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Barrera, J., Lagos, G. (2020). Limit distributions of the upper order statistics for the Levyfrailty MarshallOlkin distribution. Extremes, 23, 603–628.
Abstract: The MarshallOlkin (MO) distribution is 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 Levyfrailty subfamily of the MarshallOlkin (LFMO) distribution, since it has few parameters and because the nontrivial dependency structure is driven by an underlying Levy subordinator process. The main contribution of this work is that we derive the precise asymptotic behavior of the upper order statistics of the LFMO distribution. More specifically, we consider a sequence ofnunivariate random variables jointly distributed as a multivariate LFMO distribution and analyze the order statistics of the sequence asngrows. Our main result states that if the underlying Levy subordinator is in the normal domain of attraction of a stable distribution with index of stability alpha then, after certain logarithmic centering and scaling, the upper order statistics converge in distribution to a stable distribution if alpha> 1 or a simple transformation of it if alpha <= 1. Our result can also give easily computable confidence intervals for the last failure times, provided that a proper convergence analysis is carried out first.

Barrera, J., Carrasco, R. A., & Moreno, E. (2020). Realtime fleet management decision support system with security constraints. TOP, 28(3), 728–748.
Abstract: Intelligent transportation, and in particular, fleet management, has been a forefront concern for a plethora of industries. This statement is especially true for the production of commodities, where transportation represents a central element for operational continuity. Additionally, in many industries, and in particular those with hazardous environments, fleet control must satisfy a wide range of security restrictions to ensure that risks are kept at bay and accidents are minimum. Furthermore, in these environments, any decision support tool must cope with noisy and incomplete data and give recommendations every few minutes. In this work, a fast and efficient decision support tool is presented to help fleet managers oversee and control ore trucks, in a mining setting. The main objective of this system is to help managers avoid interactions between ore trucks and personnel buses, one of the most critical security constraints in our case study, keeping a minimum security distance between the two at all times. Furthermore, additional algorithms are developed and implemented, so that this approach can work with reallife noisy GPS data. Through the use of historical data, the performance of this decision support system is studied, validating that it works under the reallife conditions presented by the company. The experimental results show that the proposed approach improved truck and road utilization significantly while allowing the fleet manager to control the security distance required by their procedures.

Barrera, J., Carrasco, R. A., Mondschein, S., Canessa, G., & RojasZalazar, D. (2020). Operating room scheduling under waiting time constraints: the Chilean GES plan. Ann. Oper. Res., 286(12), 501–527.
Abstract: In 2000, Chile introduced profound health reforms to achieve a more equitable and fairer system (GES plan). The reforms established a maximum waiting time between diagnosis and treatment for a set of diseases, described as an opportunity guarantee within the reform. If the maximum waiting time is exceeded, the patient is referred to another (private) facility and receives a voucher to cover the additional expenses. This voucher is paid by the health provider that had to do the procedure, which generally is a public hospital. In general, this reform has improved the service for patients with GES pathologies at the expense of patients with nonGES pathologies. These new conditions create a complicated planning scenario for hospitals, in which the hospital's OR Manager must balance the fulfillment of these opportunity guarantees and the timely service of patients not covered by the guarantee. With the collaboration of the Instituto de Neurocirugia, in Santiago, Chile, we developed a mathematical model based on stochastic dynamic programming to schedule surgeries in order to minimize the cost of referrals to the private sector. Given the large size of the state space, we developed an heuristic to compute good solutions in reasonable time and analyzed its performance. Our experimental results, with both simulated and real data, show that our algorithm performs close to optimum and improves upon the current practice. When we compared the results of our heuristic against those obtained by the hospital's OR manager in a simulation setting with real data, we reduced the overtime from occurring 21% of the time to zero, and the nonGES average waiting list's length from 71 to 58 patients, without worsening the average throughput.
Keywords: Scheduling; Operating theater; Operating room scheduling

Barrera, J., Moreno, E., & Varas, S. (2020). A decomposition algorithm for computing income taxes with passthrough entities and its application to the Chilean case. Ann. Oper. Res., 286(12), 545–557.
Abstract: Income tax systems with “passthrough” entities transfer a firm's income to shareholders, which are taxed individually. In 2014, a Chilean tax reform introduced this type of entity and changed to an accrual basis that distributes incomes (but not losses) to shareholders. A crucial step for the Chilean taxation authority is to compute the final income of each individual given the complex network of corporations and companies, usually including cycles between them. In this paper, we show the mathematical conceptualization and the solution to the problem, proving that there is only one way to distribute income to taxpayers. Using the theory of absorbing Markov chains, we define a mathematical model for computing the taxable income of each taxpayer, and we propose a decomposition algorithm for this problem. This approach allows us to compute the solution accurately and to efficiently use computational resources. Finally, we present some characteristics of Chilean taxpayers' network and the computational results of the algorithm using this network.
Keywords: Income taxes; Markov processes; Networks; Algorithms

Matus, O., Barrera, J., Moreno, E., & Rubino, G. (2019). On the MarshallOlkin Copula Model for Network Reliability Under Dependent Failures. IEEE Trans. Reliab., 68(2), 451–461.
Abstract: The MarshallOlkin (MO) copulamodel has emerged as the standard tool for capturing dependence 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, because the number of parameter of the model grows exponentially with the number of components, MO suffers of the “curse of dimensionality.” MO 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 an MO model, we propose an optimization approach to define the shock's parameters in the MO copula, in order to match marginal failures probabilities and correlations between these failures. To deal with the exponential number of parameters of this problem, we use a columngeneration technique. We also discuss additional criteria that can be incorporated to obtain a suitable model. Our computational experiments show that the resulting MO model produces a close estimation of the network reliability, especially when the correlation between component failures is significant.

