Mean Value Function Research Articles

NHPP Software Reliability Model with Rayleigh Fault Detection Rate and Optimal Release Time for Operating Environment Uncertainty

Software is used in diverse settings and depends on development and testing environments. Software development should improve the reliability, quality, cost, and stability of software, making the software testing period crucial. We proposed a software reliability model (SRM) that considers the uncertainty of software environments and the fault detection rate function as a Rayleigh distribution, with an explicit mean value function solution in the model. The goodness-of-fit of the proposed model relative to several existing nonhomogeneous Poisson process (NHPP) SRMs is presented based on three software application failure datasets. Further, a cost model is also presented that addresses the error-removal risk level and required time. The optimal testing release policy for minimizing the expected total cost (ETC) is also determined for NHPP SRMs. The impact of the software environment is studied by varying it, and the optimal release times and minimum ETCs are compared. The goodness-of-fit comparison confirmed that the proposed model has more accurate prediction values than other models. Further, whereas the existing models applied to the cost model do not change after a certain operation period, the proposed model yields changes in release time even for long operating periods.

Analysis of incorporating modified Weibull model fault detection rate function into software reliability modeling

When software systems are introduced, they are typically deployed in field environments similar to those used during development and testing. However, these systems may also be used in various other locations with different environmental conditions, making it challenging to improve software reliability. Factors such as the specific operating environment and the location of bugs in the code contribute to this difficulty. In this paper, we propose a new software reliability model that accounts for the uncertainty of operating environments. We present the explicit closed-form mean value function solution for the proposed model. The model's goodness of fit is demonstrated by comparing it to the nonhomogeneous Poisson process (NHPP) model based on Weibull model, using four sets of failure data sets from software applications. The proposed model performs well under various estimation techniques, making it a versatile tool for practitioners and researchers alike. The proposed model outperforms other existing NHPP Weibull based in terms of fitting accuracy under two different methods of estimation and provides a more detailed and precise evaluation of software reliability. Additionally, sensitivity analysis shows that the parameters of the suggested distribution significantly impact the mean value function.

Developing an Innovative Imperfect Debugging Software Reliability Growth Model with Enhanced Testing Coverage Strategies

In this paper, we introduce an innovative Software Reliability Growth Model (SRGM) designed to tackle the pivotal challenges associated with software reliability in the contemporary digital landscape, where the prevalence of online systems is ubiquitous. This SRGM integrates Imperfect Debugging (ID), Testing Coverage (TC), Testing Effort (TE), and error generation into a cohesive framework. Employing a sigmoid function to encapsulate TE, it incorporates three distinct TC functions: Delayed S-shaped, Exponential, and Logistic. This model relies on foundational assumptions, including the proportionality of fault detection rates to remaining faults, the introduction of new faults during debugging, and the intricate connection between fault detection and code coverage. The Mean Value Function (MVF) is computed through these differential equations, and the resultant MVFs are systematically tabulated for all models. An examination of the sigmoid TE function and the Weibull TE function across diverse datasets, utilizing a range of goodness-of-fit criteria including Mean Square Error (MSE), Pham’s Criterion (PC), Predictive Risk Ratio (PRR), Bayesian Information Criterion (BIC), and Akaike’s Information Criterion (AIC), reveals the superior performance of the sigmoid TE function over the Weibull counterpart across various datasets and evaluation criteria. In conclusion, this paper introduces a groundbreaking SRGM that seamlessly integrates ID, TC, and TE, offering valuable insights for assessing software reliability in the dynamic landscape of modern digital systems.

Parametric estimations of the mean value and variance functions in geometric process

The paper introduces novel parametric estimators for the mean value function (MVF) and variance function (VF) in geometric processes (GPs) based on a random sample which belongs to a GP. Specifically, it addresses scenarios where the distribution function of the first interarrival time of GP is analytically known, but its parameters, along with the ratio parameter of the GP, are unknown. It establishes a solid theoretical foundation for the estimators proposed and investigates some statistical properties such as strong consistency and asymptotic unbiasedness of the estimators. These results generalize some of the findings acquired in previous studies. To evaluate the performance of the proposed estimators, we conduct a comprehensive Monte Carlo simulation study. This involves using various lifetime distributions, such as gamma, Weibull, and lognormal considering different sample sizes and parameter values. A discussion of an application to software engineering is also provided to illustrate the applicability of the methods developed in this study.

