A State Space Model for Software Reliability

In this paper, we propose a non-Gaussion state space model to apply in software reliability. This model assumes an exponential distribution for the failure time in every test-debugging stage, conditionally on the state parameter — the number of faults in the program. It is a generalized JM model which can be applied to the imperfect debugging situation as well as in evolving programs. By examining a set of data on evolving program failures, the effect of evolving program model is amply proved.     

一种新的基于不完全排错的最优软件发行策略

软件的最优发行管理问题是软件可靠性研究的一个关键问题.现有的最优软件发行模型大都假定软件排错过程是完全的,并且在排错过程中没有新的故障引入,这种假设在很多情况下是不合理的.本文提出了一种新的最优软件发行管理模型,该模型既考虑了软件的不完全排错过程,又考虑了在排错过程中可能会引入新的故障,同时还考虑了由于排错经验的不断积累,软件的完全排错概率会增加的情况.本文同时给出了该模型的解.     

Imperfect debugging in software reliability: A Bayesian approach

The objective of studying software reliability is to assist software engineers in understanding more of the probabilistic nature of software failures during the debugging stages and to construct reliability models. In this paper, we consider modeling of a multiplicative failure rate whose components are evolving stochastically over testing stages and discuss its Bayesian estimation. In doing so, we focus on the modeling of parameters such as the fault detection rate per fault and the number of faults. We discuss how the proposed model can account for “imperfect debugging” under certain conditions. We use actual inter-failure data to carry out inference on model parameters via Markov chain Monte Carlo methods and present additional insights from Bayesian analysis.     

NHPP software reliability model considering the uncertainty of operating environments with imperfect debugging and testing coverage

In this paper, we propose a testing-coverage software reliability model that considers not only the imperfect debugging (ID) but also the uncertainty of operating environments based on a non-homogeneous Poisson process (NHPP). Software is usually tested in a given control environment, but it may be used in different operating environments by different users, which are unknown to the developers. Many NHPP software reliability growth models (SRGMs) have been developed to estimate the software reliability measures, but most of the underlying common assumptions of these models are that the operating environment is the same as the developing environment. But in fact, due to the unpredictability of the uncertainty in the operating environments for the software, environments may considerably influence the reliability and software's performance in an unpredictable way. So when a software system works in a field environment, its reliability is usually different from the theory reliability, and also from all its similar applications in other fields. In this paper, a new model is proposed with the consideration of the fault detection rate based on the testing coverage and examined to cover ID subject to the uncertainty of operating environments. We compare the performance of the proposed model with several existing NHPP SRGMs using three sets of real software failure data based on seven criteria. Improved normalized criteria distance (NCD) method is also used to rank and select the best model in the context of a set of goodness-of-fit criteria taken all together. All results demonstrate that the new model can give a significant improved goodness-of-fit and predictive performance. Finally, the optimal software release time based on cost and reliability requirement and its sensitivity analysis are discussed.     

Enhancing software reliability modeling and prediction through the introduction of time-variable fault reduction factor

Over the past three decades, many software reliability models with different parameters, reflecting various testing characteristics, have been proposed for estimating the reliability growth of software products. We have noticed that one of the most important parameters controlling software reliability growth is the fault reduction factor (FRF) proposed by Musa. FRF is generally defined as the ratio of net fault reduction to failures experienced. During the software testing process, FRF could be influenced by many environmental factors, such as imperfect debugging, debugging time lag, etc. Thus, in this paper, we first analyze some real data to observe the trends of FRF, and consider FRF to be a time-variable function. We further study how to integrate time-variable FRF into software reliability growth modeling. Some experimental results show that the proposed models can improve the accuracy of software reliability estimation. Finally, sensitivity analyses of various optimal release times based on cost and reliability requirements are discussed. The analytic results indicate that adjusting the value of FRF may affect the release time as well as the development cost.     

