Existing software reliability models (SRMs) that describe software fault detection during the testing phase can be unified under the framework of self-exciting point processes. However, it remains unclear whether such generalized self-excitation models can outperform traditional non-homogeneous Poisson process (NHPP) and non-homogeneous Markov process (NHMP)-based SRMs in terms of goodness-of-fit and predictive capabilities. In this paper, we propose a family of Hawkes process (HKP)-based SRMs that incorporate time-varying base intensity functions, distinguishing them from conventional HKPs with constant base intensity. Specifically, we introduce a Pareto-type base intensity and explore ten different impact (kernel) functions in the stochastic intensity part, and then compare the performance of the proposed HKP-based models with that of representative NHPP and NHMP-based SRMs. Experimental results on eight software development project datasets demonstrate that the HKP-based models with self-excitation generally achieve superior goodness-of-fit and predictive performance.

In this paper, we consider three point-prediction methods for the number of software bugs detetced in future with the bug count data experienced in past, where the underlying software bug-detection process is described by a non-homogeneous Poisson process. In general, it is known that the past probability distribution of number of software bugs is not always identical to the future one. Nevertheless, the commonly used technique is to predict the bug counts under a strong assumption that the probability distribution with model parameters estimated from the past observation holds even in the future evolution. Since such a plug-in prediction does not often work well to guarantee the higher prediction accuracy, investigating more accurate prediction of software bug counts is an emerging issue in software reliability engineering. We propose the so-called maximum likelihood predictors to predict the future bug-detection processes and a different model selection scheme from the common information criteria. Through a numerical example with an actual software bug count data set, we compare our new prediction methods with the existing plug-in predictor.

Over the past few decades, a huge number of software reliability models (SRMs) have been proposed in the literature. Then, it is common to select an appropriate SRM with highest goodness-of-fit to the underlying software fault-count data based on any information criteria such as Akaike Information Criterion (AIC). However, it has been known that the best SRM with the minimum AIC is not always equivalent to the best prediction model for the future software fault-count process. In this paper, we propose refined software reliability prediction methods with a bagging-based ranking and model averaging technique. In numerical experiments with actual software development project data, it is shown that our bagging-based software reliability prediction models enabled to improve the predictive performances in the early and middle software testing phases, comparing with the single use of the best SRM with the minimum AIC.

Although a huge number of software reliability models (SRMs) have been proposed in the past literature, there is no unique SRM with satisfactory prediction accuracy, because it has been known that the best goodness-of-fit SRM to the underlying data is not always equivalent to the best prediction model for the future software fault-count process. It is common to select an appropriate SRM with highest goodness-of-fit to the underlying software fault-count data based on any information criteria such as Akaike information criterion (AIC). In this paper we focus on a prediction SRM consisting with linearly weighted combinational (LWC) non-homogeneous Poisson process (NHPP)-based SRMs, and overview an AdaBoosting-based approach by Li et al. (2012) to determine the optimal weights for the LWC NHPP-based SRMs. Next, we propose a refined AdaBoosting technique to keep the steps of original AdaBoosting, in order to present the original predictive performance of AdaBoosting in software reliability.

Abstract Boundary value analysis (BVA) is a common technique in software testing that uses input values that lie at the boundaries where significant changes in behavior are expected. This approach is widely recognized and used as a natural and effective strategy for testing software. Test coverage is one of the criteria to measure how much the software execution paths are covered by the set of test cases. This paper focuses on evaluating test coverage with respect to BVA by defining a metric called boundary coverage distance (BCD). The BCD metric measures the extent to which a test set covers the boundaries. In addition, based on BCD, we consider the optimal test input generation to minimize BCD under the random testing scheme. We propose three algorithms, each representing a different test input generation strategy, and evaluate their fault detection capabilities through experimental validation. The results indicate that the BCD-based approach has the potential to generate boundary values and improve the effectiveness of software testing.

This article analyzes a network of servers that cover mobile phone calls in a given convex service area. All servers work with the same reliability, and can cover all call requests within a circle of the same given radius. When working servers cover the whole area, the network system is working. System reliability is the probability that working servers can cover the whole area. Without loss of generality, we assume that each server's location, coverage radius, and reliability are known. Monte Carlo and Voronoi diagram methods are used to check the system state, and a binary search method is proposed to obtain the system reliability. Also, a simulation method is used to evaluate system reliability in special cases. Finally, numerical examples are studied to investigate the effects of the model parameters on system reliability.