A Method of Reliability Assessment Based on Trend Analysis for Open Source Software

Software reliability growth model (SRGM) is used as one of the reliability assessment methods to assess the software reliability. In SRGM, the degree of reliability growth may fluctuate greatly according to change in the internal state of the software. It is called the change point (CP). Several researchers proposed the SRGM considering CP. In the open source software (OSS), there are many projects that continue development even after the software is released. Therefore, major updates with breaking changes may occur in it. The major updates can be a factor that causes a CP because it greatly changes the internal state of the OSS. This paper focuses on the relationship between CP and software updates. We collect OSS fault data from a bug tracking system. Moreover, we examine the behavior of SRGM before and after software updates. Furthermore, we discuss the applicability of SRGM for CP in OSS. Also, we compare the proposed model based on CP with the model without CP. As a result, we have confirmed that the SRGM can evaluate the reliability in the environment with major updates. Moreover, the proposed method performs better than without considering CP model. Especially, the exponential model’s mean value function is the suitable method to assess the OSS reliability for the proposed method.

PREDICTING THE LEARNING PATH TO LEARNER’S OPTIMUM COMPREHENSION

The essence of learning is for the learner to attain a significant level of comprehension after the learning process is completed. The quest to achieve this singular purpose has led to the introduction of several learning techniques in the conventional learning environment, such as asking questions and conducting test after class, just to mention a few. Additionally, technology has been introduced in learning. Even with technological advancements, the learning experience still faces the challenge of learners not attaining the optimum comprehension state after the learning process. This is due to the present systems' inability to model the learner to determine the best methods for achieving maximum comprehension. Hence, this research paper focuses on deriving an improved mathematical model for predicting the learning path to a learner’s optimum comprehension. The paper presented three learning instructional media (learning paths); textual, audio and a hybrid of audio and video, which this research uses in modelling the learner. This is to enable the improved system predict the best learning path to optimum comprehension for learners. This research paper adopted Reinforcement Learning and the Markov decision process, specifically the Markov Chain approach, in developing an improved model for prediction. The evaluation of this research involved brainstorming on the Bellman’s equation with the aid of the Markov Chain transition state framework, resulting in an improved mean value function of 71.7. This indicates an enhanced comprehension state for the learning students compared to the existing mean value function of 46.0. The results obtained from this research clearly demonstrate that the improved model was able to predict and assign the best learning path to achieve optimum comprehension state for learners.

A finite failure software reliability model using extended log-logistic distribution

Software reliability engineering has been developed to examine software systems over the years. Non-homogeneous Poisson process (NHPP) models are widely employed for analyzing software reliability data, as they have the ability to describe the expected number of failures over time using the mean value function. In addition, these models have an intensity function representing the failure rate over time. This article proposes a new non-homogeneous Poisson process model based on extended log-logistic (NHPP ELL) distribution with three parameters. The model properties were derived and represented graphically. Furthermore, the maximum likelihood estimation method was utilized to estimate the model parameters, which were computed numerically using the Newton–Raphson method. Three real software reliability datasets were utilized to evaluate the performance of the new NHPP ELL model based on three distinct criteria: the mean square error, the coefficient of determination (R2), the Theil statistic, the predictive-ratio risk, and the predictive power. Moreover, a comparative study was conducted between the proposed model and several well-known NHPP models regarding goodness-of-fit and predictive performance. All the evaluation measures indicate that the proposed model outperforms the other models across all the considered datasets. In addition, the graphical plots depicting the actual data and the prediction results for all the fitted models show that the proposed model performs well in all three considered software reliability datasets compared to other models.

