We develop a continuous-time Markov chain model of a dependability system operating in a randomly changing environment and subject to probabilistic cascading failures. A cascading failure can be thought of as a rooted tree. The root is the component whose failure triggers the cascade, its children are those components that the root's failure immediately caused, the next generation are those components whose failures were immediately caused by the failures of the root's children, and so on. The amount of cascading is unlimited. We consider probabilistic cascading in the sense that the failure of a component of type i causes a component of type j to fail simultaneously with a given probability, with all failures in a cascade being mutually independent. Computing the infinitesimal generator matrix of the Markov chain poses significant challenges because of the exponential growth in the number of trees one needs to consider as the number of components failing in the cascade increases. We provide a recursive algorithm generating all possible trees corresponding to a given transition, along with an experimental study of an implementation of the algorithm on two examples. The numerical results highlight the effects of cascading on the dependability of the models.

There are several available generic sorting algorithms that are highly optimized and are provided as part of standard programming libraries such as the qsort implementation of quicksort available through the C Standard library. In this work we present a practical algorithmic transformation that uses random sampling and ordered search, that can convert an existing implementation of any sorting algorithm X into a sorting algorithm Y of higher observed efficiency that still utilizes X for sorting. The resulting algorithm Y is a randomized algorithm of faster running time with high probability. Experimental results based on an implementation of the transformation support the efficiency claim. The transformation could be used as a wrapper for sorting implementations to increase their efficiency and eliminate spikes of poor performance of such implementations for certain input instances.

We present new BSP algorithms for deterministic sorting and randomized median finding. We sort n general keys by using a partitioning scheme that achieves the requirements of efficiency (one-optimality) and insensitivity against data skew (the accuracy of the splitting keys depends solely on the step distance, which can be adapted to meet the worstcase requirements of our application). Although we employ sampling in order to realize efficiency, we can give a precise worst-case estimation of the maximum imbalance which might occur. We also investigate optimal randomized BSP algorithms for the problem of finding the median of n elements that require, with high-probability, 3n=(2p) + o(n=p) number of comparisons, for a wide range of values of n and p. Experimental results for the two algorithms are also presented.