Non-homogeneous Poisson Process Models Research Articles

A NHPP based software reliability model and optimal release policy with logistic–exponential test coverage under imperfect debugging

Software reliability models are used to estimate and predict the reliability, number of remaining faults, failure intensity, total software development cost, etc., of a software. Testing coverage is very important for both software developers and customers of software products. Testing converge is a measure that enables software developers to evaluate the quality of tested software and determine how much additional effort is needed to improve the reliability of the software. This paper proposes a software reliability growth model based on a non-homogeneous Poisson process (NHPP) that incorporates a logistic–exponential testing coverage function with imperfect debugging. The proposed model relates the test coverage to fault detection phenomena in debugging. Goodness-of-fit test of the proposed model is conducted using different criteria for two sets of software failure data. The proposed model is compared with other existing NHPP models. A software cost model incorporating testing coverage and an optimal release policy based on the number of remaining faults are developed.

System unavailability analysis based on window‐observed recurrent event data

Many service industries require high level of system availability to be competitive. An appropriate system unavailability metric is important for business decisions and to minimize the operation risks. In practice, a system can be unavailable for service because of multiple types of events, and the durations of these events can also vary. In addition, the data that record the system operating history often have a complicated structure. In this paper, we develop a framework for estimating system unavailability metric based on historical data of a fleet of heavy‐duty industry equipment, which we call System A. During the useful life of System A, repairs and maintenance actions are performed. However, not all repairs or maintenance actions were recorded. Specifically, the information on event times, types, and durations is available only for certain time intervals (i.e., observation windows), instead of the entire useful life span of the system. Thus, the data structure is window‐observed recurrent event with multiple event types. We use a nonhomogeneous Poisson process model with a bathtub intensity function to describe the recurrent events, and a truncated lognormal distribution to describe the event durations. We then define a conservative metric for system unavailability, obtain an estimate of this metric, and quantify the statistical uncertainty. Copyright © 2013 John Wiley & Sons, Ltd.

Different Analysis Procedures for Computing the Reliability of Repairable Systems / Różne Procedury Analizy Dla Obliczania Niezawodności Systemów Podlegających Naprawie

Abstract This paper will review the different analysis scenarios that can be used when examining the reliability of repairable systems. Five different methods will be reviewed, specifically: 1) Using the mean value of the times between failure (TBFs) at the system level. 2) Using the distributions of the components of the system and creating a reliability block diagram (RBD). 3) Using the distribution of the times between failure (TBFs) at the system level. 4) Using the system level failure data and fitting the non-homogeneous Poisson process (NHPP) model. 5) Using the system level failure data and fitting the general renewal process (GRP) model. In the paper, the five different methods will be discussed (brief background), and conclusions about the applicability and accuracy of each method will be given. In addition, this paper can be used to further understand the assumptions behind each analysis method, the data required and the type of results that can be obtained, thus assisting analysts in choosing the most appropriate method.

Analysis and Ranking of Software Reliability Models Based on Weighted Criteria Value

Many software reliability growth models (SRGMs) have been analyzed for measuring the growth of software reliability. Selection of optimal SRGMs for use in a particular case has been an area of interest for researchers in the field of software reliability. All existing methodologies use same weight for each comparison criterion. But in reality, it is the fact that all the parameters do not have the same priority in reliability measurement. Keeping this point in mind, in this paper, a computational methodology based on weighted criteria is presented to the problem of performance analysis of various non-homogenous Poisson process (NHPP) models. It is relatively simple and requires less calculation. A set of twelve comparison criteria has been formulated and each criterion has been assigned different weight to rank the software reliability growth models proposed during the past 30 years. Case study results show that the weighted criteria value method offers a very promising technique in software reliability growth models comparison.

A Systematic Review of Software Reliability and Software Cost Models

Software reliability assessment is important to evaluate and predict the reliability and performance of software system. The models applicable to the assessment of software reliability are called Software Reliability Growth Models (SRGMs). An SRGM provides a mathematical relationship between time span of testing or using the software and the cumulative number of faults detected. It is used to assess the reliability of the software during testing and operational phases. An important class of SRGM that has been widely studied is Non Homogeneous Poisson Process (NHPP). NHPP models are useful in describing failure processes, providing trends such as reliability growth and fault-content. Different NHPP models have been developed for different applications. In this paper, we described several existing software reliability growth models based on Non Homogeneous Poisson processes (NHPPs). This paper also addresses cost estimation models and cost functions that can be used to evaluate the cost of software during its development. Keywords: Software Engineering; Software Reliability; SRGM; NHPP; MVF.

