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## Section: New Results

### Monte Carlo

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

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.

MCMC. The current popular method for approximate simulation from the posterior distribution of the linear Bayesian LASSO is a Gibbs sampler. It is well-known that the output analysis of an MCMC sampler is difficult due to the complex dependence among the states of the underlying Markov chain. Practitioners can usually only assess the convergence of MCMC samplers using heuristics. In [42] we construct a method that yields an independent and identically distributed (iid) draws from the LASSO posterior. The advantage of such exact sampling over the MCMC sampling is that there are no difficulties with the output analysis of the exact sampler, because all the simulated states are independent. The proposed sampler works well when the dimension of the parameter space is not too large, and when it is too large to permit exact sampling, the proposed method can still be used to construct an approximate MCMC sampler

Rare event simulation. We develop in [48] simulation estimators of measures associated with the tail distribution of the hitting time to a rarely visited set of states of a regenerative process. In various settings, the distribution of the hitting time divided by its expectation converges weakly to an exponential as the rare set becomes rarer. This motivates approximating the hitting-time distribution by an exponential whose mean is the expected hitting time. As the mean is unknown, we estimate it via simulation. We then obtain estimators of a quantile and conditional tail expectation of the hitting time by computing these values for the exponential approximation calibrated with the estimated mean. Similarly, the distribution of the sum of lengths of cycles before the one hitting the rare set is often well-approximated by an exponential, and we analogously exploit this to estimate tail measures of the hitting time. Numerical results demonstrate the effectiveness of our estimators.

In rare event simulation, the two main approaches are Splitting and Importance Sampling.

Concerning Splitting, we study in [52] the behavior of a method for sampling from a given distribution conditional on the occurrence of a rare event. The method returns a random-sized sample of points such that unconditionally on the sample size, each point is distributed exactly according to the original distribution conditional on the rare event. For a cost function which is nonzero only when the rare event occurs, the method provides an unbiased estimator of the expected cost, but if we select at random one of the returned points, its distribution differs in general from the exact conditional distribution given the rare event. However, we prove that if we repeat the algorithm, the distribution of the selected point converges to the exact one in total variation.

Splitting and another technique based on a conditional Monte Carlo approach have been applied and compared in [73] for the reliability estimation for networks whose links have random capacities and in which a certain target amount of flow must be carried from some source nodes to some destination nodes. Each destination node has a fixed demand that must be satisfied and each source node has a given supply. We want to estimate the unreliability of the network, defined as the probability that the network cannot carry the required amount of flow to meet the demand at all destination nodes. When this unreliability is very small, which is our main interest in this paper, standard Monte Carlo estimators become useless because failure to meet the demand is a rare event. We find that the conditional Monte Carlo technique is more effective when the network is highly reliable and not too large, whereas for a larger network and/or moderate reliability, the splitting approach is more effective. In [65] we presented the main ideas behind another approach for the same kind of problem, where we generalize the Creation Process idea to a multi-level setting, on top of which we explore the behavior of the Splitting method, with very good results when the system is highly reliable.

Importance sampling (IS) is the main used other technique, but it requires a fine tuning of parameters. This has been applied in [60] to urban passenger rail systems, that are large scale systems comprising highly reliable redundant structures and logistics (e.g., spares or repair personnel availability, inspection protocols, etc). To meet the strict contractual obligations, steady state unavailability of such systems needs to be accurately estimated as a measure of a solution’s life cycle costs. We use Markovian Stochastic Petri Nets models to conveniently represent the systems. We propose a multi-level Cross-Entropy optimization scheme, where we exploit the regenerative structure in the underlying continuous time Markov chain and to determine optimal (IS) rates in the case of rare events. The CE scheme is used in a pre-simulation and applied to failure transitions of the Markovian SPN models only. The proposed method divides a rare problem into a series of less rare problems by considering increasingly rare component failures. In the first stage a standard regenerative simulation is used for non-rare system failures. At each subsequent stage, the rarity is progressively increased (by decreasing the failure rates of components) and the IS rates of transitions obtained from the previous problem are used at the current stage. The final pre-simulation stage provides a vector of IS rates that are optimized and are used in the main simulation. The experimental results showed bounded relative error property as the rarity of the original problem increases, and as a consequence a considerable variance reduction and gain (in terms of work normalized variance).

In [68] and [29] we introduced the idea that the problem with the standard estimator in the case of rare events is not the estimator itself but its usual implementation, and we describe an efficient way of implementing it in order to be able to perform estimations that, otherwise, are out of reach following the crude approach as it is usually coded. The idea is to reduce the time needed to sample the standard estimator and not the variance. The interest in taking this viewpoint is also discussed.

In [30] we gave a tutorial on Monte Carlo techniques for rare event analysis, where the basic Splitting and Importance Sampling families of estimators are presented, together with the Zero Variance subfamily of the first class of techniques, plus other methods such as the Recursive Variance Reduction approach.

Taking dependence into account. The Marshall-Olkin copula model has emerged as the standard tool for capturing dependency between components in failure analysis in reliability. In this model shocks arise at exponential random times, that affect one or several components inducing a natural correlation in the failure process. However, the method presents the classic “curse of dimensionality” problem. Marshall-Olkin 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 such a MO model, we propose in [19] an optimization approach in order to define the shock’s parameters in the copula, with the goal of matching marginal failures probabilities and correlations between these failures. To deal with the exponential number of parameters of this 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 approach produces a close estimation of the network reliability, especially when the correlation between component failures is significant.

Random variable generation. Random number generators were invented before there were symbols for writing numbers, and long before mechanical and electronic computers. All major civilizations through the ages found the urge to make random selections, for various reasons. Today, random number generators, particularly on computers, are an important (although often hidden) ingredient in human activity. We study in [18] the lattice structure of random number generators of the MIXMAX family, a class of matrix linear congruential generators that produce a vector of random numbers at each step. These generators were initially proposed and justified as close approximations to certain ergodic dynamical systems having the Kolmogorov $K$-mixing property, which implies a chaotic (fast-mixing) behavior. But for a $K$-mixing system, the matrix must have irrational entries, whereas for the MIXMAX it has only integer entries. As a result, the MIXMAX has a lattice structure just like linear congruential and multiple recursive generators. Its matrix entries were also selected in a special way to allow a fast implementation and this has an impact on the lattice structure. We study this lattice structure for vectors of successive and non-successive output values in various dimensions. We show in particular that for coordinates at specific lags not too far apart, in three dimensions, all the nonzero points lie in only two hyperplanes. This is reminiscent of the behavior of lagged-Fibonacci and AWC/SWB generators. And even if we skip the output coordinates involved in this bad structure, other highly structured projections often remain, depending on the choice of parameters.