Software availability is defined as the attribute that the software intensive system is operable and available whenever we want to use it and one of the user-oriented quality characteristics of software systems. This paper surveys the current research in software availability assessment techniques. At first we discuss the basic ideas of software availability modeling and the difference between modeling for software and hardware systems. Markov processes are applied to the descriptions of the time-dependent behaviors of software systems alternating between up and down states; then the processes of software reliability growth and the increase in difficulty of restorations are incorporated as well. We also refer to several extended models reflecting the software failure-occurrence phenomenon and the restoration scenario peculiar to the user-operational phase, considering computation performance, and combining a hardware and a software subsystem. Furthermore, numerical illustrations for software availability measurement with respective models are presented.

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The problem of evaluating the goodness of statistical models is fundamental and of importance in various fields of statistics, natural sciences, neural networks, engineering, economics, etc. Akaike's . information criterion, known as AIC, provides a useful tool for constructing statistical models, and a number of successful applications of AIC in statistical data analysis have been reported. AIC is a criterion for evaluating the models estimated by the maximum likelihood method. With the development of various non-linear modeling techniques, the construction of criteria which enable us to evaluate various types of statistical models has been required. The aim of this paper is to give a systematic account of some recent developments in model evaluation criteria from information-theoretic and Bayesian points of views. We intend to provide a basic expository account of the fundamental principles behind information criteria. We also discuss the application of the bootstrap methods in model evaluation problems.

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We review recently developed numerical algorithm for finite-temperature density-matrix renormalization groups (DMRG). Physical quantities of one-dimensional quantum systems are efficiently calculated by this method. Mathematically, an extreme eigenvalue and its eigenvector can be obtained for non-Hermitian matrices of mammoth size (typically up to 2^<100>) , by reducing to a tractable size: generalization of the original DMRG algorithm proposed for Hermitian matrices. Special focus is on the tensorproduct structure of the vector space and the matrices, which reflects the physics of the original problem, and how and why this method works efficiently is briefly discussed.

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An efficient simulation method is developed for time-dependent system reliability analyses, where extremely small probability of system failure is desired. The method is based upon an importance sampling technique, in which the Girsanov theorem is applied to determine an importance sampling measure on a function space. First, expressing the probability of system failure as a functional integral on the function space, we mathematically formulate an importance sampling simulation scheme, where Newton's method to determine the sampling measure is used. Next, a new method to select the optimally sampling measure proposed by the author is briefly surveyed. Finally, two simple numerical examples are shown to examine the computational efficiency of the proposed method, which indicate that the proposed method can give quite accurate estimations for the probability of system failure even if the number of generated samples are small.

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