Modelling of Indian Stock Prices using Nonhomogeneous Poisson Processes with Time Trends

The financial sector in India has undergone radical reforms, particularly in the stock market segment, since early 1990s. Testing duration in stock markets concerns the ability to predict the turning points of bull and bear cycles. This article study some point process models to fit the data from Indian stock market cycles. We have considered the BSE 30 (SENSEX) data from January, 1991 to August, 2012 for bull and bear markets. The duration dependence of stock market cycles can help to pinpoint the peaks and troughs in these cycles. Upon carrying out various statistical procedures and goodness of fit tests, we found that the Nonhomogeneous Poisson Process models like Power Law Process, Modulated Power Law Process, Log-linear process and other models are some of the possible alternative models to describe the data. We provide estimates, confidence interval estimates and tests of hypothesis for the parameters involved in a particular model.

A Bootstrapping Approach for Software Reliability Measurement Based on a Discretized NHPP Model

Discrete software reliability measurement has a proper characteristic for describing a software reliability growth process which depends on a unit of the software fault-detection period, such as the number of test runs, the number of executed test cases. This paper discusses discrete software reliability measurement based on a discretized nonhomogeneous Poisson process (NHPP) model. Especially, we use a bootstrapping method in our discrete software reliability measurement for discussing the statistical inference on parameters and software reliability assessment measures of our model. Finally we show numerical examples of interval estimations based on our bootstrapping method for the several software reliability assessment measures by using actual data.

Numerical method for Weibull generalized renewal process and its applications in reliability analysis of NC machine tools

In generalized renewal process (GRP) reliability analysis for repairable systems, Monte Carlo (MC) simulation method instead of numerical method is often used to estimate model parameters because of the complexity and the difficulty of developing a mathematically tractable probabilistic model. In this paper, based on the conditional Weibull distribution for repairable systems, using negative log-likelihood as an objective function and adding inequality constraints to model parameters, a nonlinear programming approach is proposed to estimate restoration factor for the Kijima type GRP model I, as well as the model II. This method minimizes the negative log-likelihood directly, and avoids solving the complex system of equations. Three real and different types of field failure data sets with time truncation for NC machine tools are analyzed by the proposed numerical method. The sampling formulas of failure times for the GRP models I and II are derived and the effectiveness of the proposed method is validated with MC simulation method. The results show that the GRP model is superior to the ordinary renewal process (ORP) and the power law non-homogeneous Poisson process (PL-NHPP) model.

A database and model to support proactive management of sediment-related sewer blockages

Due to increasing customer and political pressures, and more stringent environmental regulations, sediment and other blockage issues are now a high priority when assessing sewer system operational performance. Blockages caused by sediment deposits reduce sewer system reliability and demand remedial action at considerable operational cost. Consequently, procedures are required for identifying which parts of the sewer system are in most need of proactive removal of sediments. This paper presents an exceptionally long (7.5 years) and spatially detailed (9658 grid squares – 0.03 km2 each – covering a population of nearly 7.5 million) data set obtained from a customer complaints database in Bogotá (Colombia). The sediment-related blockage data are modelled using homogeneous and non-homogeneous Poisson process models. In most of the analysed areas the inter-arrival time between blockages can be represented by the homogeneous process, but there are a considerable number of areas (up to 34%) for which there is strong evidence of non-stationarity. In most of these cases, the mean blockage rate increases over time, signifying a continual deterioration of the system despite repairs, this being particularly marked for pipe and gully pot related blockages. The physical properties of the system (mean pipe slope, diameter and pipe length) have a clear but weak influence on observed blockage rates. The Bogotá case study illustrates the potential value of customer complaints databases and formal analysis frameworks for proactive sewerage maintenance scheduling in large cities.

Forecasting warranty claims for recently launched products

Forecasting warranty claims for recently launched products that have short histories of claim records is vitally important for manufacturers in preparing their fiscal plans. Since the amount of historical claim data for such products is not large enough, developing forecasting models with good performance has been a difficult problem.The objective of this paper is to develop an algorithm for forecasting the number of warranty claims of recently launched products. A two-phase modelling algorithm is developed: in Phase I, we estimate the upper and the lower bounds of the warranty claim rates of the reference products that have been in the market for a longer time; in Phase II, we build forecasting models for the recently launched products and assume that their future claim rates are subject to the bound constraints derived from Phase I. Based on this algorithm, we use the NHPP (non-homogeneous Poisson process) and the constrained maximum likelihood estimation to build forecasting models on artificially generated data as well as warranty claim data collected from an electronics manufacturer. The results show that the proposed algorithm outperforms commonly used NHPP models.