Multivariate Random Fields Evolving Temporally Over Hyperbolic Spaces

AbstractGaussian random fields are completely characterised by their mean value and covariance function. Random fields on hyperbolic spaces have been studied to a limited extent only, namely for the case of scalar-valued fields that are not evolving over time. This paper challenges the problem of the second-order characteristics of multivariate (vector-valued) random fields that evolve temporally over hyperbolic spaces. Specifically, we characterise the continuous space–time covariance functions that are isotropic (radially symmetric) over space (the hyperbolic space) and stationary over time (the real line). Our finding is the analogue of recent findings that have been shown for the case where the space is either the n-dimensional sphere or more generally a two-point homogeneous space. Our main result can be read as a spectral representation theorem, and we also detail the main result for the subcase of covariance functions having a spectrum that is absolutely continuous with respect to the Lebesgue measure (technical details are reported below).

Software testing and release decision at different statistical confidence levels with consideration of debuggers’ learning and negligent factors

This research delves into the software testing process and its environmental factors to uncover the core elements influencing software reliability. Specifically, it focuses on the learning and negligent aspects of the software reliability growth model. The learning factor accelerates reliability growth, leading to an S-shaped curve in the mean value function, while the negligent factor highlights the occurrence of imperfect debugging. The study also uses Brownian motion and stochastic differential equations to establish statistical confidence intervals for reliability and costs. These intervals aid software managers in assessing potential release risks at various confidence levels, allowing them to make informed decisions considering resource constraints and desired system reliability levels across different scenarios.

Local polynomial software reliability models and their application

In this paper, we propose local polynomial software reliability models (SRMs), which can be categorized into a semi-parametric modeling framework. Our models belong to the common non-homogeneous Poisson process (NHPP)-based SRMs, but possess a flexible structure to approximate an arbitrary mean value function by controlling the polynomial degree. More specifically, we develop two types of local polynomial NHPP-based SRMs; finite-failure (type-I) and infinite-failure (type-II) SRMs, which are substantial extensions of the existing NHPP-based SRMs in the similar categories. We also develop two maximum likelihood estimation algorithms in both estimation and prediction phases, where the former is used for the testing period experienced in the past, and the latter for the prediction in the future. In numerical experiments with actual 8 software fault count time-interval data sets, we compare our local polynomial NHPP-based SRMs with the well-known existing parametric NHPP-based SRMs in terms of goodness-of-fit and predictive performances. Finally, it can be concluded that our local polynomial NHPP-based SRMs with lower polynomial degrees could outperform the existing NHPP-based SRMs in several cases and should be listed as candidates for the representative NHPP-based SRMs in software reliability analysis.

Topology optimization of stability‐constrained structures with simple/multiple eigenvalues

AbstractThis work focuses on topology optimization formulations with linear buckling constraints wherein eigenvalues of arbitrary multiplicities can be canonically considered. The non‐differentiability of multiple eigenvalues is addressed by a mean value function which is a symmetric polynomial of the repeated eigenvalues in each cluster. This construction offers accurate control over each cluster of eigenvalues as compared to the aggregation functions such as ‐norm and Kreisselmeier–Steinhauser (K–S) function where only approximate maximum/minimum value is available. This also avoids the two‐loop optimization procedure required by the use of directional derivatives (Seyranian et al. Struct Optim. 1994;8(4):207‐227.). The spurious buckling modes issue is handled by two approaches—one with different interpolations on the initial stiffness and geometric stiffness and another with a pseudo‐mass matrix. Using the pseudo‐mass matrix, two new optimization formulations are proposed for incorporating buckling constraints together with the standard approach employing initial stiffness and geometric stiffness as two ingredients within generalized eigenvalue frameworks. Numerical results show that all three formulations can help to improve the stability of the optimized design. In addition, post‐nonlinear stability analysis on the optimized designs reveals that a higher linear buckling threshold might not lead to a higher nonlinear critical load, especially in cases when the pre‐critical response is nonlinear.

