a b s t r a c tGompertz curve has been used to estimate the number of residual faults in testing phases of software development, especially by Japanese software development companies. Since the Gompertz curve is a deterministic function, the curve cannot be applied to estimating software reliability which is the probability that software system does not fail in a prefixed time period. In this article, we propose a stochastic model called the Gompertz software reliability model based on non-homogeneous Poisson processes. The proposed model can be derived from the statistical theory of extreme-value, and has a similar asymptotic property to the deterministic Gompertz curve. Also, we develop an EM algorithm to determine the model parameters effectively. In numerical examples with software failure data observed in real software development projects, we evaluate performance of the Gompertz software reliability model in terms of reliability assessment and failure prediction.

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Software rejuvenation is a preventive maintenance tech-nique that has been extensively studied in the recent literature. In this paper; we extend the classical result by Huang, Kintala, Kolettis and Fulton (1995), and in addition propose a mod$ed stochastic model to generate the sojhare rejuvenation schedule. More precisely, the software rejuvenation models are formulated via the semi-Markov process, and the optimal software rejuvenation schedule which minimizes the expected costs per unit time in the steadystate are derived analytically for respective cases. Furthel; we develop non-parametric algorithms to estimate the optimal software rejuvenation schedules, provided that the statistical complete (unsensored) sample data of failure time is given. In numerical examples, we compare two models in terms of economic justification, and examine asymptotic properties f o r the statistical estimation algorithms.