On the prediction of underground water pipe failures: zero inflation and pipe-specific effects

The prediction of pipe failures in water distribution systems is an essential planning tool for water companies. Previous methods focus on the prediction of either future failure numbers or aspects of pipe condition. However, most of these only predict at the level of large pipe groups (of similar characteristics) and often cannot provide uncertainty bounds. Here, a new statistical method is developed to predict the probability of failure at the single pipe level. The method extends the Non-Homogeneous Poisson Process (NHPP) in two ways: firstly, it incorporates pipe-specific random effects to account for unmeasured information on the factors affecting the pipe failures. Secondly, the method explicitly accounts for zero inflation, that is the possibility that more zero failures occur than expected from a simple Poisson assumption. This zero-inflated NHPP (ZINHPP) model was applied to two real-life datasets, one from North America and one from New Zealand. The results clearly demonstrate improved prediction capability, especially in the New Zealand data, which contain a much larger percentage of pipes with zero failures.

Modelling number, vertical distribution, and size of live branches on coniferous tree species in British Columbia

A compound, nonhomogeneous Poisson process was used to model the number, vertical distribution, and size of branches on four coniferous tree species: 134 western hemlock ( Tsuga heterophylla (Raf.) Sarg.) on six sites, 45 amabilis fir ( Abies amabilis Douglas ex J. Forbes) (three sites), 60 lodgepole pine ( Pinus contorta var. latifolia Engelm. ex S. Watson) (six sites), and 60 white spruce ( Picea glauca (Moench) Voss) trees (five sites) and two varieties: 66 coastal Douglas-fir ( Pseudotsuga menziesii var. menziesii (Mirb.) Franco) (five sites) and 50 interior Douglas-fir ( Pseudotsuga menziesii var. glauca (Mayr) Franco) (four sites). Branches of these species are typically more or less clustered and have a characteristic, nonuniform vertical distribution along annual shoots. Total number and relative positions of clusters varied with shoot age. Clustering patterns in three of four species and two varieties appeared to scale proportionally with shoot length. However, in lodgepole pine, which has fewer clusters per metre and more branches per cluster, the vertical distribution of clusters along shoots ≤5 years old was consistent with a gamma-Poisson model but converged to a nonhomogeneous Poisson process model in shoots >5 years old. Separate mixed-effect regression models were developed for each species relating length and diameter of live branches to tree (crown), shoot, and branch (cluster) predictor variables.

Exponential Regression-Based Software Reliability Model and Its Computational Aspect

An exponential regression-based model with stochastic intensity is developed to describe the software reliability growth phenomena, where the software testing metrics depend on the intensity process. For such a generalized modeling framework, the common maximum likelihood method cannot be applied any more to the parameter estimation. In this paper, we propose to use the pseudo maximum likelihood method for the parameter estimation and to seek not only the model parameters but also the software reliability measures approximately. It is shown in numerical experiments with real software fault data that the resulting software reliability models based on four parametric approximations provide the better goodness-of-fit performance than the common non-homogeneous Poisson process models without testing metric information.

Maximizing the probability of stopping on any of the last m successes in independent Bernoulli trials with random horizon

We consider the problem of maximizing the probability of stopping on any of the last m successes in independent Bernoulli trials with random horizon of length N, where m is a predetermined integer. A prior is given for N. It is known that, when N is degenerate, i.e. P{N = n} = 1 for a given n > m, the sum-the-multiplicative-odds theorem gives the solution and shows that the optimal rule is a threshold rule, i.e. it stops on the first success appearing after a given stage. However, when N is nondegenerate, the optimal rule is not necessarily a threshold rule. So our main concern in Section 2 is to give a sufficient condition for the optimal rule to be a threshold rule when N is a bounded random variable such that P{N ≤ n} = 1. Application will be made to the usual (discrete arrival time) secretary problem with a random number N of applicants in Section 3. When N is uniform or curtailed geometric, the optimal rules are shown to be threshold rules and their asymptotic results are obtained. We also examine, as a nonhomogeneous Poisson process model, an intermediate prior that allows N to be uniform or degenerate. In Section 4 we consider a continuous arrival time version of the secretary problem with a random number M of applicants. It is shown that, whatever the distribution of M, we can win with probability greater than or equal to u m *, where u m * is, as given in (1.4), the asymptotic win probability of the usual secretary problem when N degenerates to n and n → ∞.

