive Decomposition estimator—chooses the first working link on cuts uniformly, whereas the second—the 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 (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.

Reliability Evaluation of Communication Networks

Abstract In the evaluation of the capacity of a communication network architecture to resist to the possible faults of some of its components, several reliability metrics are currently used. The evaluation of these metrics is in general a very costly task since most of them are, as algorithmic problems, classed in the NP-hard family. As a consequence, many different techniques have been proposed to solve them. We discuss here a promising class of methods called “factorization” and some of the implementation issues. An alternative approach to these exact techniques is to perform statistical estimations using a Monte Carlo simulation. It allows to deal with larger networks (having, say, hundreds of components) if the user accepts probabilistic answers. In the case of highly reliable networks, the standard Monte Carlo technique is also prohibitively expensive and variance reduction techniques must be used. We propose here a new Monte Carlo algorithm specifically designed to this context. For both approaches, exact and simulation algorithms, numerical results are provided allowing a comparison and giving an idea of the performances that can be expected.

On the robustness of fishman's bound-based method for the network reliability problem

Static network unreliability computation is an NP-hard problem, leading to the use of Monte Carlo techniques to estimate it. The latter, in turn, suffer from the rare event problem, in the frequent situation where the systems unreliability is a very small value. As a consequence, specific rare event event simulation techniques are relevant tools to provide this estimation. We focus here on a method proposed by Fishman making use of bounds on the structure function of the model. The bounds are based on the computation of (disjoint) mincuts disconnecting the set of nodes and (disjoint) minpaths ensuring that they are connected. We analyze the robustness of the method when the unreliability of links goes to zero. We show that the conditions provided by Fishman, based on a bound, are only sufficient, and we provide more insight and examples on the behavior of the method.

A Monte Carlo Estimator Based on a State Space Decomposition Methodology for Flow Network Reliability

The exact evaluation of the probability that the maximum st-flow is greater than or equal to a fixed value d in a stochastic flow network is an NP-hard problem. This limitation leads to consider Monte Carlo alternatives. In this paper, we show how to exploit the state space decomposition methodology of Doulliez and Jamoulle for deriving a Monte Carlo simulation algorithm. We show that the resulting Monte Carlo estimator belongs to the variance-reduction family and we give a worst-case bound on the variance-reduction ratio that can be expected when compared with the standard sampling. We illustrate by numerical comparisons that the proposed simulation algorithm allows substantial variance-reduction with respect to the standard one and it is competitive when compared to a previous work in this context.