Equation of Finite Change and Structural Analysis of Mean Value

This paper describes a problem of finding the contributions of multiple variables to a change in their function. Such a problem is well known in economics, for example, in the decomposition of a change in the mean price via the varying in time prices and volumes of multiple products. Commonly, it is considered by the tools of index analysis, the formulae of which present rather heuristic constructs. As shown in this work, the multivariate version of the Lagrange mean value theorem can be seen as an equation of the function’s finite change and solved with respect to an interior point whose value is used in the estimation of the contribution of the independent variables. Consideration is performed on the example of the weighted mean value function, which is the main characteristic of statistical estimation in various fields. The solution for this function can be obtained in the closed form, which helps in the analysis of results. Numerical examples include the cases of Simpson’s paradox, and practical applications are discussed.

Statistical Characteristics of Geometry, Density and Porosity of Individual Ore Particles: A Case Study

This study aims to develop a methodology to describe and predict the statistical characteristics of individual ore particles in terms of length, width, height, volume, mass, area, circularity, aspect ratio, density, and porosity. The mean value, standard deviation, and appropriate distribution function were calculated or identified for each data set of a given particle property in a given size fraction. It was found that the mean value and the standard deviation of the same particle property can either be predicted from particle size or be approximated by a constant. The best-fit distribution of each kind of particle property was identified by the Anderson–Darling test using Minitab software. Generally, the data sets with the same particle property but different size fractions and ore types follow the same distribution. A methodology was developed to predict the distribution of individual particle properties in a given size fraction by particle size, and the fitting quality is good in most cases. The statistical characteristics of individual ore particles can improve the precise processing of ore feed in concentrators, the preparation of feed samples for lab-scale testing, the calibration of image analysis of ore particle size distribution, etc.

Uncertainty Quantification Method of Crosstalk Involving Braided-Shielded Cable

In this paper, in view of the uncertainty of geometric parameters of braided-shielded cable and structural parameters of the shield in practical problems, polynomial chaos expansions (PCE) is used to explore the uncertainty quantification of a crosstalk calculation model of braided-shielded cable. First, the model based on multi-conductor transmission line theory is expanded by PCE. Second, the truncation degrees of polynomials and sample size of PCE are determined by the leave-one-out method, and the statistical characteristic parameters of crosstalk are obtained by combining coefficients of polynomials. Compared with the calculation results of the Monte Carlo method, the mean value, standard deviation and probability density function obtained by the two methods are basically consistent, but the PCE method has obvious advantages in calculation efficiency. Finally, the influence of input variable parameters in the crosstalk calculation model of braided-shielded cable are calculated by combining the PCE and the Sobol’s global sensitivity analysis method, which provides theoretical guidance for the electromagnetic compatibility design of electrical and electronic equipment using braided-shielded cable.

Annual Mean Temperature and Rain Precipitation in North America Using NHPP to Detect Climate Changes

In this study, non-homogeneous Poisson processes (NHPP) are assumed to analyze annual average temperatures and rain precipitations, considering climate data for some regions of North America reported for a long period. A power law process (PLP) is assumed for the intensity function (derivative of the mean value function) or rate \(\lambda\) (t), t \(\ge\) 0 of the NHPP which the Poisson events occur considering data (accumulated number of years in a given time interval [0,t) where the climate measure is above a threshould given by the overal average in the assumed period) in presence or not of a change-point. The parameters of the assumed model are estimated under a Bayesian approach and using MCMC (Markov Chain Monte Carlo) methods. Alternatively to the use of a PLP process, we also assume a polynomial parametrical form for the mean value function of the NHPP process where a simple Bayesian inference approach is proposed to get better fit for the intensity and mean value functions of the NHPP process. From the fitted models it was possible to to detect the years where climate changes occurred.