A cost model for N-version programming with imperfect debugging

One of the challenging problems for software companies is to find the optimal time of release of the software so as to minimize the total cost expended on testing and potential penalty cost due to unresolved faults. If the software is for a safety critical system, then the software release time becomes more important. The criticality of a failure caused by a fault also becomes an important issue for safety critical software. In this paper we develop a total cost model based on criticality of the fault and cost of its occurrence during different phases of development for N-version programming scheme, a popular fault-tolerant architecture. The mathematical model is developed using the reliability growth model based on the non-homogeneous Poisson process. The models for optimal release time under different constraints are developed under the assumption that the debugging is imperfect and there is a penalty for late release of the software. The concept of Failure Mode Effects and Criticality Analysis is used for measuring criticality.     

An imperfect debugging model with two types of hazard rates for software reliability measurement and assessment

We propose a software reliability model which assumes that there are two types of software failures. The first type is caused by the faults latent in the system before the testing; the second type is caused by the faults regenerated randomly during the testing phase. The former and latter software failure-occurrence phenomena are described by a geometrically decreasing and a constant hazard rate, respectively. Further, this model describes the imperfect debugging environment in which the fault-correction activity corresponding to each software failure is not always performed perfectly. Defining a random variable representing the cumulative number of faults successfully corrected up to a specified time point, we use a Markov process to formulate this model. Several quantitative measures for software reliability assessment are derived from this model. Finally, numerical examples of software reliability analysis based on the actual testing data are presented.     

Markovian Imperfect Software Debugging Model and Its Performance Measures

Abstract

This paper considers a Markovian imperfect software debugging model incorporating two types of faults and derives several measures including the first passage time distribution. When a debugging process upon each failure is completed, the fault which causes the failure is either removed from the fault contents with probability p or is remained in the system with probability 1 ? p. By defining the transition probabilities for the debugging process, we derive the distribution of first passage time to a prespecified number of fault removals and evaluate the expected numbers of perfect debuggings and debugging completions up to a specified time. The availability function of a software system, which is the probability that the software is in working state at a given time, is also derived and thus, the availability and working probability of the software system are obtained. Throughout the paper, the length of debugging time is treated to be random and thus its distribution is assumed. Numerical examples are provided for illustrative purposes.     

A Markov modulated Poisson model for software reliability

In this paper, we consider a latent Markov process governing the intensity rate of a Poisson process model for software failures. The latent process enables us to infer performance of the debugging operations over time and allows us to deal with the imperfect debugging scenario. We develop the Bayesian inference for the model and also introduce a method to infer the unknown dimension of the Markov process. We illustrate the implementation of our model and the Bayesian approach by using actual software failure data.     

A discrete time model for software reliability with application to a flight control software

This work is motivated by a particular software reliability problem in a unit of flight control software developed by the Indian Space Research Organization (ISRO), in which the testing of the software is carried out in multiple batches, each consisting of several runs. As the errors are found during the runs within a batch, they are noted, but not debugged immediately; they are debugged only at the end of that particular batch of runs. In this work, we introduce a discrete time model suitable for this type of periodic debugging schedule and describe maximum likelihood estimation for the model parameters. This model is used to estimate the reliability of the software. We also develop a method to determine the additional number of error‐free test runs required for the estimated reliability to achieve a specific target with some high probability. We analyze the test data on the flight control software of ISRO. Copyright © 2011 John Wiley & Sons, Ltd.     

An inventory model with reliability in an imperfect production process

The paper analyzes an economic manufacturing quantity (EMQ) model with price and advertising demand pattern in an imperfect production process under the effect of inflation. If the machine goes through a long-run process, it may shift from in-control state to out-of-control state. As a result, the system produces imperfect items. The imperfect items are reworked at a cost to make it as new. The production of imperfect quality items increases with time. To reduce the production of the imperfect items, the systems have to more reliable and the produced items depend on the reliability of the machinery system. In this direction, the author considers that the development cost, production cost, material cost are dependent on reliability parameter. Considering reliability as a decision variable, the author constructs an integrated profit function which is maximized by control theory. A numerical example along with graphical representation and sensitivity analysis are provided to illustrate the model.     

