arXiv subjects

Guido Lagos

Publications and source records attributed to Guido Lagos.

6 recordsLinked to original sources

arXiv2026

Simple repair policies and decompositions for semi-coherent systems with simultaneous failures

We consider semi-coherent binary systems that are subject to simultaneous failures of its components. These are systems whose components can be either working or failed; the system can also be working or failed depending on the state of the components; and repairing a component cannot cause the system to fail. We consider that one or more components can fail simultaneously, allowing us to model external shocks and disasters. For this, we use the L\'evy-frailty Marshall-Olkin (LFMO) multivariate distribution to model the failure times of the components. We aim to answer in which states of the system we should repair the components. This is a challenging question, as the number of repair policies grows super-exponentially in the number of components. To tackle this, we propose a simple family of repair policies, which we call $r$-out-of-$n$:R repair policies, where one repairs all failed components when the system fails or when there are $r$ or more failed components. Our main contribution is that we derive exact and simple expressions for key performance-evaluation quantities of the system operating under our proposed repair policies. That is, we give explicit expressions for the mean time-to-failure of the system, mean time-to-repair, probability of system-failure before repair, and component- and system-repair rate. We also give expressions for the expected cost and long-term average cost, when there are components' and system repair cost. The only relevant parameters involved in the derived expressions are the structural signature of the system, and the Laplace exponent associated to the LFMO distribution.

math.PR↗

arXiv2025

Limiting behavior of mixed coherent systems with L\'evy-frailty Marshall-Olkin failure times

In this paper we show a limit result for the reliability function of a system -- that is, the probability that the whole system is still operational after a certain given time -- when the number of components of the system grows to infinity. More specifically, we consider a sequence of mixed coherent systems whose components are homogeneous and non-repairable, with failure-times governed by a L\'evy-frailty Marshall-Olkin (LFMO) distribution -- a distribution that allows simultaneous component failures. We show that under integrability conditions the reliability function converges to the probability of a first-passage time of a L\'evy subordinator process. To the best of our knowledge, this is the first result to tackle the asymptotic behavior of the reliability function as the number of components of the system grows. To illustrate our approach, we give an example of a parametric family of reliability functions where the system failure time converges in distribution to an exponential random variable, and give computational experiments testing convergence.

math.PR↗

arXiv2024

Forecasting Outside the Box: Application-Driven Optimal Pointwise Forecasts for Stochastic Optimization

We study a class of two-stage stochastic programs, namely, those with fixed recourse matrix and fixed costs, and linear second stage. We show that, under mild assumptions, the problem can be solved with just one scenario, which we call an ``optimal scenario.'' Such a scenario does not have to be unique and may fall outside the support of the underlying distribution. Although finding an optimal scenario in general might be hard, we show that the result can be particularly useful in the case of stochastic optimization problems with contextual information, where the goal is to optimize the expected value of a certain function given some contextual information (e.g., previous demand, customer type, etc.) that accompany the main data of interest. The contextual information allows for a better estimation of the quantity of interest via machine learning methods. We focus on a class of learning methods -- sometimes called in the literature decision-focused learning -- that integrate the learning and optimization procedures by means of a bilevel optimization formulation, which determines the parameters for pointwise forecasts. By using the optimal scenario result, we prove that when such models are applied to the class of contextual two-stage problems considered in this paper, the pointwise forecasts computed from the bilevel optimization formulation actually yield asymptotically the best approximation of an optimal scenario within the modeler's pre-specified set of parameterized forecast functions. Numerical results conducted with inventory problems from the literature (with synthetic data) as well as a bike-sharing problem with real data demonstrate that the proposed approach performs well when compared to benchmark methods from the literature.

math.OC↗

arXiv2019

Limit distributions of the upper order statistics for the Lévy-frailty Marshall-Olkin distribution

The Marshall-Olkin (MO) distribution has been considered a key model in reliability theory and in risk analysis, where it is used to model the lifetimes of dependent components or entities of a system and dependency is induced by "shocks" that hit one or more components at a time. Of particular interest is the Lévy-frailty subfamily of the Marshall-Olkin (LFMO) distribution, since it has few parameters and because the nontrivial dependency structure is driven by an underlying Lévy subordinator process. The main contribution of our work is that we derive the precise asymptotic behavior of the upper order statistics of the LFMO distribution. More specifically, we consider a sequence of $n$ univariate random variables jointly distributed as a multivariate LFMO distribution and analyze the order statistics of the sequence as $n$ grows. Our main result states that if the underlying Lévy subordinator is in the normal domain of attraction of a stable distribution with index of stability $α$ then, after certain logarithmic centering and scaling, the upper order statistics converge in distribution to a stable distribution if $α>1$ or a simple transformation of it if $α\leq1$. Our result is especially useful in network reliability and systemic risk, when modeling the lifetimes of components in a system using the LFMO distribution, as it allows to understand the behavior of systems that rely on its last working components. Our result can also give easily computable confidence intervals for these components, provided that a proper convergence analysis is carried out first.

math.PR↗

arXiv2017

On the Euler discretization error of Brownian motion about random times

In this paper we derive weak limits for the discretization errors of sampling barrier-hitting and extreme events of Brownian motion by using the Euler discretization simulation method. Specifically, we consider the Euler discretization approximation of Brownian motion to sample barrier-hitting events, i.e. hitting for the first time a deterministic "barrier" function; and to sample extreme events, i.e. attaining a minimum on a given compact time interval or unbounded closed time interval. For each case we study the discretization error between the actual time the event occurs versus the time the event occurs for the discretized path, and also the discretization error on the position of the Brownian motion at these times. We show limits in distribution for the discretization errors normalized by their convergence rate, and give closed-form analytic expressions for the limiting random variables. Additionally, we use these limits to study the asymptotic behaviour of Gaussian random walks in the following situations: (1.) the overshoot of a Gaussian walk above a barrier that goes to infinity; (2.) the minimum of a Gaussian walk compared to the minimum of the Brownian motion obtained when interpolating the Gaussian walk with Brownian bridges, both up to the same time horizon that goes to infinity; and (3.) the global minimum of a Gaussian walk compared to the global minimum of the Brownian motion obtained when interpolating the Gaussian walk with Brownian bridges, when both have the same positive drift decreasing to zero. In deriving these limits in distribution we provide a unified framework to understand the relation between several papers where the constant $-ζ(1/2)/\sqrt{2 π}$ has appeared, where $ζ$ is the Riemann zeta function. In particular, we show that this constant is the mean of some of the limiting distributions we derive.

math.PR↗

arXiv2017

A Dichotomy for Sampling Barrier-Crossing Events of Random Walks with Regularly Varying Tails

We study how to sample paths of a random walk up to the first time it crosses a fixed barrier, in the setting where the step sizes are iid with negative mean and have a regularly varying right tail. We introduce a desirable property for a change of measure to be suitable for exact simulation. We study whether the change of measure of Blanchet and Glynn (2008) satisfies this property and show that it does so if and only if the tail index $α$ of the right tail lies in the interval $(1, \, 3/2)$.

math.PR↗