In this paper, we discuss the hierarchical modeling in model-based dependability evaluation. The hierarchical modeling is a modeling approach that combines non-state-space models such as fault trees and state-space models such as Markov chains. It can mitigate the state explosion problem for state-space models. The fundamental idea is that state-space models are used to represent the dynamic behavior for basic events of non-state-space model. This enables us to represent the dynamic behavior of the system. On the computation of dependability indices such as system reliability, we utilize the non-state-space model. This structure reduces the number of states to be evaluated. The existing hierarchical approaches deal with binary-type reliability models, focusing only on the working and failure of the system/components. In this paper, we extend the existing hierarchical approach to handle the computation of dependability indices using the multi-state system instead of a fault tree, thereby accommodating both reliability and certain performance models, rather than being limited to binary-type models. Additionally, we introduce the computation method with MtMDD (multi-terminal multi-valued decision diagram). Numerical results demonstrated that the proposed hierarchical modeling can efficiently describe multi-state systems and effectively obtain desired dependability indices. It outperforms existing approaches such as Kronecker representation.

In this article, we focus on nonhomogeneous Markov processes (NHMPs), which are generalizations of the well-known homogeneous Markov processes (HMPs) and nonhomogeneous Poisson processes, and compare two software reliability models (SRMs) which can be classified into a generalized binomial process (GBP) and a generalized Polya process (GPP). GBP and GPP are also characterized, respectively, as a Markov inverse death process and a Markov birth process, with state- and time-dependent transition rates. We develop a unified software reliability modeling framework based on the NHMPs and apply it to the software reliability prediction. Through numerical examples with the fault count data observed in actual closed-source software (CSS) and open-source software (OSS) development projects, we compare two SRMs (GBP and GPP) in terms of the goodness-of-fit and predictive performances, in addition to the quantitative software reliability assessment. We also consider software release problems with these generalized SRMs, and investigate the impact on the software release decision.

The test case generation technique from formal specifications called the Vibration Testing Method has been put forward. This technique is aimed at gaining coverage of program paths and detecting bugs, even though the test cases are generated only based on specifications. Since it lacks a supporting tool currently, its application is inefficient and errorprone. In this paper, we tackle this problem by describing a supporting tool for the method that we have developed over the last two years. The tool does not only automatically generate test cases based on the Vibration Method, but also can automatically analyze test results. Further, it can also automatically “prove” theorems to support practical formal verification of program properties. During the development of the tool, we have made some important improvements to the techniques of the method for automatic test case generation. We have conducted a small experiment to evaluate our tool and the improved Vibration Method. The experiment result shows that a 12% improvement on the previous method is made.

Wavelet shrinkage estimation is a non-parametric technique to estimate the non-homogeneous Poisson process (NHPP)-based software reliability growth model (SRGM), and provides better goodness-of-fit performances than the common parametric approach by means of the maximum likelihood estimation. However, it has a serious drawback that not only the long-term prediction but also the quantification of software reliability for an arbitrary testing/operational period were difficult. In this paper we propose a long-term prediction approach for the NHPP-based SRGM with the wavelet shrinkage estimation, where several denoising and data transform techniques are applied to estimate the NHPP discrete intensity function. We develop a wavelet-based software reliability assessment tool; WSRAT, to enable a lightweight and automatic software reliability prediction, in addition to implement the existing denoising and data transform techniques. W-SRAT is a unique software reliability assessment tool based on the wavelet shrinkage estimation, and is a web-based freeware to predict the cumulative number of software faults and the quantitative software reliability. We demonstrate how to use the W-SRAT in actual software reliability management.

Intrusion tolerance is an ability to keep the correct service by masking the intrusion based on fault-tolerant techniques. With the rapid development of virtualization, the virtual machine (VM)-based intrusion tolerance scheme has been developed according to the concept of state machine replication with Byzantine fault tolerant technique. In this article, we present the quantitative security evaluation of the VM-based intrusion tolerant system with the time to security failure. We assume that the arrival stream follows a Markovian arrival process (MAP), which is one of the most general stochastic processes, and analytically derive the Laplace-Stieltjes transform of time to security failure based on the analysis of the MAP/G/1/∞ queue.

This paper presents the computation method for the dependability measure in fault trees when the uncertainty of model parameters is considered. The propagation of uncertainty of model parameters can be estimated by regarding the model parameters as random variables. However, the computation cost of expected values is now so low in such situation. In this paper, we focus on the moment-based approximation method for uncertainty propagation in fault trees. In particular, to obtain the moment-based approximation, we discuss the computation of first and second derivatives of dependability measures in fault trees with BDD (binary decision diagram) representation.

Software reliability models (SRMs) are the stochastic processes of the number of faults detected in the development phase, and are utilized to estimate the quantitative reliability measures of software. In the reliability evaluation with SRM, after estimating model parameters from the observed the number of detected faults, we should test the estimated SRM is fitted to the observed data statistically, i.e., we perform the goodness-of-fit test for the estimated SRM. In the past literature, Kolmogorov-Smirnov (KS) test has been used as the goodness-of-fit test for SRM. In this paper, we revisit the KS test for SRM in the case where the model parameters of SRM are estimated from grouped data of the number of detected faults.

