Michael G. Thomason

A Markov chain model for statistical software testing

Statistical testing of software establishes a basis for statistical inference about a software systems expected field quality. This paper describes a method for statistical testing based on a Markov chain model of software usage. The significance of the Markov chain is twofold. First, it allows test input sequences to be generated from multiple probability distributions, making it more general than many existing techniques. Analytical results associated with Markov chains facilitate informative analysis of the sequences before they are generated, indicating how the test is likely to unfold. Second, the test input sequences generated from the chain and applied to the software are themselves a stochastic model and are used to create a second Markov chain to encapsulate the history of the test, including any observed failure information. The influence of the failures is assessed through analytical computations on this chain. We also derive a stopping criterion for the testing process based on a comparison of the sequence generating properties of the two chains. >

Convergence of Powers of a Fuzzy Matrix

Abstract A Boolean matrix is a matrix with elements having values of either 1 or 0; a fuzzy matrix is a matrix with elements having values in the closed interval [0, 1]. Fuzzy matrices occur in the modeling of various fuzzy systems, with products usually determined by the “max(min)” rule arising from fuzzy set theory. In this paper, some sufficient conditions for convergence under “max(min)” products of the powers of a square fuzzy matrix and of a fuzzy state process are established.

A practical analysis of low-density parity-check erasure codes for wide-area storage applications

As peer-to-peer and widely distributed storage systems proliferate, the need to perform efficient erasure coding, instead of replication, is crucial to performance and efficiency. Low-density parity-check (LDPC) codes have arisen as alternatives to standard erasure codes, such as Reed-Solomon codes, trading off vastly improved decoding performance for inefficiencies in the amount of data that must be acquired to perform decoding. The scores of papers written on LDPC codes typically analyze their collective and asymptotic behavior. Unfortunately, their practical application requires the generation and analysis of individual codes for finite systems. This paper attempts to illuminate the practical considerations of LDPC codes for peer-to-peer and distributed storage systems. The three main types of LDPC codes are detailed, and a huge variety of codes are generated, then analyzed using simulation. This analysis focuses on the performance of individual codes for finite systems, and addresses several important heretofore unanswered questions about employing LDPC codes in real-world systems.

Processor Allocation and Checkpoint Interval Selection in Cluster Computing Systems

Performance prediction of checkpointing systems in the presence of failures is a well-studied research area. While the literature abounds with performance models of checkpointing systems, none addresses the issue of selecting runtime parameters other than the optimal checkpointing interval. In particular, the issue of processor allocation is typically ignored. In this paper, we present a performance model for long-running parallel computations that execute with checkpointing enabled. We then discuss how it is relevant to todays parallel computing environments and software, and present case studies of using the model to select runtime parameters.

Small parity-check erasure codes - exploration and observations

Erasure codes have profound uses in wide- and medium-area storage applications. While infinite-size codes have been developed with optimal properties, there remains a need to develop small codes with optimal properties. In this paper, we provide a framework for exploring very small codes, and we use this framework to derive optimal and near-optimal ones for discrete numbers of data bits and coding bits. These codes have heretofore been unknown and unpublished, and should be useful in practice. We also use our exploration to make observations about upper bounds for these codes, in order to gain a better understanding of them and to spur future derivations of larger, optimal and near-optimal codes.

Dynamic programming alignment of sequences representing cyclic patterns

String alignment by dynamic programming is generalized to include cyclic shift and corresponding optimal alignment cost for strings representing cyclic patterns. A guided search algorithm uses bounds on alignment costs to find all optimal cyclic shifts. The bounds are derived from submatrices of an initial dynamic programming matrix. Algorithmic complexity is analyzed for major stages in the search. The applicability of the method is illustrated with satellite DNA sequences and circularly permuted protein sequences. >

Syntactic Recognition of Imperfectly Specified Patterns

The methods developed in this correspondence represent an approach to the problem of handling error-corrupted syntactic pattern strings, an area generally neglected in the numerous techniques for linguistic pattern description and recognition which have been reported. The basic approach consists of applying error transformations to the productions of context-free grammars in order to generate new grammars (also context-free) capable of describing not only the original error-free patterns, but also patterns containing specific types of errors such as deleted, added, and interchanged symbols which arise often in the pattern-scanning process. Theoretical developments are illustrated in the framework of a syntactic recognition system for chromosome structures.

Dynamic Programming Inference of Markov Networks from Finite Sets of Sample Strings

Inference of Markov networks from finite sets of sample strings is formulated using dynamic programming. Strings are installed in a network sequentially via optimal string-to-network alignments computed with a dynamic programming matrix, the cost function of which uses relative frequency estimates of transition probabilities to emphasize landmark substrings common to the sample set. Properties of an inferred network are described and the method is illustrated with artificial data and with data representing banded human chromosomes.

The Activity of a Variable and Its Relation to Decision Trees

The construction of sequential testing procedures from functions of discrete arguments is a common problem in switching theory, software engineering, pattern recognition, and management. The concept of the activity of an argument is introduced, and a theorem is proved which relates it to the expected testing cost of the most general type of decision trees. This result is then extended to trees constructed from relations on finite sets and to decision procedures with cycles. These results are used, in turn, as the basis for a fast heuristic selection rule for constructing testing procedures. Finally, some bounds on the performance of the selection rule are developed.

A Markov chain model for predicting the reliability of multi-build software

Abstract In previous work we developed a method to model software testing data, including both failure events and correct behavior, as a finite-state, discrete-parameter, recurrent Markov chain. We then showed how direct computation on the Markov chain could yield various reliability related test measures. Use of the Markov chain allows us to avoid common assumptions about failure rate distributions and allows both the operational profile and test coverage of behavior to be explicitly and automatically incorporated into reliability computation. Current practice in Markov chain based testing and reliability analysis uses only the testing (and failure) activity on the most recent software build to estimate reliability. In this paper we extend the model to allow use of testing data on prior builds to cover the real-world scenario in which the release build is constructed only after a succession of repairs to buggy pre-release builds. Our goal is to enable reliability prediction for future builds using any or all testing data for prior builds. The technique we present uses multiple linear regression and exponential smoothing to merge multi-build test data (modeled as separate Markov chains) into a single Markov chain which acts as a predictor of the next build of testing activity. At the end of the testing cycle, the predicted Markov chain represents field use. It is from this chain that reliability predictions are made.