A generalized JM model with applications to imperfect debugging in software reliability

For their nice mathematical properties, state space models have been widely used, especially for forecasting. Over the last decades, the study of tracking software reliability by statistical models has attracted scientists’ attention. However, most of models focus on perfect debugging although practically imperfect debugging arises everywhere. In this paper, a non-Gaussian state space model is modified to predict software failure time with imperfect debugging. In fact, this model is very flexible so that we can modify the system equation in this model to satisfy the various situations. Besides, this model is suitable for tracking software reliability, and applied to two well known datasets on software failures.     

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.     

Risk programming analysis with imperfect information

A Monte Carlo procedure is used to demonstrate the dangers of basing (farm) risk programming on only a few states of nature and to study the impact of applying alternative risk programming methods. Two risk programming formulations are considered, namely mean-variance (E,V) programming and utility efficient (UE) programming. For the particular example of a Norwegian mixed livestock and crop farm, the programming solution is unstable with few states, although the cost of picking a sub-optimal plan declines with increases in number of states. Comparing the E,V results with the UE results shows that there were few discrepancies between the two and the differences which do occur are mainly trivial, thus both methods gave unreliable results in cases with small samples.     

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.     

Transient solutions of a software model with imperfect debugging and generation of errors by two servers

In this paper, we obtain the transient solution of probabilities of error in the software, mean number of faults and the expected number of failures remaining at time t, under the assumption that the number of faults is finite, the failure rate is proportional to the number of faults present in the software at any time, debugging is imperfect and error generation will never lead the software to have infinite errors. Moreover, the software is tested by two servers with the first M errors being debugged by first server and the remaining errors (M +1 ≤n ≤N) by the second server. Also, when a failure occurs, instantaneously repair starts with the following probabilities.

• 1.(a) The fault content is reduced by one by the first (second) server with probability μ₁(μ₂),μ₂ ≥μ₁
• 2.(b) The fault content remains unchanged with probability Ψ.
• 3.(c) The fault content is increased by one by the first (second) server with probability λ₁(λ₂), λ₁ ≥λ₂ where μ₁ + Ψ + λ₁ = 1, μ₂ + Ψ + λ₂ = 1, μ₁ ⪢ Ψ ⪢ λ₁, μ₂ ⪢ Ψ ⪢ λ₂. Finally, a numerical example is presented for the transient probabilities for the number of errors in the software, mean number of faults and the expected number of failures remaining in the software.

     

A novel repair model for imperfect maintenance

^** Email: shaomin.wu{at}reading.ac.uk Commonly used repair rate models for repairable systems in thereliability literature are renewal processes, generalised renewalprocesses or non-homogeneous Poisson processes. In additionto these models, geometric processes (GP) are studied occasionally.The GP, however, can only model systems with monotonously changing(increasing, decreasing or constant) failure intensities. Thispaper deals with the reliability modelling of failure processesfor repairable systems where the failure intensity shows a bathtub-typenon-monotonic behaviour. A new stochastic process, i.e. an extendedPoisson process, is introduced in this paper. Reliability indicesare presented, and the parameters of the new process are estimated.Experimental results on a data set demonstrate the validityof the new process.     

A consumption model with separability and imperfect decision makers

A consumption model is considered, where agents may have a simplified image of reality and a simplified way of reaching decisions. Some rules of behaviour are assumed, considering, in particular, the decision-theoretic ideas of decentralization and price aggregation. The consumers have a price-independent relative income distribution. The existence and unicity of temporary equilibria are investigated.      

Linear programming model for finding optimal roadway grades that minimize earthwork cost

The cost of earthwork involved in road construction can vary widely based on the roadway grades chosen by the designer. There is almost an infinite number of feasible grades available to the designer to choose from, all of which satisfy the geometric specification of the road. For every feasible grade selected, a transportation problem must be solved, which is a very tedious task. The roadway grade selection is usually considered as a stage and the earthwork (transportation) is another stage. These two stages have always been treated separately in the literature. The model presented here combines the roadway grade selection stage and the earthwork transportation stage in a single linear programming problem, thus guaranteeing global optimality.     

A model of imperfect preventive maintenance with dependent failure modes

Consider a system subject to two modes of failures: maintainable and non-maintainable. A failure rate function is related to each failure mode. Whenever the system fails, a minimal repair is performed. Preventive maintenances are performed at integer multiples of a fixed period. The system is replaced when a fixed number of preventive maintenances have been completed. The preventive maintenance is imperfect because it reduces the failure rate of the maintainable failures but does not affect the failure rate of the non-maintainable failures. The two failure modes are dependent in the following way: after each preventive maintenance, the failure rate of the maintainable failures depends on the total of non-maintainable failures since the installation of the system. The problem is to determine an optimal length between successive preventive maintenances and the optimal number of preventive maintenances before the system replacement that minimize the expected cost rate. Optimal preventive maintenance schedules are obtained for non-decreasing failure rates and numerical examples for power law models are given.     

A compromise programming model for developing the cost of including carbon pools and flux into forest management

Annals of Operations Research - Policy makers need research based decision analysis models that include carbon sequestration and forest products in order to make policies that are both economically...     