One of open-source software (OSS) is that it can be used for a long term by repeating version-up iteratively in the operational phase, so that OSS possesses a different software bug detection and correction profiles from the closed source software (CSS) products. More specifically, the software bug fixing process of OSS can be considered to show effects of the long-term operation and/or periodicity due to the multiple version-up procedures, in addition to the common reliability growth phenomenon observed in the relatively short-term software testing. In this article we propose a stochastic point process approach to represent the long-term effect and the periodicity effect of OSS with the actual OSS bug fixing data. By conducting the reliability analysis of OSS, it is possible to assess the operational reliability of OSS quantitatively and to share the published quality indicators of OSS by the whole OSS community.

In this paper we consider a statistical method to prioritize software test cases with operational profile, where the system behavior is described by a Markov reward model. Especially, we introduce software code metrics as reward parameters and apply the resulting Markov reward model to the test case prioritization problem, where our research question is set as how software code metrics affect the test case prioritization. In a numerical example with a real application software, we embed some seeding faults in advance and carry out 1,000 random test experiments. It is shown that our metrics-based test case prioritization can reduce a large amount of testing effort efficiently.

In this paper, we consider a stochastic model to evaluate the system availability of an intrusion tolerant system (ITS), where the system undergoes the patch management with a periodic vulnerability checking strategy, i.e., a pull-type patch management. Based on the model, this paper discusses the appropriate timing for patch applying. In particular, the paper models the attack behavior of adversary and the system behaviors under reactive defense strategies by a composite stochastic reward net (SRN). Furthermore, we formulate the interval availability by applying the phase-type (PH) approximation to solve the Markov regenerative process (MRGP) models derived from the SRNs. Numerical experiments are conducted to study the sensitivity of the system availability with respect to the number of checking.

While the number of test runs (test cases) is often used to define the time scale to measure quantitative software reliability, the common calendar-time modeling with non-homogeneous Poisson processes (NHPPs) is approximately applied to describe the time scale and the software fault-count phenomena as well. In this paper we give a conjecture that such an approximate treatment is not theoretically justified, and propose a simple test-run reliability modeling framework based on non-homogeneous binomial processes (NHBPs). We show that the Poisson-binomial distribution plays a central role in the software test-run reliability modeling, and apply it to the software release decision. In numerical experiments with seven software fault count data we compare the NHBP based software reliability models (SRMs) with their corresponding NHPP based SRMs and refer to an applicability of NHBP based software test-run reliability modeling.

Marshall-Olkin type distribution is defined as a maximum or minimum value distribution of N i.i.d. (secondary) random variables, where N is a geometric distributed random variable. Gopal, and Damondaran (2011) and Xiao (2015) considered the special cases where the underlying random variables are exponentially and Weibull distributed, respectively, and investigated an applicability of these specific Marshall-Olkin type distributions to software reliability modeling with non-homogeneous Poisson process. In this paper, we further generalize the existing software reliability growth model (SRGM) to more general ones by assuming that the secondary probability distribution is given by one of eleven representative distributions with positive support. For these new SRGMs, we develop efficient parameter estimation algorithms based on the EM (Expectation-Maximization) principle, and provide two stable statistical inference schemes in the respective cases where the software fault-detection time and its grouped data are available, respectively. In numerical examples with sixteen data sets (eight for each of detection time data or grouped data), the resulting Marshall-Olkin type SRGMs are compared with the existing eleven SRGMs in terms of goodness-of-fit performance, and reliability prediction.

Traditional software reliability analysis utilizes only the fault count data observed in testing phase, and is done independently of the source code itself. Recently, it is known that utilization of software metrics in software reliability modeling and analysis can lead to more accurate reliability estimation and fault prediction through many empirical studies. However, such a metrics-based modeling also requires a careful selection of software metrics and their measurement, which are often troublesome and cost-consuming in practice. In this paper, we propose a kernel-based approach to estimate the quantitative software reliability, where two cases are considered; multiple software metrics are used and not. In the former case, we combine the kernel regression with the well-known non-homogeneous Poisson process-based software reliability growth model (SRGM), and propose a new metrics-based SRGM. In the latter case, we perform a similarity-based analysis through a source code transformation algorithm and try to estimate the quantitative software reliability from the source code directly without measuring multiple software metrics. Numerical examples with real application programs are presented to validate our kernel-based approach in the above two cases.

