## 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.

**Rare event simulation**.
However, when the events of interest are rare,
simulation requires a special attention, to accelerate the occurrence
of the event and get unbiased estimators of the event of
interest with a sufficiently small relative variance
(see our book [108] for a global introduction to the field). This is the
main problem in the area. Dionysos' work focuses then on dealing with
the rare event situation, with a particular focus on dependability [40].

A non-negligible part of our activity on the application of rare event simulation was about the evaluation of static network reliability models. In a static network reliability model one typically assumes that the failures of the components of the network are independent. This simplifying assumption makes it possible to estimate the network reliability efficiently via specialized Monte Carlo algorithms. Hence, a natural question to consider is whether this independence assumption can be relaxed, while still attaining an elegant and tractable model that permits an efficient Monte Carlo algorithm for unreliability estimation. In [12], we provide one possible answer by considering a static network reliability model with dependent link failures, based on a Marshall-Olkin copula, which models the dependence via shocks that take down subsets of components at exponential times, and propose a collection of adapted versions of permutation Monte Carlo (PMC, a conditional Monte Carlo method), its refinement called the turnip method, and generalized splitting (GS) methods, to estimate very small unreliabilities accurately under this model. The PMC and turnip estimators have bounded relative error when the network topology is fixed while the link failure probabilities converge to zero, whereas GS does not have this property. But when the size of the network (or the number of shocks) increases, PMC and turnip eventually fail, whereas GS works nicely (empirically) for very large networks, with over 5000 shocks in our examples. In [73], we propose a methodology for calibrating a dependent failure model to compute the reliability in a telecommunication network, following a similar starting point (that is, using Marshall-Olkin copulas). In practice, this model is difficult to calibrate because it requires the estimation of a number of parameters that is exponential in the number of links. We formulate an optimization problem for calibrating a Marshall-Olkin copula model to attain given marginal failure probabilities for all links and the correlations between them. Using a geographic failure model, we calibrate various Marshall-Olkin copula models using our methodology, we simulate them, and we benchmark the reliabilities thus obtained. Our experiments show that considering the simultaneous failures of small and connected subsets of links is the key for obtaining a good approximation of reliability, confirming what it is suggested by the telecommunication literature.

A related problem is when links have random capacities and a certain target amount of flow must be carried from some source nodes to some destination nodes is considered in [47]. Each destination node has a fixed demand that must be satisfied and each source node has a given supply. The goal is to estimate the unreliability of the network, defined as the probability that given the realized link capacities, the network cannot carry the required amount of flow to meet the demand at all destination nodes. We adapt GS and PMC to this context. In [55], we explore other methods designed to reduce the variance of the estimators in this context. All of them are adaptations of methods originally developed to make reliability estimations on different network models. These methods are introduced together with a brief review of the algorithms on which they are based.

A new application of our previously designed zero-variance approximation importance sampling method has been developed in [76]: To accurately estimate the reliability of highly reliable rail systems and comply with contractual obligations, rail system suppliers such as ALSTOM require efficient reliability estimation techniques. While in our previous works, the studied graph models were dealing with failing links, we propose an adaptation of the algorithm to evaluate the reliability of real transport systems where nodes are the failing components. This is more representative of railway telecommunication system behavior. Robustness measures of the accuracy of the estimates, bounded or vanishing relative error properties, are discussed and results from a real network (Data Communication System used in automated train control system) showing bounded relative error property, are presented.

**Random variable generation.**
Simulation requires the use of pseudo-random generators. In [18], we examine the requirements and the available methods and software to
provide (or imitate) uniform random numbers in parallel computing environments. In this context, for the great majority of applications, independent
streams of random numbers are required, each being computed on a single
processing element at a time. Sometimes, thousands or even millions of such
streams are needed. We explain how they can be produced and managed.
We devote particular attention to multiple streams for GPU devices.

Sampling from the Normal distribution truncated to some finite or semi-infinite interval is of particular interest for certain applications in Bayesian statistics, such as to perform exact posterior simulations for parameter inference. We study and compare in [46] various methods to generate such random variables, with special attention to the situation where the interval is far in the tail. The algorithms are implemented and available in Java, R, and MATLAB, and the software is freely available.

**Quasi-Monte Carlo (QMC).**
Finally, we have continued our work on QMC methods. In [15], we review the Array-RQMC method, its variants, sorting strategies, and convergence results. We are interested in the convergence rate of measures of discrepancy of the states at a given step of the chain, as a function of the sample
size, and also the convergence rate of the variance of the sample average of
a (cost) function of the state at a given step, viewed as an estimator of the expected cost. We summarize known convergence rate results and show empirical
results that suggest much better convergence rates than those that are proved.
We also compare different types of multivariate sorts to match the chains with
the RQMC points, including a sorting procedure based on a Hilbert curve.

The description of a new software tool and library named Lattice Builder, written in C++, that implements a variety of construction algorithms for good rank-1 lattice rules (a familiy of sequences used in QMC methods) is provided in [17]. The library is extensible, thanks to the decomposition of the algorithms into decoupled components, which makes it easy to implement new types of weights, new search domains, new figures of merit, etc.