An NHPP software reliability model with a fault detection rate function of the Burr type XII distribution

The software reliability model is developed considering the number of software failures and the interval time between failures, and is a very useful model for predicting software failures that may occur in the future. In most existing software reliability model, the mean value function is considered to follow a non-homogeneous Poisson process (NHPP). NHPP software reliability model has a mean value function that considers the failure intensity, detection rate, number of remaining failures, and various environments. The Burr distribution can fit a wide range of empirical data. In addition, it was proposed several types of the cumulative functions and has stressed the advantages obtained by the direct use of the cumulative function. The Burr type XII distribution is a continuous probability distribution for a non-negative random variable. In this paper, we propose a new software reliability model with a fault detection rate function of the Burr type XII distribution based on NHPP. We show that the NHPP software reliability model proposed through various criteria shows much better the goodness-of-fit than other existing NHPP software reliability models.

Modelling of NHPP Based Software Reliability Growth Model from the Perspective of Testing Coverage, Error Propagation and Fault Withdrawal Efficiency

This paper presents a software reliability growth model framework modeled from a non-homogenous poisson process (NHPP). Invariably, software testing continues to be a paramount measure for validating the standard of software. Test coverage measures appraise and estimate the proportion and gradation of testing in software. Therefore, presenting a correct picture of the test coverage becomes a prime requisite to guarantee software reliability. As an enhancement over the existing models, the proposed model integrates testing coverage (TC), error propagation, and fault withdrawal efficiency while keeping the number of parameters restrained to make the framework more reliable for parameter estimation. A relative analysis to assess the efficacy of the proposed model and some existing models has been carried out on the failure data obtained from three real-world software applications using six comparison criteria. Finally, the weighted criteria rank method has been used to rank the models and assess their performance. In addition, sensitivity analysis has been carried out to demonstrate the effect of the parameters of the proposed model on the mean value function.

Active Learning for Contextual Search with Binary Feedback

In this paper, we study the learning problem in contextual search, which is motivated by applications such as crowdsourcing and personalized medicine experiments. In particular, for a sequence of arriving context vectors, with each context associated with an underlying value, the decision maker either makes a query at a certain point or skips the context. The decision maker will only observe the binary feedback on the relationship between the query point and the value associated with the context. We study a probably approximately correct learning setting, where the goal is to learn the underlying mean value function in context with a minimum number of queries. To address this challenge, we propose a trisection search approach combined with a margin-based active learning method. We show that the algorithm only needs to make [Formula: see text] queries to achieve an ε-estimation accuracy. This sample complexity significantly reduces the required sample complexity in the passive setting where neither sample skipping nor query selection is allowed, which is at least [Formula: see text]. This paper was accepted by J. George Shanthikumar, data science. Funding: X. Chen and Q. Liu were supported by the National Science Foundation [Grant IIS-1845444].

Interference Aggregation Involving Near- to Far-Field Propagation of Disturbance and its Impact on Emission Limits for Radio Protection

To introduce the interference aggregation effect into the emission-limit-setting model, aggregate interference power arising from randomly distributed disturbance sources is studied considering the near- to far-field propagation of radiated disturbance. It is assumed that point sources are uniformly distributed in a two-dimensional model and that the distance to the victim receiver ranges from the near-field region to the far-field region. A piecewise approximation is applied to the distance attenuation curve with a corner distance that indicates the border between the near and far fields. The mean value, standard deviation, and probability density function (PDF) of the decibel-scaled aggregate interference power are theoretically and numerically analyzed at various source densities. The results show that the shape of the PDF markedly changes as the source density increases because of the mixed contribution of disturbing sources both from the near and far fields. Accordingly, the variations in the mean value and standard deviation of the aggregate interference power are strongly dependent on the distance attenuation at which major interfering sources are found. When introducing the interference aggregation effect into the emission requirements, both the distance attenuation characteristics and the estimated source density in a real environment should be carefully considered.

Some new bounds for the mean value function of the residual lifetime process

In this study, mean residual lifetime (MRL) function of the residual lifetime process is investigated. Various two-sided bounds are obtained for this function by using the monotone convergence of the sequence of the functions. Since many important features of any stochastic process depend on classifications of distributions, improved bounds are found for MRL function by considering the taxonomy of the distributions. Moreover, some new bounds are achieved by using the asymptotic expression of MRL function and its boundary relation with the MRL of a single component at age t. Three examples are given to illustrate the results proposed.