This paper presents a generalized modeling framework of fault detection and correction processes with bivariate distributions. The presented framework includes almost all existing software reliability growth models, namely the models in which both fault detection and correction processes are described by non-homogeneous Poisson processes. In our framework, the time dependency of fault correction time corresponds to the correlation between fault detection and correction times. Moreover, we propose a new fault detection and correction process model with hyper-Erlang distributions, and develop the model parameter estimation algorithm via EM (expectation-maximization) algorithm. In numerical examples, we demonstrate the data fitting ability of hyper-Erlang model with actual fault detection and correction data of open source projects.

This paper considers the statistical approach to model software degradation process from time series data of system attributes. We first develop the continuous-time Markov chain (CTMC) model to represent the degradation level of system. By combining the CTMC with system attributes distributions, a continuous-time hidden Markov model (CT-HMM) is proposed as the basic model to represent the degradation level of system. To estimate model parameters, we develop the EM algorithm for CT-HMM. The advantage of this modeling is that the estimated model is directly applied to existing CTMC-based software aging and rejuvenation models. In numerical experiments, we exhibit the performance of our method by simulated data and also demonstrate estimating the software degradation process with experimental data in MySQL database system.

Software rejuvenation is one of the proactive fault management techniques to prevent system performance degradation, which may lead to the system failure caused by software aging. In the design of software rejuvenation, it is important to determine the optimal timing of triggering the rejuvenation in terms of the system overhead. In this paper, we consider six software rejuvenation policies, which are categorized into time-based and workload-based policies, under the environment where the arrival stream of system follows a Markovian arrival process (MAP). After building the stochastic models with respective rejuvenation policies, we formulate the loss probability of transaction and the upper bound of mean response time as the system performance indices. In the numerical illustrations, we exhibit a comprehensive study to compare six software rejuvenation policies numerically and show that the proposed rejuvenation policies called wait-time policies are superior to the others under the MAP arrival stream.

This paper proposes a unified modeling framework for software reliability assessment in open source project. We combine the classical non-homogeneous Poisson process based software reliability growth model (SRGM) with a familiar regression scheme called the generalized linear model (GLM), and develop a novel framework not only to estimate software reliability measures, but also to investigate impacts of software metrics on the fault-detection process. The resulting GLM-based SRGM involves the common SRGMs as well as some existing metrics-based SRMs, such as logistic-regression-based SRGM and Poisson-regression-based SRGM, and possesses a great data fitting ability. We also provide an effective parameter estimation algorithm based on the EM (Expectation-Maximization) principle. Finally, it is shown through numerical experiments with actual open source project data that our approach can estimate software reliability measures with higher accuracy and can feedback the analysis results to improve the current software development projects.

Border effect in communication network area is one of the most important problems to quantify accurately the performance/dependability of wireless ad hoc networks (WAHNs), because the assumption on uniformity of network node density is often unrealistic to describe the actual communication area. This problem appears in modeling the node behavior of WAHNs and in quantification of their network survivability. In this paper we focus on the border effects, and reformulate the network survivability models based on a semi-Markov process, where two kinds of communication network areas are considered, square area and circular area. Based on some geometric ideas, we improve the quantitative network survivability measures for three stochastic models taking account of the border effects. In numerical examples, we compare our new models with the existing ones without border effects in both steady-state and transient network survivability analyses. We show through a simulation study that the border effects are significant factors in quantifying network survivability.

Component importance analysis is to measure the effect on system reliability of component reliabilities, and it can be used to the design of system from the reliability point of view. In this paper, we consider the component importance analysis of real-time computing systems in the presence of common-cause failures (CCFs) (i.e., failure dependencies). Although the CCFs are known as a risk factor of degradation of system reliability, it is difficult to evaluate the component importance measures in the presence of CCFs analytically. This paper introduces a continuous-time Markov chain (CTMC) model for real-time computing system, and applies the CTMC-based component-wise sensitivity analysis which can evaluate the component importance measures without any structure function of system. Also, in numerical experiments, we evaluate the effect of CCFs by the comparison of system performance measures and component importance in the case of system with CCFs with those in the case that there is no CCF in the system.

This paper considers the model-based software reliability evaluation using the information on software testing inputs. Concretely, we define the distance between two software test cases by means of their testing inputs, and estimate the probability that the domain for a test input has already been covered by already-executed test cases. Based on the probability, we formulate the fault-detection probability in the software reliability growth model. In numerical experiments, we compare the proposed model with an existing non-homogeneous Poisson process based model with the distance of 5,000 test inputs in a real software application, and discuss the effect of test information on the software reliability evaluation.