Maximizing the probability of stopping on any of the last m successes in independent Bernoulli trials with random horizon

We consider the problem of maximizing the probability of stopping on any of the last m successes in independent Bernoulli trials with random horizon of length N, where m is a predetermined integer. A prior is given for N. It is known that, when N is degenerate, i.e. P{N = n} = 1 for a given n > m, the sum-the-multiplicative-odds theorem gives the solution and shows that the optimal rule is a threshold rule, i.e. it stops on the first success appearing after a given stage. However, when N is nondegenerate, the optimal rule is not necessarily a threshold rule. So our main concern in Section 2 is to give a sufficient condition for the optimal rule to be a threshold rule when N is a bounded random variable such that P{N ≤ n} = 1. Application will be made to the usual (discrete arrival time) secretary problem with a random number N of applicants in Section 3. When N is uniform or curtailed geometric, the optimal rules are shown to be threshold rules and their asymptotic results are obtained. We also examine, as a nonhomogeneous Poisson process model, an intermediate prior that allows N to be uniform or degenerate. In Section 4 we consider a continuous arrival time version of the secretary problem with a random number M of applicants. It is shown that, whatever the distribution of M, we can win with probability greater than or equal to um*, where um* is, as given in (1.4), the asymptotic win probability of the usual secretary problem when N degenerates to n and n → ∞.

A Bayesian Hierarchical Power Law Process Model for Multiple Repairable Systems with an Application to Supercomputer Reliability

Los Alamos National Laboratory was home to the Blue Mountain supercomputer, which at one point was the world's fastest computer. This paper presents and analyzes hardware failure data from Blue Mountain. Nonhomogeneous Poisson process models are fit to the data within a hierarchical Bayesian framework using Markov chain Monte Carlo methods. The implementation of these methods is convenient and flexible. Simulations are used to demonstrate strong frequentist properties and provide comparisons between time-truncated and failure-count designs and demonstrate the benefits of hierarchical modeling of multiple repairable systems over the modeling of such systems separately.

Estimation and Analysis of Some Generalized Multiple Change-Point Software Reliability Models

Software typically undergoes debugging during both a testing phase before product release, and an operational phase after product release. But it is noted that the fault detection and removal processes during software development and operation are different. For example, the fault removal during operation occurs generally at a slower pace than development. In this paper, we derive a powerful, easily deployable technique for software reliability prediction and assessment in the testing and operational phases. We first review how several existing software reliability growth models (SRGM) based on non- homogeneous Poisson processes (NHPP) can be readily derived from a unified theory. With the unified theory, we further incorporate the concept of multiple change-points, i.e. points in time when the software environment changes, into software reliability modeling. Several models are proposed and discussed under both ideal and imperfect debugging conditions. We estimate the parameters of the proposed models by employing real software failure data, and give a fair comparison with some existing SRGM. Numerical results show that the proposed models can provide good software reliability prediction in the various stages of software development and operation. Our approach is flexible; we can model various environments ranging from exponential-type to S-shaped NHPP models.

Reliability analysis and optimal version-updating for open source software

Context Although reliability is a major concern of most open source projects, research on this problem has attracted attention only recently. In addition, the optimal version-dating for open source software considering its special properties is not yet discussed. Objective In this paper, the reliability analysis and optimal version-updating for open source software are studied. Method A modified non-homogeneous Poisson process model is developed for open source software reliability modeling and analysis. Based on this model, optimal version-updating for open source software is investigated as well. In the decision process, the rapid release strategy and the level of reliability are the two most important factors. However, they are essentially contradicting with each other. In order to consider these two conflicting factors simultaneously, a new decision model based on multi-attribute utility theory is proposed. Results Our models are tested on the real world data sets from two famous open source projects: Apache and GNOME. It is found that traditional software reliability models provide overestimations of the reliability of open source software. In addition, the proposed decision model can help management to make a rational decision on the optimal version-updating for open source software. Conclusion Empirical results reveal that the proposed model for open source software reliability can describe the failure process more accurately. Furthermore, it can be seen that the proposed decision model can assist management to appropriately determine the optimal version-update time for open source software.

Monte Carlo exact goodness-of-fit tests for nonhomogeneous Poisson processes

Nonhomogeneous Poisson processes (NHPPs) are often used to model failure data from repairable systems, and there is thus a need to check model fit for such models. We study the problem of obtaining exact goodness-of-fit tests for parametric NHPPs. The idea is to use conditional tests given a sufficient statistic under the null hypothesis model. The tests are performed by simulating conditional samples given the sufficient statistic. Algorithms are presented for testing goodness-of-fit for the power law and the log-linear law NHPP models. It is noted that while exact algorithms for the power law case are well known in the literature, the availability of such algorithms for the log-linear case seems to be less known. A data example, as well as simulations, are considered. Copyright © 2010 John Wiley & Sons, Ltd.

Integrable difference equations for software reliability assessment and their applications

This paper discusses software reliability growth models (SRGMs) based on integrable difference equations, which are derived from the original nonhomogeneous Poisson process models by using Hirota’s bilinearization methods. After that, goodness-of-fit comparisons of our discrete models with existing deterministic discrete models are performed by using actual data sets. Further, we disucss discrete optimal software release problems under simultaneous cost and reliability requirements based on our discrete SRGMs.