Mohammad G.M. Khan

The exponentiated Weibull software reliability growth model with various testing‐efforts and optimal release policy: A performance analysis

Purpose – The purpose of this research is to incorporate the exponentiated Weibull testing‐effort functions into software reliability modeling and to estimate the optimal software release time.Design/methodology/approach – This paper suggests a software reliability growth model based on the non‐homogeneous Poisson process (NHPP) which incorporates the exponentiated Weibull (EW) testing‐efforts.Findings – Experimental results on actual data from three software projects are compared with other existing models which reveal that the proposed software reliability growth model with EW testing‐effort is wider and effective SRGM.Research limitations/implications – This paper presents a SRGM using a constant error detection rate per unit testing‐effort.Practical implications – Software reliability growth model is one of the fundamental techniques to assess software reliability quantitatively. The results obtained in this paper will be useful during the software testing process.Originality/value – The present schem...

A study of testing‐effort dependent inflection S‐shaped software reliability growth models with imperfect debugging

Purpose – The purpose of this paper is to investigate how to incorporate the exponentiated Weibull (EW) testing‐effort function (TEF) into inflection S‐shaped software reliability growth models (SRGMs) based on non‐homogeneous Poisson process (NHPP). The aim is also to present a more flexible SRGM with imperfect debugging.Design/methodology/approach – This paper reviews the EW TEFs and discusses inflection S‐shaped SRGM with EW testing‐effort to get a better description of the software fault detection phenomenon. The SRGM parameters are estimated by weighted least square estimation (WLSE) and maximum‐likelihood estimation (MLE) methods. Furthermore, the proposed models are also discussed under imperfect debugging environment.Findings – Experimental results from three actual data applications are analyzed and compared with the other existing models. The findings reveal that the proposed SRGM has better performance and prediction capability. Results also confirm that the EW TEF is suitable for incorporating...

Theory & Methods: An Optimal Multivariate Stratified Sampling Design Using Dynamic Programming

Numerous optimization problems arise in survey designs. The problem of obtaining an optimal (or near optimal) sampling design can be formulated and solved as a mathematical programming problem. In multivariate stratified sample surveys usually it is not possible to use the individual optimum allocations for sample sizes to various strata for one reason or another. In such situations some criterion is needed to work out an allocation which is optimum for all characteristics in some sense. Such an allocation may be called an optimum compromise allocation. This paper examines the problem of determining an optimum compromise allocation in multivariate stratified random sampling, when the population means of several characteristics are to be estimated. Formulating the problem of allocation as an all integer nonlinear programming problem, the paper develops a solution procedure using a dynamic programming technique. The compromise allocation discussed is optimal in the sense that it minimizes a weighted sum of the sampling variances of the estimates of the population means of various characteristics under study. A numerical example illustrates the solution procedure and shows how it compares with Cochrans average allocation and proportional allocation.

A note on optimum allocation in multivariate stratified sampling

In stratified random sampling when several characteristics are to be estimated simultaneously, an allocation that is optimum for one characteristic may be far away from optimum for others. To resolve this conflict the authors formulate the problem of determining optimum compromise allocation as a nonlinear programming problem (NLPP). The allocation obtained is optimum in the sense that it minimizes the sum of weighted variances of the estimated population means of the characteristics subject to a fixed sampling cost. The formulated NLPP is treated as multistage decision problem and solved using dynamic programming technique. A numerical example is presented to illustrate the computational details.

Software Reliability Modeling Incorporating Log‐Logistic Testing‐Effort with Imperfect Debugging

Reference [5] have proposed the log‐logistic SRGM that can capture the increasing/decreasing nature of the failure occurrence rate per fault. Therefore, in this paper, we will investigate how to incorporate the log‐logistic testing‐effort function (TEF) into inflection S‐shaped software reliability growth models based on non‐homogeneous Poisson process (NHPP). The models parameters are estimated by least square estimation (LSE) and maximum likelihood estimation (MLE) methods. The methods of data analysis and comparison criteria are presented and the experimental results from actual data applications are analyzed. Results are compared with the other existing models to show that the proposed models can give fairly better predictions. It is shown that the log‐logistic TEF is suitable for incorporating into inflection S‐shaped NHPP growth models. In addition, the proposed models are also discussed under imperfect debugging environment.

Determining Optimum Strata Boundaries and Sample Sizes for Skewed Population with Log-Normal Distribution

The method of choosing the best boundaries that make strata internally homogenous as far as possible is known as optimum stratification. To achieve this, the strata should be constructed in such a way that the strata variances for the characteristic under study be as small as possible. If the frequency distribution of the study variable x is known, the optimum strata boundaries (OSB) could be obtained by cutting the range of the distribution at suitable points. If the frequency distribution of x is unknown, it may be approximated from the past experience or some prior knowledge obtained at a recent study. Many skewed populations have log-normal frequency distribution or may be assumed to follow approximately log-normal frequency distribution. In this article, the problem of finding the OSB and the optimum sample sizes within the stratum for a skewed population with log-normal distribution is studied. The problem of determining the OSB is redefined as the problem of determining optimum strata widths (OSW) and is formulated as a Nonlinear Programming Problem (NLPP) that seeks minimization of the variance of the estimated population mean under Neyman allocation subject to the constraint that the sum of the widths of all the strata is equal to the range of the distribution. The formulated NLPP turns out to be a multistage decision problem that can be solved by dynamic programming technique. A numerical example is presented to illustrate the application and computational details of the proposed method. A comparison study is conducted to investigate the efficiency of the proposed method with other stratification methods, viz., Dalenius and Hodges’ cum method, geometric method by Gunning and Horgan, and Lavallée–Hidiroglou method using Kozak’s algorithm available in the literature. The study reveals that the proposed technique is efficient in minimizing the variance of the estimate of the population mean and is useful to obtain OSB for a skewed population with log-normal frequency distribution.