In designing information communication devices such as real-time embedded systems, it is quite important to maximize the system performance under some hard energy constraints. As a useful technology to reduce the energy consumption in computer-based systems, the dynamic voltage and frequency scaling (DVFS) is becoming very popular. In this paper, we consider an optimal DVFS allocation problem by maximizing the system reliability subject to the hard real-time and energy constraints. More specifically, we formulate a reliability maximization problem when the task processing is probabilistic and is described by a discrete-time Markov chain. Two approximate formulas are proposed to calculate the system reliability efficiently. We perform the sensitivity analysis of model parameters in numerical examples, and also give a case study to design a Wi-Fi subsystem in terms of the DVFS allocation.

This paper discusses parallel algorithms for transient analysis of continuous-time Markov chains (CTMCs). In dependable computing, it is used for evaluating the rare events such as failure based on CTMC models. The uniformizaton is a well-known algorithm for obtaining the transient solution of CTMC. However, the computation cost of uniformization is not low in the case of large-sized and stiff CTMCs. This paper considers parallelization of the uniformization algorithm. Particularly, we propose a coarse-grained parallel uniformization which is appropriate for multicore processors. This method enables us to analyze the large-sized and stiff CTMCs efficiently. In numerical examples, we examine the effectiveness of the proposed parallel algorithms with multicore processors.

A multi-version concurrent control (MVCC) is widely spread as an important scheme to ensure the data integrity even in on-line storage services such as Dropbox as well as database management system. The scheme of MVCC is to make a snapshot of the current data before access to the data, and to compare the snapshot with the original data just before the update is committed. This discipline is called first-committer-wins rule (FCWR). This paper focuses on the performance evaluation of MVCC based on a stochastic model. According to queueing analysis, we derive quantitative performance measures from the model analytically. Further, we reveal the effectiveness of MVCC by comparing with the serializable policy.

Software reliability models (SRMs) are used to assess software reliability and to control quantitatively software testing. In this paper we consider metrics-based SRMs and tackle a statistical estimation of both software test process and product reliability simultaneously. The basic idea is to apply the Markov-dependent Poisson regression to describe the random testing environment by a discrete-time Markov chain. We formulate four Markov-dependent Poisson regression-based SRMs and develop the EM (expectation-maximization) algorithms to estimate the maximum likelihood estimates of model parameters. Numerical examples with real software project data show that our approach is useful to quantify both of software test process and software product reliability, and can answer the question why the stability of test process can lead to the improvement of software product reliability.

This paper presents a software reliability assessment tool which is built as a Microsoft Excel AddIn. The tool handles 11 types of software reliability growth models (SRGMs) and easily provides the maximum likelihood estimates (MLEs) from input data on spreadsheets. The tool consists of two components: the user interface written by Visual Basic for Application (VBA) and the dynamic link library (DLL) written by C language. In the DLL, the state-of-the-art parameter estimation algorithms for SRGMs are implemented, which are developed from the EM (expectation-maximization) algorithm. These algorithms enable us to derive the MLEs for any patterns of data.

Incremental development of software becomes much popular and enables to reduce the development cost effectively. On the other hand, it has not been known yet that the incremental development can really contribute to guarantee the software reliability more than the waterfall development paradigm. In this paper we estimate quantitative software reliability with both of static fault count data and static metrics data for incremental development processes. Since the measurement of software development project data is often expensive, we encounter the situation where the time series data are not always available. We develop metrics-based software reliability models based on the non-homogeneous Poisson processes for the purpose of reliability assessment in the incremental development, and compare them with an elementary approach with multiple linear regression model. Numerical examples are given with real software project data to show that our proposed methods outperform the common multiple linear regression model under the assumption on independent incremental testing phases.

Multifactor software reliability modeling with software test metrics data is well known to be useful for predicting the software reliability with higher accuracy, because it utilizes not only software fault count data but also software testing metrics data observed in the development process. In this paper we generalize the existing Cox proportional hazards regression-based software reliability model by introducing more generalized hazards representation, and improve the goodness-of-fit and predictive performances. In numerical examples with real software development project data, we show that our generalized model can significantly outperform several logistic regression-based models as well as the existing Cox proportional hazards regression-based model.

In this paper, we propose a probabilistic approach to finding failure-causing inputs based on Bayesian estimation. According to our probabilistic insights of software testing, the test case generation algorithms are developed by Markov chain Monte Carlo (MCMC) methods. Dissimilar to existing random testing schemes such as adaptive random testing, our approach can also utilize the prior knowledge on software testing. In experiments, we compare effectiveness of our MCMC-based random testing with both ordinary random testing and adaptive random testing in real program sources. These results indicate the possibility that MCMC-based random testing can drastically improve the effectiveness of software testing.

