Section: New Results

Monte Carlo

Participants: Bruno Tuffin, 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. We have published an introduction to Monte Carlo methods on Insterstices, including animations https://interstices.info/jcms/int_69164/la-simulation-de-monte-carlo.

Rare event simulation. The mean time to failure (MTTF) of a stochastic system is often estimated by simulation. One natural estimator, which we call the direct estimator, simply averages independent and identically distributed copies of simulated times to failure. When the system is regenerative, an alternative approach is based on a ratio representation of the MTTF. The purpose of [42] is to compare the two estimators. We first analyze them in the setting of crude simulation (i.e., no importance sampling), showing that they are actually asymptotically identical in a rare-event context. The two crude estimators are inefficient in different but closely related ways: the direct estimator requires a large computational time because times to failure often include many transitions, whereas the ratio estimator entails estimating a rare-event probability. We then discuss the two approaches when employing importance sampling; for highly reliable Markovian systems, we show that using a ratio estimator is advised.

Another problem studied in [40] is the estimation of the tail of the distribution of the sum of correlated log-normal random variables. While a number of theoretically efficient estimators have been proposed for this setting, using a few numerical examples we illustrate that these published proposals may not always be useful in practical simulations. As a remedy to this defect, we propose a new estimator and we demonstrate that, not only is our novel estimator theoretically efficient, but, more importantly, its practical performance is significantly better than that of its competitors.

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. In the invited paper [32], we give a historical account on the design, implementation, and testing of uniform random number generators used for simulation.

We study in [68] the lattice structure of random number generators of the specific 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. 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.

Quasi-Monte Carlo (QMC). In [5], which appeared in 2017, we survey basic ideas and results on randomized quasi-Monte Carlo (RQMC) methods, discuss their practical aspects, and give numerical illustrations. RQMC can improve accuracy compared with standard Monte Carlo (MC) when estimating an integral interpreted as a mathematical expectation. RQMC estimators are unbiased and their variance converges at a faster rate (under certain conditions) than MC estimators, as a function of the sample size. Variants of RQMC also work for the simulation of Markov chains, for function approximation and optimization, for solving partial differential equations, etc. In this introductory survey, we look at how RQMC point sets and sequences are constructed, how we measure their uniformity, why they can work for high-dimensional integrals, and how can they work when simulating Markov chains over a large number of steps.

General presentations. Finally, in two general presentations, we described state-of-the-art technologies available to deal with rare events by means of Monte Carlo techniques, including several methods produced inside Dionysos. In the tutorial [33], we gave an overview of the field, with a focus on dependability analysis applications. The keynote [36] described specific procedures taken from our monograph  [72], that were adapted to the needs of the micro-simulation community.