## 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.