Software rejuvenation is a preventive and proactive maintenance solution that is particularly useful for counteracting the phenomenon of software aging. Hence, it should be ideally triggered adaptively without the complete knowledge on system failure (degradation) time distribution in operational phase. In this paper we consider an operational software system with multiple degradation levels and derive the optimal software rejuvenation policy maximizing the steady-state system availability, via the semi-Markov decision process. We develop a statistically non-parametric algorithm to estimate the optimal software rejuvenation schedule. Then, the reinforcement learning algorithm, called Q learning, is used for developing an on-line adaptive algorithm. A numerical example is presented to investigate asymptotic behavior of the resulting on-line adaptive algorithm.

Quantifying the effects of software testing metrics such as the number of test runs on the fault detection ability is quite important to design and manage effective software testing. This paper focuses on the regression model which represents the causal relationship between the software testing metrics and the fault detection probability. In a numerical experiment, we perform the quantitative estimation of the causal relationship through the quantization of software testing metrics.

This paper proposes an unified modeling framework of Markov-type software reliability models (SRMs) using Markovian arrival processes (MAPs). The MAP is defined as a point process whose inter-arrival time follows a phase-type distribution incorporating the correlation between successive two arrivals. This paper presents MAP representation of Markov-type SRMs, called MAP-based SRMs. This framework enables us to use generalized formulas for several reliability measures such as the expected number of failures and the software reliability which can be applied to all the Markov-type SRMs. In addition, we discuss the parameter estimation for the MAP-based SRMs from grouped failure data and find maximum likelihood estimates of all the Markov-type SRMs. The resulting MAP-based SRM is a novel approach to unifying the model-based software reliability evaluation using failure data.

This paper proposes a multi-factor software reliability model based on logistic regression and its effective statistical parameter estimation method. The proposed parameter estimation algorithm is composed of the algorithm used in the logistic regression and the EM (expectation-maximization) algorithm for discrete-time software reliability models. The multi-factor model deals with the metrics observed in testing phase (testing environmental factors), such as test coverage and the number of test workers, to predict the number of residual faults and other reliability measures. In general, the multi-factor model outperforms the traditional software reliability growth model like discrete-time non-homogeneous models in terms of data-fitting and prediction abilities. However, since it has a number of parameters, there is the problem in estimating model parameters. Our modeling framework and its estimation method are quite simpler than the existing methods, and are promising for expanding the applicability of multi-factor software reliability model. In numerical experiments, we examine data-fitting ability of the proposed model by comparing with the existing multi-factor models. The proposed method provides the similar fitting ability to existing multi-factor models, although the computation effort of parameter estimation is low.

Software system developed for a specific user under contract undergoes a period of testing by the user before acceptance. This is known as user acceptance testing and is useful to debug the software in the user's operational circumstance. In this paper we first present a simple non-homogeneous Poisson process (NHPP)-based software reliability model to assess the quantitative software reliability under the user acceptance test, where the idea of an accelerated life testing model is introduced to represent the user's operational phase and to investigate the impact of user's acceptance test. This idea is applied to the reliability assessment of web applications in a different testing environment, where two stress tests with normal and higher workload conditions are executed in parallel. Through numerical examples with real software fault data observed in actual user acceptance and stress tests, we show the applicability of the software accelerated life testing model to two different software testing schemes.

This paper refines statistical inference of computer virus propagation with maximum likelihood (ML) estimation. In particular, in order to utilize actual infection data that are opened in Web sites, we reformulate classical stochastic models by Markovian arrival processes (MAPs). The reformulated models lead to plausible parameter estimation based on the ML estimation. We propose efficient algorithms to compute the ML estimates of epidemic models using the EM (expectation-maximization) algorithm. Experiments illustrate the estimation of virus propagation with real infection data by our methods. Finally we refer to characterization of virus propagation from the view point of stochastic modeling.

This paper proposes compositional stochastic models for an intrusion tolerant system to evaluate quantitative security measures for security. The models are described by MRSPNs (Markov regenerative stochastic Petrinets), which have high representation ability. Through modeling of an intrusion tolerant system, we present standardized components using MRSPNs, which would be useful to evaluate security measures for other systems as well as an intrusion tolerant system. In a numerical example, we carry out transient analysis by using developed compositional MRSPN-based models and evaluate the point availability.

This paper proposes a patch management model with non-homogeneous vulnerability-discovery processes to find the optimal security patch release times. The proposed model is an extension of Cavusoglu et al.\ (2006, 2008) by applying non-homogeneous vulnerability-discovery processes which are based on a vulnerability life-cycle model, and provides the optimal schedule for security patch release times over a software life cycle by means of cost analysis. In numerical examples, we show that the optimal patch release policy becomes an aperiodic release strategy, and compare the minimum cost under the optimal policy with that under a periodic release strategy. In addition, based on opened vulnerability data, we illustrate the optimal security patch release policy for a real software product.