A sequential software release policy

Most existing studies on software release policies use models based on the non-homogeneous Poisson process. In this paper, we discuss a software release policy based on a state space model. The state space model has a Gamma-Gamma type invariant conditional distribution. A cost model that removes errors in software systems and risk cost due to software failure is used. The optimal release time to minimize the expected cost in every test-debugging stage is discussed.     

股票指数的时间序列模型分析

借助于SA S软件将工程中的K a lm an滤波方法与时间序列的状态空间模型结合对上海A股指数进行了拟合与预测分析,通过对拟合与预测误差的计算可以发现这种模型是可行的;然后还把与滤波结合的状态空间模型的分析结果和常见的时间序列模型如:AR IM A模型、逐步自回归模型以及指数平滑模型的分析结果进行比较,比较的结果说明结合滤波的状态空间模型分析的结果比后三种的结果更加精确.结果为时间序列数据分析提供了一个较好的分析工具.     

Interval reliability for aggregated Markov repairable system with repair time omission

In this paper, Markov models of repairable systems with repair time omission are considered whose finite state space is grouped into two sets, the set of working states, W, and the set of failed states, F. If the system enters failed states from a working state at any instance, and sojourns at the failed states F less than a given nonnegative critical value τ, then the repair interval can be omitted from downtime records. Otherwise, If the system enters failed states from a working state at any instance, and sojourns at the failed states F more than the given nonnegative critical value τ, then the repair interval cannot be omitted from downtime records. In terms of the assumption, a new model is developed. The focus of attention is the new model’s availability, interval reliability and interval unreliability. Several results are derived for these reliability indexes for the new model. Some special cases and numerical examples are given to illustrate the results obtained by using Maple software in the paper.     

Sensitivity analysis of release time of software reliability models incorporating testing effort with multiple change-points

To accurately model software failure process with software reliability growth models, incorporating testing effort has shown to be important. In fact, testing effort allocation is also a difficult issue, and it directly affects the software release time when a reliability criteria has to be met. However, with an increasing number of parameters involved in these models, the uncertainty of parameters estimated from the failure data could greatly affect the decision. Hence, it is of importance to study the impact of these model parameters. In this paper, sensitivity of the software release time is investigated through various methods, including one-factor-at-a-time approach, design of experiments and global sensitivity analysis. It is shown that the results from the first two methods may not be accurate enough for the case of complex nonlinear model. Global sensitivity analysis performs better due to the consideration of the global parameter space. The limitations of different approaches are also discussed. Finally, to avoid further excessive adjustment of software release time, interval estimation is recommended for use and it can be obtained based on the results from global sensitivity analysis.     

Optimal maintenance scheduling for a complex manufacturing system subject to deterioration

We address the problem of determining inspection strategy and replacement policy for a deteriorating complex multi-component manufacturing system whose state is partially observable. We develop inspection and replacement scheduling models and other simple maintenance scheduling models via employing an imperfect repair model coupled with a damage process induced by operational conditions. The system state in performance of the imperfectly repaired system is modelled using a proportional intensity model incorporating a damage process and a virtual age process caused by repair. The system is monitored at periodic times and maintenance actions are carried out in response to the observed system state. Decisions to perform imperfect repair and replacement are based on the system state and crossing of a replacement threshold. The model proposed here aims at joint determination of a cost-optimal inspection and replacement policy along with an optimal level of maintenance which result in low maintenance cost and high operational performance and reliability of the system. To demonstrate the use of the model in practical applications a numerical example is provided. Solutions to optimal system parameters are obtained and the response of the model to these parameters is examined. Finally some features of the model are demonstrated. The approach presented provides a framework so that different scenario can be explored.     

