Juan A. Carrasco

An algorithm to find minimal cuts of coherent fault-trees with event-classes, using a decision tree

A new algorithm, the Carrasco-Sune minimal cuts (CS-MC) algorithm for computing the minimal cuts of s-coherent fault trees is presented. Input events of the fault tree are assumed classified into classes, where events of the same class are indistinguishable. This allows capturing some symmetries which some systems exhibit. CS-MC uses a decision tree. The search implemented by the decision tree is guided by heuristics which try to make the CS-MC algorithm as efficient as possible. In addition, an irrelevance test on the inputs of the fault tree is used to prune the search. The performance of CS-MC is illustrated and compared with the basic top-down and bottom-up algorithms using a set of fault trees, some of which are very difficult. The CS-MC performs very well even in the difficult examples, and the memory requirements of CS-MC are small.

Transient analysis of some rewarded Markov models using randomization with quasistationarity detection

Rewarded homogeneous continuous-time Markov chain (CTMC) models can be used to analyze performance, dependability and performability attributes of computer and telecommunication systems. In this paper, we consider rewarded CTMC models with a reward structure including reward rates associated with states and two measures summarizing the behavior in time of the resulting reward rate random variable: the expected transient reward rate at time t and the expected averaged reward rate in the time interval [0, t]. Computation of those measures can be performed using the randomization method, which is numerically stable and has good error control. However, for large stiff models, the method is very expensive. Exploiting the existence of a quasistationary distribution in the subset of transient states of discrete-time Markov chains with a certain structure, we develop a new variant of the (standard) randomization method, randomization with quasistationarity detection, covering finite CTMC models with state space S/spl cup/{f/sub 1/,f/sub 2/,..., f/sub A/}, A/spl ges/1, where all states in S are transient and reachable among them and the states f/sub i/ are absorbing. The method has the same good properties as the standard randomization method and can be much more efficient. We also compare the performance of the method with that of regenerative randomization.

Efficient transient simulation of failure/repair Markovian models

Simulation methods for the solution of the extremely large Markovian dependability models which result from complex fault-tolerant computer systems have recently been developed. The author presents efficient simulation methods for the estimation of transient reliability/availability metrics for repairable fault-tolerant computer systems which combine estimator decomposition techniques with an efficient importance sampling technique. Comparison with simulation methods previously proposed for the same type of metrics and models shows that the proposed methods are orders of magnitude faster.<<ETX>>

A parameterized multi-step Newton method for solving systems of nonlinear equations

We construct a novel multi-step iterative method for solving systems of nonlinear equations by introducing a parameter θ to generalize the multi-step Newton method while keeping its order of convergence and computational cost. By an appropriate selection of θ, the new method can both have faster convergence and have larger radius of convergence. The new iterative method only requires one Jacobian inversion per iteration, and therefore, can be efficiently implemented using Krylov subspace methods. The new method can be used to solve nonlinear systems of partial differential equations, such as complex generalized Zakharov systems of partial differential equations, by transforming them into systems of nonlinear equations by discretizing approaches in both spatial and temporal independent variables such as, for instance, the Chebyshev pseudo-spectral discretizing method. Quite extensive tests show that the new method can have significantly faster convergence and significantly larger radius of convergence than the multi-step Newton method.

Higher order multi-step Jarratt-like method for solving systems of nonlinear equations

This paper proposes a multi-step iterative method for solving systems of nonlinear equations with a local convergence order of 3m4, where m(2) is the number of steps. The multi-step iterative method includes two parts: the base method and the multi-step part. The base method involves two function evaluations, two Jacobian evaluations, one LU decomposition of a Jacobian, and two matrixvector multiplications. Every stage of the multi-step part involves the solution of two triangular linear systems and one matrixvector multiplication. The computational efficiency of the new method is better than those of previously proposed methods. The method is applied to several nonlinear problems resulting from discretizing nonlinear ordinary differential equations and nonlinear partial differential equations.

Bounding steady-state availability models with group repair and phase type repair distributions

We propose an algorithm to obtain bounds for the steady-state availability using Markov models in which only a small portion of the state space is generated. The algorithm is applicable to models with group repair and phase type repair distributions and involves the solution of only four linear systems of the size of the generated state space, independently on the number of “return” states. Numerical examples are presented to illustrate the algorithm and compare it with a previous bounding algorithm.

Transient analysis of large Markov models with absorbing states using regenerative randomization

ABSTRACT In this article, we develop a new method, called regenerative randomization, for the transient analysis of continuous time Markov models with absorbing states. The method has the same good properties as standard randomization: numerical stability, well-controlled computation error, and ability to specify the computation error in advance. The method has a benign behavior for large t and is significantly less costly than standard randomization for large enough models and large enough t. For a class of models, class C, including typical failure/repair reliability models with exponential failure and repair time distributions and repair in every state with failed components, stronger theoretical results are available assessing the efficiency of the method in terms of “visible” model characteristics. A large example belonging to that class is used to illustrate the performance of the method and to show that it can indeed be much faster than standard randomization.

Markovian Dependability/Performability Modeling of Fault-tolerant Systems

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Efficient exploration of availability models guided by failure distances

Recently, a method to bound the steady-state availability using the failure distance concept has been proposed. In this paper we refine that method by introducing state space exploration techniques. In the methods proposed here, the state space is incrementally generated based on the contributions to the steady-state availability band of the states in the frontier of the currently generated state space. Several state space exploration algorithms are evaluated in terms of bounds quality and memory and CPU time requirements. The more efficient seems to be a waved algorithm which expands transition groups. We compare our new methods with the method based on the failure distance concept without state exploration and a method proposed by Souza e Silva and Ochoa which uses state space exploration but does not use the failure distance concept. Using typical examples we show that the methods proposed here can be significantly more efficient than any of the previous methods.

Failure Transition Distance-Based Importance Sampling Schemes for theSimulation of Repairable Fault-Tolerant Computer Systems

Markov models are often used to evaluate dependability attributes of fault-tolerant computer systems. The use in practice of Markov models is, however, hampered by the well-known state space explosion problem. Simulation alleviates the problem. For Markov models of repairable fault-tolerant systems, standard simulation of dependability measures tends to be expensive due to the rarity of the system failure event. Importance sampling can speed up the simulation. This paper develops two importance sampling schemes, called failure transition distance biasing & balanced failure transition distance biasing, which exploit the failure transition distance concept in an attempt to improve the efficiency of two other schemes, failure biasing & and balanced failure biasing. The schemes require the computation of the so-called failure transition distances, and procedures to perform those computations are developed. The presentation is tied to a previously proposed measure-specific simulation method for the steady-state unavailability. An optimization method of the parameters of the importance sampling schemes is also developed. For the simulation of the steady-state unavailability, failure transition distance biasing has (as failure biasing) the bounded relative error property for balanced fault-tolerant systems & balanced failure transition distance biasing has (as balanced failure biasing) the bounded relative error property for both balanced & unbalanced fault-tolerant systems. It is proved that, for balanced fault-tolerant systems, both failure transition distance biasing & balanced failure transition distance biasing can indeed improve the efficiency of failure biasing & balanced failure biasing. In addition, numerical experiments seem to indicate that, for unbalanced fault-tolerant systems, balanced failure transition distance biasing can also improve the efficiency of balanced failure biasing. The application of the failure transition distance-based importance sampling schemes is, however, limited to systems not having too many minimal failure covers, or, at least, not having too many minimal failure covers of small cardinality. A minimal failure cover is a minimal bag of failure bags such that the failure of its components implies the failure of the system; a failure bag is any non-empty bag of component classes which can fail simultaneously.