This paper proposes a hyper-Erlang software reliability model (HErSRM) in the framework of non-homogeneous Poisson process (NHPP) modeling. The proposed HErSRM is a generalized model which contains some existing NHPP-based SRMs like Goel-Okumoto SRM and Delayed S-shaped SRM, and can represent a variety of software fault-detection patterns. Such characteristics are useful to solve the model selection problem arising in the practical use of NHPP-based SRMs. More precisely, we discuss the statistical inference of HErSRM based on the EM (expectation-maximization) algorithm. In numerical experiments, we show that the HErSRM outperforms conventional NHPP-based SRMs with respect to fitting ability.

In this paper, we develop bivariate software fault-detection models with two time measures: calendar time (day) and test-execution time (CPU time) and incorporate both of them to assess the quantitative software reliability with higher accuracy. The resulting stochastic models are characterized by a simple binomial process and the bivariate order statistics of software fault-detection times with different time scales.

In this paper, we present a variational Bayesian (VB) approach to computing the interval estimates for nonhomogeneous Poisson process (NHPP) software reliability models. This approach is an approximate method that can produce analytically tractable posterior distributions. We present simple iterative algorithms to compute the approximate posterior distributions for the parameters of the gamma-type NHPP-based software reliability model using either individual failure time data or grouped data. In numerical examples, the accuracy of this VB approach is compared with the interval estimates based on conventional Bayesian approaches, i.e., Laplace approximation, Markov chain Monte Carlo (MCMC) method, and numerical integration. The proposed VB approach provides almost the same accuracy as MCMC, while its computational burden is much lower.

This paper presents statistical inference of computer virus propagation using non-homogeneous Poisson processes (NHPPs). Under some mathematical assumptions, the number of infected hosts can be modeled by an NHPP. In particular, this paper applies a framework of mixed-type NHPPs to the statistical inference of periodic virus propagation. The mixed-type NHPP is defined by a superposition of NHPPs. In numerical experiments, we examine a goodnessof-fit criterion of NHPPs on fitting to real virus infection data, and discuss the effectiveness of the model-based prediction approach for computer virus propagation.

The software fault correction profiles play significant roles to assess the quality of software testing as well as to keep the good software maintenance activity. In this paper we develop a quantitative method to evaluate the software maintainability based on a stochastic model. The model proposed here is a queueing model with an infinite number of servers, and is related to the software fault- detection/correction profiles. Based on the familiar maximum likelihood estimation, we estimate quantitatively both the software reliability and maintainability with real project data, and refer to their applicability to the software maintenance practice.

In this paper, we develop a parameter estimation method to Markovian software reliability models. When software fault-detection rates change in the software testing phase, fault-detection processes can be generally modeled by Markov modulated processes. This paper deals with a unified parameter estimation method for Markov modulated software reliability models as well as the typical pure birth process models. In numerical examples, we evaluate a goodness-of-fit for the Markov modulated software reliability models with real fault data, and show numerically that the Markov modulated software reliability models are superior to the existing pure birth process models in the viewpoint of information criterion

This paper presents a unified framework for software reliability modeling with non-homogeneous Poisson processes, where each software fault-detection time obeys the phase-type distribution and the initial number of inherent faults is given by a Poisson distributed random variable. However, it is worth noting that the resulting software reliability models, called phase-type software reliability models, generalize the existing models but may involve a number of model parameters in the phase-type software reliability model, so that the usual maximum likelihood estimation based on the Newton's method or quasi-Newton's method does not often function well. In this paper, we develop EM (expectation-maximization) algorithms for the phase-type software reliability models with two types of fault data: fault-detection time data and grouped data with arbitrary time intervals. In numerical examples, we compare the EM algorithms with the quasi-Newton's method and illustrate the effectiveness on our unified model and parameter estimation method

In this paper, we propose some reliability estimation methods in software testing. The proposed methods are based on the familiar Bayesian statistics, and can be characterized by using test outcomes in input domain models. It is shown that the resulting approaches are capable of estimating software reliability in the case where the detected software faults are removed. In numerical examples, we compare the proposed methods with the existing method, and investigate the effect of fault removal on the reliability estimation in software testing. We show that the proposed methods can give more accurate estimates of software reliability

In this paper, we consider two kinds of sequential checkpoint placement problems with infinite/finite time horizon. For these problems, we apply approximation methods based on the variational principle and develop computation algorithms to derive the optimal checkpoint sequence approximately. Next, we focus on the situation where the knowledge on system failure is incomplete, i.e., the system failure time distribution is unknown. We develop the so-called min-max checkpoint placement methods to determine the optimal checkpoint sequence under an uncertain circumstance in terms of the system failure time distribution. In numerical examples, we investigate quantitatively the proposed distribution-free checkpoint placement methods, and refer to their potential applicability in practice.