A production–inventory model in an imperfect production process

The paper develops a model to determine the optimal product reliability and production rate that achieves the biggest total integrated profit for an imperfect manufacturing process. The basic assumption of the classical Economic Manufacturing Quantity (EMQ) model is that all manufacturing items are of perfect quality. The assumption is not true in practice. Most of the production system produces perfect and imperfect quality items. In some cases the imperfect quality (non conforming) items are reworked at a cost to restore its quality to the original one. Rework cost may be reduced by improvements in product reliability (i.e., decreasing in product reliability parameter). Lower value of product reliability parameter results in increase development cost of production and also smaller quantity of nonconforming products. The unit production cost is a function of product reliability parameter and production rate. As a result, higher development cost increases unit production cost. The problem of optimal planning work and rework processes belongs to the broad field of production–inventory model which deals with all kinds of reuse processes in supply chains. These processes aim to recover defective product items in such a way that they meet the quality level of ‘good item’. The benefits from imperfect quality items are: regaining the material and value added on defective items and improving the environment protection. In this point of view, a model is introduced here to guide a firm/industry in addressing variable product reliability factor, variable unit production cost and dynamic production rate for time-varying demand. The paper provides an optimal control formulation of the problem and develops necessary and sufficient conditions for optimality of the dynamic variables. In this purpose, the Euler–Lagrange method is used to obtain optimal solutions for product reliability parameter and dynamic production rate. Finally, numerical examples are given to illustrate the proposed model.     

Parametric Reduction of FE Models with Variable Mesh Topology

Designing whole machines or processes you may need both, an integrated dynamic simulation of all components on system level and a detailed analysis of how the macroscopic behavior of a component depends on geometry and material parameters. The former analysis is usually based on systems of differential algebraic equations representing a component by not more than a few hundred states and requires tools like Matlab-Simulink® or Dymola®. The latter analysis solves discretized partial differential equations with several 100,000 degrees of freedom using finite element software like Ansys® or Comsol®. Model reduction bridges the gap between the two worlds providing small state space models with approximately the same input-output behavior as the original large finite element models. Building systems from generic components, e.g. a gas transport network from pipeline models with variable length, or optimizing the design of a device with respect to mechanical or thermal properties, we need parametric reduced models. The idea is to reduce FE models offline for selected parameter sets and to generate models for new parameters by cheap interpolation rather than expensive reduction. The different approaches to parametric linear model reduction may be divided into three classes [1]. Interpolation of transfer functions is well suited for parabolic or highly damped hyperbolic problems. However, poles are duplicated rather than shifted, which is unacceptable for weakly damped hyperbolic problems like in mechanics. The second class of methods look for a basis of state space covering system behavior over the full range of parameters. They share the critical assumption that number and meaning of states do not change with the parameters. In terms of finite elements this means that the meshes for different design parameters are morphed variants of the same reference mesh.This may become a severe restriction in practice when automatic meshing is to be applied to complicated geometries. Therefore, we propose a method from the third class, which is based on interpolating reduced system matrices [2]. Only those parts of the mesh need to share a constant topology where nodal inputs are applied and outputs are collected. The inner mesh, however, may change for different parameters. The main challenges arise from the fact that state space representations of a system are unique only up to a change of basis and that interpolating matrices which refer to non-fitting bases may cause arbitrary errors. In the article we will show how problems like leaving and entering modes or eigenvalue crossing can be overcome by using normal forms and eigenvalue tracking in parameter space. The method, which is implemented in the Fraunhofer Model Reduction Toolbox, is applied to a parametric model of a mechanical device the eigenfrequencies of which have to be kept away from some dominant excitation frequencies. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一类虚拟寿命不完全维修模型的统计模拟分析

本文通过利用统计分析的方法分析了一类虚拟寿命不完全维修模型,在此类模型中,系统的报废时间由系统的虚拟寿命决定。由于在分析此类模型时,大多数情况下系统的运行时间之间相互不独立,直接导出相关的可靠性指标的计算公式比较困难,因此采用统计模拟的方法得出相关的可靠性指标。