A nonlinear programming model with implicit variables for packing ellipsoids

The problem of packing ellipsoids is considered in the present work. Usually, the computational effort associated with numerical optimization methods devoted to packing ellipsoids grows quadratically with respect to the number of ellipsoids being packed. The reason is that the number of variables and constraints of ellipsoids’ packing models is associated with the requirement that every pair of ellipsoids must not overlap. As a consequence, it is hard to solve the problem when the number of ellipsoids is large. In this paper, we present a nonlinear programming model for packing ellipsoids that contains a linear number of variables and constraints. The proposed model finds its basis in a transformation-based non-overlapping model recently introduced by Birgin et al. (J Glob Optim 65(4):709–743, 2016). For solving large-sized instances of ellipsoids’ packing problems with up to 1000 ellipsoids, a multi-start strategy that combines clever initial random guesses with a state-of-the-art (local) nonlinear optimization solver is presented. Numerical experiments show the efficiency and effectiveness of the proposed model and methodology.     

A hybrid maintenance model with imperfect inspection for a system with deterioration and Poisson failure

A new maintenance model for a system with both deterioration and Poisson failures is proposed. In this model, at any time-instant G _S and when the system is operating, one of the following decisions may be taken: (1) stop the system to perform a scheduled minimal maintenance; (2) stop the system to perform an inspection; and (3) no action and allow the system to go on with its operation. Following an inspection, based on the deterioration condition of the system, one of the following decisions may be taken: (a) if the system is in a ‘good’ condition, no maintenance action is taken and a number of periodic minimal maintenance activities are scheduled, starting T1 later; (b) if the system is in an ‘intermediate’ condition, a minimal maintenance is performed and an inspection is scheduled for T2 later (T2 < T1); and (c) if the system is in a ‘bad’ condition, a major maintenance is performed and a number of periodic minimal maintenances are scheduled, starting T1 later. In addition, a deterioration failure is restored by a major repair and a Poisson failure is restored by a minimal repair. Generalised stochastic Petri nets are used to represent and analyse the model, which represents a ‘composite’ maintenance strategy. Based on maximisation of the throughput of the system the benefit of this model compared to (1) an equivalent periodic inspection model and (2) an equivalent planned scheduled maintenance model, is demonstrated. This study presents a new hybrid model with a general framework for incorporating various types of maintenance policies. Also by incorporation of a number of features, this model will be more applicable to real world technical systems (complex systems), although it can be applied to individual components that are part of a complex system.     

A probabilistic programming model for blending aggregates

The paper introduces the formulation of a probabilistic programming model to find the optimum mix proportion of aggregates to meet the specific grading requirement in order to minimize the cost which consists of the material cost and the expected penalty cost. The model is probabilistic since the gradation, which is the major parameter, is a random variable. A linear programming model is first formulated. Using the LP solution as initial value, a direct search technique is then employed to solve the problem. The model is expected to be applicable to any problem of aggregates blending. In this paper, however, the mixing aggregates of an asphalt mixing plant is exemplified to test the applicability of the model.     

An optimal software debugging model

This paper deals with the question of optimal debugging times for large-scale computer programs, given certain assumptions concerning the distribution of error within the program, the cost that is incurred whenever a fault occurs, and the cost of debugging. Modeling takes place within the framework of independent Poisson processes describing faults caused by errors. The optimization aim is to minimize the long-run total expected discounted cost. Some general results concerning the optimal polity are presented.This research has been supported by NATO Grant No. RG.86/0024, by the Alexander von Humboldt Foundation, and by the University of Passau, Germany.     

An imperfect maintenance model with block replacements

We consider a model in which when a device fails it is either repaired to its condition prior to failure or replaced. Moreover, the device is replaced at times kT, k = 1, 2, … The decision to repair or replace the device at failure depends on the age of the device at failure. We find the optimal block time, T₀, that minimizes the long-run average cost of maintenance per unit time. Our results are shown to extend many of the well known results for block replacement policies.     

A quadratic programming model for optimal data distribution

In this paper, a quadratic programming model is developed to take into consideration a number of factors that can influence the process of optimal allocation of data among the nodes in a distributed database. The factors include communication costs, translation costs, congestion costs and storage costs. Beale's method is used to solve the resulting quadratic program. Some numerical examples are presented and the potentials of such an approach in the design and analysis of distributed databases are discussed.This work was partially supported by a grant from Natural Science and Engineering Research Council of Canada.     

A stochastic programming model for aggregate production planning

This paper looks at a problem of production planning at the level of the work centre, rather than of the individual machine. The approach is based on methodology which combines aspects of decision analysis with mathematical programming procedures.A model is developed which can take into account various real life factors in the planning phase, including delaying production, and the probability of production being scrapped, or of extra work being required. An extension to the multiple objective situation is outlined, and some computational results are reported.     

A stochastic programming model for water resource planning

A framework is offered for the evaluation of electricity generation and water supply for agricultural irrigation. This assessment is conducted through the construction of an appropriate stochastic optimization model. A recursive least squares algorithm is incorporated in the model which enables more accurate estimation of model parameters.     

A quadratic programming model for product configuration optimization

The various features of any product are differentially appealing to the various portions of the consumer population and have differential costs of production and marketing. This paper considers the problem of “overall product optimization”, or more specifically, “optimal product configuration”. Product price is, of course, a part of this configuration.     

A generalized linear programming model for nurse scheduling

This paper presents a 0–1 column generation model with a resource constrained shortest path auxiliary problem for nurse scheduling. The master problem finds a configuration of individual schedules to satisfy the demand coverage constraints while minimizing salary costs and maximizing both employee preferences and team balance. A feasible solution of the auxiliary problem is an acceptable schedule for a given nurse, with respect to collective agreement requirements such as seniority, workload, rotations and days off. We define a new resource structure in the auxiliary problem in order to take into account the complex collective agreement rules specific to the nurse scheduling problem. This model generalizes further the previous formulations discussed in the literature and can be viewed as a general scheme for complex personnel scheduling problems, especially in the context of organizations which operate around the clock. Solution methods and preliminary test results are discussed.