In this article, we propose a stochastic model called the Gompertz software reliability model based on the familiar non-homogeneous Poisson process. It is shown that the proposed model can be derived from the well-known statistical theory of extreme-value and has the quite similar asymptotic property to the classical Gompertz curve. In a numerical example with the software failure data observed in a real software development project, we apply the Gompertz software reliability model to assess the software reliability and to predict the number of initial fault contents. We empirically conclude that our new model may function better than the existing models and is attractive in terms of goodness-of-fit test based on information criteria and mean squared error.

Long running software systems are known to experience an aging phenomenon called software aging, one in which the accumulation of errors during the execution of software leads to performance degradation and eventually results in failure. To counteract this phenomenon a proactive fault management approach, called software rejuvenation, is particularly useful. It essentially involves gracefully terminating an application or a system and restarting it in a clean internal state. In this paper, we perform the dependability analysis of a client/server software system with rejuvenation under the assumption that the requests arrive according to the Markov modulated Poisson process. Three dependability measures, steady-state availability, loss probability of requests and mean response time on tasks, are derived through the hidden Markovian analysis based on the time-based software rejuvenation scheme. In numerical examples, we investigate the sensitivity of some model parameters to the dependability measures.

Propagation of Internet worms is a serious problem in our highly information oriented society. In this paper, we propose a stochastic model for Internet worm propagation to evaluate its dependability measures quantitatively. More precisely, the deterministic kill-signal model is reformulated based on a continuous-time Markov chain. We define some dependability measures and derive the recursive computation algorithms to assess them. In numerical experiments, we investigate the behavior of actual Internet worms with real infection data and characterize their propagation.

In this paper we consider two kinds of sequential checkpoint placement problems with infinite/finite time horizon. For these problems, we apply the approximation methods based on the variational principle and develop the computation algorithms to derive the optimal checkpoint sequence approximately. Next, we focus on the situation where the knowledge on system failure is incomplete, i.e. the system failure time distribution is unknown. We develop the so-called min-max checkpoint placement methods to determine the optimal checkpoint sequence under the uncertain circumstance in terms of the system failure time distribution. In numerical examples, we investigate quantitatively the min-max checkpoint placement methods, and refer to their potential applicability in practice.

We develop a new checkpointing scheme for a uniprocess application. First, we model the checkpointing scheme by a semiMarkov decision process, and apply the reinforcement learning algorithm to estimate statistically the optimal checkpointing policy. More specifically, the representative reinforcement learning algorithm, called the Q-learning algorithm, is used to develop an adaptive checkpointing scheme. In simulation experiments, we examine the asymptotic behavior of the system overhead with adaptive checkpointing and show quantitatively that the proposed dynamic checkpoint algorithm is useful and robust under an incomplete knowledge on the failure time distribution.

This paper focuses on an estimation problem of model parameters in software reliability modeling. We introduce the EM (expectation-maximization) algorithms for software reliability models and compare them with the classical parameter estimation methods. Especially, we extensively develop the EM algorithms for two cases; (i) the time interval data of software fault detection are available, (ii) additive software reliability models based on non-homogeneous Poisson processes are used. In numerical examples, we compare the iterative schemes based on the EM algorithms with classical methods such as the Newton's method and the Fisher's scoring method and show that the EM algorithms are attractive in terms of convergence property.

Long running software systems are known to experience an aging phenomenon called software aging, one in which the accumulation of errors during the execution of software leads to performance degradation and eventually results in failure. To counteract this phenomenon an active fault management approach, called software rejuvenation, is particularly useful. It essentially involves gracefully terminating an application or a system and restarting it in a clean internal state. We deal with dependability analysis of a client/server software system with rejuvenation. Three dependability measures in the server process, steady-state availability, loss probability of requests and mean response time on tasks, are derived from the well-known hidden Markovian analysis under the time-based software rejuvenation scheme. In numerical examples, we investigate the sensitivity of some model parameters to the dependability measures.

Software rejuvenation is a preventive maintenance technique that has been extensively studied in the recent literature. We consider a generalized problem to estimate the optimal software rejuvenation schedule. More precisely, the software rejuvenation model is formulated via the semi-Markov process, and the optimal software rejuvenation schedule which minimizes the expected total discounted cost over an infinite time horizon is derived analytically. Further, we develop a statistically nonparametric algorithm to estimate the optimal software rejuvenation schedule, provided that the complete sample data of failure time is given. In numerical examples, we investigate how the discount factor affects the optimal policy and examine an asymptotic property for the statistical estimation algorithm.