Determining the optimal allocation of testing resource for modular software system using dynamic programming

Abstract Reliability is a major concern in the process of software development because unreliable software can cause failure in the computer system that can be hazardous. A way to enhance the reliability of software is to detect and remove the faults during the testing phase, which begins with module testing wherein modules are tested independently to remove a substantial number of faults within a limited resource. Therefore, the available resource must be allocated among the modules in such a way that the number of faults is removed as much as possible from each of the modules to achieve higher software reliability. In this article, we discuss the problem of optimal resource allocation of the testing resource for a modular software system, which maximizes the number of faults removed subject to the conditions that the amount of testing-effort is fixed, a certain percentage of faults is to be removed and a desired level of reliability is to be achieved. The problem is formulated as a non linear programming problem (NLPP), which is modeled by the inflection S-shaped software reliability growth models (SRGM) based on a non homogeneous Poisson process (NHPP) which incorporates the exponentiated Weibull (EW) testing-effort functions. A solution procedure is then developed using a dynamic programming technique to solve the NLPP. Furthermore, three special cases of optimum resource allocations are also discussed. Finally, numerical examples using three sets of software failure data are presented to illustrate the procedure developed and to validate the performance of the strategies proposed in this article. Experimental results indicate that the proposed strategies may be helpful to software project managers for making the best decisions in allocating the testing resource. In addition, the results are compared with those of Kapur et al. (2004), Huang and Lyu (2005), and Jha et al. (2010) that are available in the literature to deal the similar problems addressed in this article. It reveals that the proposed dynamic programming method for the testing-resource allocation problem yields a gain in efficiency over other methods.

Designing stratified sampling in economic and business surveys

In most economic and business surveys, the target variables (e.g. turnover of enterprises, income of households, etc.) commonly resemble skewed distributions with many small and few large units. In such surveys, if a stratified sampling technique is used as a method of sampling and estimation, the convenient way of stratification such as the use of demographical variables (e.g. gender, socioeconomic class, geographical region, religion, ethnicity, etc.) or other natural criteria, which is widely practiced in economic surveys, may fail to form homogeneous strata and is not much useful in order to increase the precision of the estimates of variables of interest. In this paper, a stratified sampling design for economic surveys based on auxiliary information has been developed, which can be used for constructing optimum stratification and determining optimum sample allocation to maximize the precision in estimate.

Calibration estimator of population mean in stratified random sampling

Stratified Sampling is one of the most widely used sampling techniques as it increases the precision of the estimate of the survey variable. On the other hand, calibration estimation is a method of adjusting the original design weights to improve survey estimates by using auxiliary information such as the known population total (or mean) of the auxiliary variables. A calibration estimator uses calibrated weights that are determined to minimize a given distance measure to the original design weights while satisfying a set of constraints related to the auxiliary information. In this paper, a new calibration estimator of population mean in stratified sampling design is proposed, which incorporates not only the population mean but also the variance stratified mean available for the auxiliary variable. The problem of determining the optimum calibrated weights is formulated as a Nonlinear Programming Problem (NLPP) that is solved using Lagrange multiplier technique. The computational details of the procedure are illustrated in the presence of one auxiliary variable. A numerical example is presented and a simulation study is carried out to illustrate the computational details and the performance of the proposed calibration estimator. The results reveal that the proposed calibration estimator is more efficient than the other calibration estimators of the population mean.

A procedure for computing optimal stratum boundaries and sample sizes for multivariate surveys

In most surveys, the target variables (items of interest) commonly resemble right-skewed distributions where the Stratified Random Sampling technique is used as a method of sampling and estimation. The methodology of constructing strata is called stratification. Over a particular characteristic chosen as the stratification variable (such as gender, geographical region, ethnicity, or any natural criteria), the survey may fail to form homogeneous strata - this would impact the precision in the estimates of the target variables. Stratification can lead to substantial improvements in the precision of sample estimators, which not only depends on the sample size, but also on the heterogeneity among the units of the population. The principal reason for stratification in the design of sample surveys is to reduce the variance of sample estimates. Surveys normally have more than one target variable with several variables both available and desirable for stratification. Stratification in such multivariate situations has not been explored to a great deal like the univariate case and requires algorithms to determine efficient stratum boundaries. This paper takes into consideration multiple survey variables and attempts to present a computational procedure to construct optimal stratum boundaries (OSB) using Dynamic Programming (DP) technique. A numerical example to determine the OSB for two main variables under study is also presented.