Víctor Suñé

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.

An ROBDD-Based Combinatorial Method for the Evaluation of Yield of Defect-Tolerant Systems-on-Chip

In this paper, we develop a combinatorial method for the evaluation of the functional yield of defect-tolerant systems-on-chip (SoC). The method assumes that random manufacturing defects are produced according to a model in which defects cause the failure of given components of the system following a distribution common to all defects. The distribution of the number of defects is arbitrary. The yield is obtained by conditioning on the number of defects that result in the failure of some component and performing recursive computations over a reduced ordered binary decision diagram (ROBDD) representation of the fault-tree function of the system. The method has excellent error control. Numerical experiments seem to indicate that the method is efficient and, with some exceptions, allows the analysis with affordable computational resources of systems with very large numbers of components.

A failure-distance based method to bound the reliability of nonrepairable fault-tolerant systems without the knowledge of minimal cuts

CTMC (continuous-time Markov chains) are a commonly used formalism for modeling fault-tolerant systems. One of the major drawbacks of CTMC is the well-known state-space explosion problem. This paper develops and analyzes a method (SC-BM) to compute bounds for the reliability of nonrepairable fault-tolerant systems in which only a portion of the state space of the CTMC is generated. SC-BM uses the failure distance concept as the method described previously by the authors (1997) but, unlike that method, which is based on the computation of exact failure distances, SC-BM uses lower bounds for failure distances, which are computed on the system fault-tree, avoiding the computation and holding of all minimal cuts as required in the earlier work. This is important because computation of all minimal cuts is NP-hard and the number of minimal cuts can be very large. In some cases SC-BM gives exactly the same bounds as the previous method; in other cases it gives less tight bounds. SC-BM computes tight bounds for the reliability of quite complex systems with an affordable number of generated states for short to quite large mission times. The analysis of several examples seems to show that the bounds obtained by SC-BM appreciably outperform those obtained by simpler methods, and, when they are not equal, are only slightly worse than the bounds obtained by the previous method. In addition, the overhead in CPU time due to computing lower bounds for failure distances seems to be reasonable.

A Numerical Method for the Evaluation of the Distribution of Cumulative Reward till Exit of a Subset of Transient States of a Markov Reward Model

Markov reward models have interesting modeling applications, particularly those addressing fault-tolerant hardware/software systems. In this paper, we consider a Markov reward model with a reward structure including only reward rates associated with states, in which both positive and negative reward rates are present and null reward rates are allowed, and develop a numerical method to compute the distribution function of the cumulative reward till exit of a subset of transient states of the model. The method combines a model transformation step with the solution of the transformed model using a randomization construction with two randomization rates. The method introduces a truncation error, but that error is strictly bounded from above by a user-specified error control parameter. Further, the method is numerically stable and takes advantage of the sparsity of the infinitesimal generator of the transformed model. Using a Markov reward model of a fault-tolerant hardware/software system, we illustrate the application of the method and analyze its computational cost. Also, we compare the computational cost of the method with that of the (only) previously available method for the problem. Our numerical experiments seem to indicate that the new method can be efficient and that for medium size and large models can be substantially faster than the previously available method.

Efficient implementations of the randomization method with control of the relative error

Randomization is a well-known numerical method for the transient analysis of continuous-time Markov chains. The main advantages of the method are numerical stability, well-controlled computation error and ability to specify the computation error in advance. Typical implementations of the method control the truncation error in absolute value, which is not completely satisfactory in some cases. Based on a theoretical result regarding the dependence on the parameter of the Poisson distribution of the relative error introduced when a weighted sum of Poisson probabilities is truncated by the right, in this paper we develop efficient and numerically stable implementations of the randomization method for the computation of two measures on rewarded continuous-time Markov chains with control of the relative error. The numerical stability of those implementations is analyzed using a small example. We also discuss the computational efficiency of the implementations with respect to simpler alternatives.

Numerical iterative methods for Markovian dependability and performability models: new results and a comparison

In this paper we deal with iterative numerical methods to solve linear systems arising in continuous-time Markov chain (CTMC) models. We develop an algorithm to dynamically tune the relaxation parameter of the successive over-relaxation method. We give a sufficient condition for the Gauss-Seidel method to converge when computing the steady-state probability vector of a finite irreducible CTMC, an a suffient condition for the Ge neralized Minimal Residual projection method not to converge to the trivial solution 0 when computing that vector. Finally, we compare several splitting-based iterative methods an a variant of the Generalized Minimal Residual projection method.

A method for the computation of reliability bounds for non-repairable fault-tolerant systems

A realistic modeling of fault-tolerant systems requires to take into account phenomena such as the dependence of component failure rates and coverage parameters on the operational configuration of the system, which cannot be properly captured using combinatoric techniques. Such dependencies can be modeled with detail using continuous-time Markov chains (CTMCs). However, the use of CTMC models is limited by the well-known state space exploitation problem. We develop a method for the computation of bounds for the reliability of non-repairable fault-tolerant systems which requires the generation of only a subset of states. The tightness of the bounds increases as more detailed states are generated. The method uses the failure distance concept and is illustrated using an example of a quite complex fault-tolerant system whose failure behavior has the above mentioned types of dependencies.

Combinatorial methods for the evaluation of yield and operational reliability of fault-tolerant systems-on-chip

Abstract In this paper we develop combinatorial methods for the evaluation of yield and operational reliability of fault-tolerant systems-on-chip. The methods assume that defects are produced according to a model in which defects are lethal and affect given components of the system following a distribution common to all defects; the method for the evaluation of operational reliability also assumes that the fault-tree function of the system is increasing. The distribution of the number of defects is arbitrary. The methods are based on the formulation of, respectively, the yield loss and the operational unreliability as the probability that a given Boolean function with multiple-valued variables has value 1. That probability is computed by analyzing a ROMDD (reduced ordered multiple-value decision diagram) representation of the function. For efficiency reasons, a coded ROBDD (reduced ordered binary decision diagram) representation of the function is built first and, then, that coded ROBDD is transformed into the ROMDD required by the methods. We present numerical experiments showing that the methods are able to cope with quite large systems in moderate CPU times.

Implicit ODE solvers with good local error control for the transient analysis of Markov models

Obtaining the transient probability distribution vector of a continuous-time Markov chain (CTMC) using an implicit ordinary differential equation (ODE) solver tends to be advantageous in terms of run-time computational cost when the product of the maximum output rate of the CTMC and the largest time of interest is large. In this paper, we show that when applied to the transient analysis of CTMCs, many implicit ODE solvers are such that the linear systems involved in their steps can be solved by using iterative methods with strict control of the 1-norm of the error. This allows the development of implementations of those ODE solvers for the transient analysis of CTMCs that can be more efficient and more accurate than more standard implementations.

A combinatorial method for the evaluation of yield of fault-tolerant systems-on-chip

In this paper we develop a combinatorial method for the evaluation of yield of fault-tolerant systems-on-chip. The method assumes that defects are produced according to a model in which defects are lethal and affect given components of the system following a distribution common to all defects. The distribution of the number of defects is arbitrary. The method is based on the formulation of the yield as 1 minus the probability that a given boolean function with multiple-valued variables has value 1. That probability is computed by analyzing a ROMDD (reduced ordered multiple-value decision diagram) representation of the function. For efficiency reasons, we first build a coded ROBDD (reduced ordered binary decision diagram) representation of the function and then transform that coded ROBDD into the ROMDD required by the method. We present numerical experiments showing that the method is able to cope with quite large systems in moderate CPU times.