K e y Words -Fault tree, Minimal cuts, Decision tree. Summary & Conclusions -A new algorithm (CS-MC)for computing the minimal cuts of scoherent 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 CS-MC 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 topdown 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. 1NTR.ODUCTION Acronym'ATPG automatic test pattern generation (automatic generation of input vectors for the test of digital circuits) (the new algorithm in this paper)CS-MC Carrasco-Sufi6 minimal-cuts algorithm DT decision tree BDD binary decision diagram Fault trees are a very popular tool in reliability engineering. The knowledge of the minimal cuts2 allows the designer to analyze the criticality of the basic events and to improve the reliability of the modeled system. It is well-known that computation of all minimal cuts of an arbitrary fault tree is NP-hard [all. In spite of this theoretical difficulty, there exist algorithms which perform reasonably well in many practical cases. Older algorithms can be classified in 2 categories: top-down and bottom-up. All minimal cuts of a 'fan-out free' fault tree (a fault tree without repeated basic events or gates branching out to 'The singular & plural of an acronym are always spelled the same. 2The term 'minimal cut' is used instead of the more common 'minimal cutset' because the minimal cuts in this paper are bags.de Catalunya, Barcelona more than one gate input) can be computed very easily by traversing the fault tree in a top-down fashion. In the basic top-down algorithm [ll], a set of cuts (often called the superset) is obtained as if the fault tree were fan-out free. The set of minimal cuts is then obtained by using in each cut the reduction rule zz 4 z and keeping those cuts which are not properly contained in any other cut. The algorithm involves N . ( N -1) inclusion tests, where N is the cardinality of the superset. These inclusion tests can be performed very efficiently by assigning different prime numbers to the basic events, and representing the cuts by the product of the constituent basic events [23]. However, even using these techniques, reduction of the superset is expensive if N is large. Also, some fault trees with a manageable number of minimal cuts have N so large that it is impossible to keep in memory the superset3.Some improvements to the basic top-down algorithm have been proposed. In [2] the size of the superset and the number of required inclusion tests is reduced by eliminating repeated e...

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 e cient 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 e ciency of the implementations with respect to simpler alternatives. ?

In this paper we develop combinatorial methods for the evaluation of yield and operational reliability of fault-tolerant systems-on-chip. The method for yield computation 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 method for the computation 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 and the operational reliability 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. * This paper is an extended version of D.

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.

Abstract-In this paper, we develop a combinatorial method for the evaluation of the functional yield of defect-tolerant systems-onchip (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.Index Terms-Combinatorial method, defect-tolerant systems-on-chip (SoC), manufacturing defects, reduced ordered binary decision diagram (ROBDD), yield.