Barrera, J., HomemDeMello, T., Moreno, E., Pagnoncelli, B. K., & Canessa, G. (2016). Chanceconstrained problems and rare events: an importance sampling approach. Math. Program., 157(1), 153–189.
Abstract: We study chanceconstrained problems in which the constraints involve the probability of a rare event. We discuss the relevance of such problems and show that the existing samplingbased algorithms cannot be applied directly in this case, since they require an impractical number of samples to yield reasonable solutions. We argue that importance sampling (IS) techniques, combined with a Sample Average Approximation (SAA) approach, can be effectively used in such situations, provided that variance can be reduced uniformly with respect to the decision variables. We give sufficient conditions to obtain such uniform variance reduction, and prove asymptotic convergence of the combined SAAIS approach. As it often happens with IS techniques, the practical performance of the proposed approach relies on exploiting the structure of the problem under study; in our case, we work with a telecommunications problem with Bernoulli input distributions, and show how variance can be reduced uniformly over a suitable approximation of the feasibility set by choosing proper parameters for the IS distributions. Although some of the results are specific to this problem, we are able to draw general insights that can be useful for other classes of problems. We present numerical results to illustrate our findings.

Barrera, J., & Ycart, B. (2014). Bounds for left and right window cutoffs. ALEALatin Am. J. Probab. Math. Stat., 11(2), 445–458.
Abstract: The location and width of the time window in which a sequence of processes converges to equilibrum are given under conditions of exponential convergence. The location depends on the side: the leftwindow and rightwindow cutoffs may have different locations. Bounds on the distance to equilibrium are given for both sides. Examples prove that the bounds are tight.
Keywords: cutoff; exponential ergodicity

Barrera, J., Cancela, H., & Moreno, E. (2015). Topological optimization of reliable networks under dependent failures. Oper. Res. Lett., 43(2), 132–136.
Abstract: We address the design problem of a reliable network. Previous work assumes that link failures are independent. We discuss the impact of dropping this assumption. We show that under a commoncause failure model, dependencies between failures can affect the optimal design. We also provide an integerprogramming formulation to solve this problem. Furthermore, we discuss how the dependence between the links that participate in the solution and those that do not can be handled. Other dependency models are discussed as well. (C) 2014 Elsevier B.V. All rights reserved.

Ruz, G. A., Timmermann, T., Barrera, J., & Goles, E. (2014). Neutral space analysis for a Boolean network model of the fission yeast cell cycle network. Biol. Res., 47, 12 pp.
Abstract: Background: Interactions between genes and their products give rise to complex circuits known as gene regulatory networks (GRN) that enable cells to process information and respond to external stimuli. Several important processes for life, depend of an accurate and contextspecific regulation of gene expression, such as the cell cycle, which can be analyzed through its GRN, where deregulation can lead to cancer in animals or a directed regulation could be applied for biotechnological processes using yeast. An approach to study the robustness of GRN is through the neutral space. In this paper, we explore the neutral space of a Schizosaccharomyces pombe (fission yeast) cell cycle network through an evolution strategy to generate a neutral graph, composed of Boolean regulatory networks that share the same state sequences of the fission yeast cell cycle. Results: Through simulations it was found that in the generated neutral graph, the functional networks that are not in the wildtype connected component have in general a Hamming distance more than 3 with the wildtype, and more than 10 between the other disconnected functional networks. Significant differences were found between the functional networks in the connected component of the wildtype network and the rest of the network, not only at a topological level, but also at the state space level, where significant differences in the distribution of the basin of attraction for the G(1) fixed point was found for deterministic updating schemes. Conclusions: In general, functional networks in the wildtype network connected component, can mutate up to no more than 3 times, then they reach a point of no return where the networks leave the connected component of the wildtype. The proposed method to construct a neutral graph is general and can be used to explore the neutral space of other biologically interesting networks, and also formulate new biological hypotheses studying the functional networks in the wildtype network connected component.

Barrera, J., & Fontbona, J. (2010). The Limiting MoveToFront SearchCost In Law Of Large Numbers Asymptotic Regimes. Ann. Appl. Probab., 20(2), 722–752.
Abstract: We explicitly compute the limiting transient distribution of the searchcost in the movetofront Markov chain when the number of objects tends to infinity, for general families of deterministic or random request rates. Our techniques are based on a “law of large numbers for random partitions,” a scaling limit that allows us to exactly compute limiting expectation of empirical functionals of the request probabilities of objects. In particular, we show that the limiting searchcost can be split at an explicit deterministic threshold into one random variable in equilibrium, and a second one related to the initial ordering of the list. Our results ensure the stability of the limiting searchcost under general perturbations of the request probabilities. We provide the description of the limiting transient behavior in several examples where only the stationary regime is known, and discuss the range of validity of our scaling limit.

Barrera, J., Bertoncini, O., & Fernandez, R. (2009). Abrupt Convergence and Escape Behavior for Birth and Death Chains. J. Stat. Phys., 137(4), 595–623.
Abstract: We link two phenomena concerning the asymptotical behavior of stochastic processes: (i) abrupt convergence or cutoff phenomenon, and (ii) the escape behavior usually associated to exit from metastability. The former is characterized by convergence at asymptotically deterministic times, while the convergence times for the latter are exponentially distributed. We compare and study both phenomena for discretetime birthanddeath chains on a"currency sign with drift towards zero. In particular, this includes energydriven evolutions with energy functions in the form of a single well. Under suitable drift hypotheses, we show that there is both an abrupt convergence towards zero and escape behavior in the other direction. Furthermore, as the evolutions are reversible, the law of the final escape trajectory coincides with the time reverse of the law of cutoff paths. Thus, for evolutions defined by onedimensional energy wells with sufficiently steep walls, cutoff and escape behavior are related by time inversion.
