Homogeneous Poisson Process Research Articles

Stochastic model of seed dispersal with homogeneous and non-homogeneous Poisson processes under habitat reduction conditions.

This study presents a stochastic model of seed dispersal based on a branching random walk (BRW) framework, incorporating both homogeneous and non-homogeneous Poisson point processes (PPP). Building on the model introduced by Coletti et al. (2023), we examine the effects of habitat reduction on seed dispersal dynamics. We analyze the phase transition behavior of the BRW model under varying conditions of habitat fragmentation, focusing on how these conditions influence the critical dispersal rate. Specifically, we study a BRW on the real line with a non-homogeneous PPP driven by a log-normal density, constrained between spatial barriers. Our simulations localize the critical dispersal rate with respect to barrier positions and compare this dependence between homogeneous and non-homogeneous models.

Advantages of Accounting for Stochasticity in the Premium Process

In this paper, we study a risk model with stochastic premium income and its impact on solvency risk management. It is assumed that both the premium arrival process and the claim arrival process are modelled by homogeneous Poisson processes, and that the premium amounts are modelled by independent and identically distributed random variables. While this model has been studied in the existing literature and certain explicit results are known under more restrictive assumptions, these results are relatively difficult to apply in practice. In this paper, we investigate the factors that differentiate this model and the classical risk model. After reviewing various known results of this model, we derive a simulation approach for obtaining the probability of ultimate ruin based on importance sampling, which does not require specific distributions for the premium and the claim. We demonstrate this approach first with examples where the distribution of the sampling random variable can be identified. We then provide additional examples where we use the fast Fourier transform to obtain an approximation of the sampling random variable. The simulated results are compared with the known results for the probability of ruin. Using the simulation approach, we apply this model to a real-life auto-insurance data set. Differences with the classical model are then discussed. Finally, we comment on the suitability and impact of using this model in the context of solvency risk management.

Using the Non-Homogeneous Poisson Process (Duane's Model) to Analyze the Number of Failures in Industrial Equipment

Objective: The aim of this paper is to show the application of Duane's Chart to analyze the time between failures in a company in the South of Rio de Janeiro. Theoretical framework: A Duane chart is a scatter plot of the cumulative failure rate versus time. Use a Duane chart to do the following: •Assess whether the data follows a power-law process or a homogeneous Poisson process. •Determine whether a repairable system is improving, getting worse or remaining stable. The fitted line on the Duane chart plots the best fitted line when the power-law process assumption is valid and the shape and scale are estimated using the least squares method. The Duane plot should be approximately linear if the power-law process or homogeneous Poisson process is appropriate. A negative slope shows an improvement in reliability, a positive slope shows a deterioration in reliability and no slope (a horizontal line) shows a stable system (Zanin et al., 2010). Method: Data was taken from a company in the South of Rio de Janeiro and a case study was carried out using Duane's Chart. Final Considerations: This result indicates that Duane's Model is a very viable option when the aim is to monitor the failure rate as a function of time to check whether a system is improving or not. In this specific Case Study, it is clear that there is a significant improvement in the system over time and that the work carried out in this company is being very effective in reducing and even eliminating failures in the long term. Implications of the research: The use cases of Duane's Chart are multiplying in scientific literature and are proving to be highly effective for interpreting data in the area of reliability. Originality/value: Despite being a well-known Statistical Tool, Duane's Chart has a specific application in monitoring the number of failures over time in a system.

Copula-based Bayesian estimation of probabilities for two dependent non homogenous compound Poisson processes

Bivariate copulas for two dependent non homogeneous compound Poisson processes (NCPP) are considered. In this scenario, the arrival times of events are assumed to follow two independent non homogeneous Poisson processes (NHPP), but the corresponding renewal variables are dependent through a bivariate copula. The article aims to provide a computational tool for the estimation of the probability that the random sum associated with the first NCPP is less than the other random sum by a truncated time. The proposed ML and Bayesian estimation methods in a general setting can be applied using bivariate copulas such as Gumbel (for upper tail dependency), Clayton (for lower tail dependency), and Frank (for symmetric dependency). Power Law (PL) intensity function for the NHPPs and Inverse-Gaussian as renewal distribution are considered as examples. Bayes estimates of NHPPs parameters use gamma priors, and for Bayes estimates of Inverse-Gaussian distribution, an MCMC algorithm is employed. Our method has a broad range of practical applications, especially in the insurance sector, where the renewal variable may represent the dollar amount associated with a claim. To illustrate our approach, we analyze injury claims in North Carolina and South Carolina over a specific period, using a dataset sourced from docs.databricks.com.

Bayesian estimation of the mean time between failures of subsystems with different causes using interval‐censored system maintenance data

AbstractEnsuring an acceptable level of reliability stands as a primary imperative for any mission‐focused operation since it serves as a critical determinant of success. Inadequate reliability can lead to severe repercussions, including substantial expenses for repairs and replacements, missed opportunities, service disruptions, and in the worst cases, safety violations and human casualties. Within national defense organizations such as the USAF, the precise assessment and maintenance of system reliability play a pivotal role in ensuring the success of mission‐critical operations. In this research, our primary objective is to model the reliability of repairable subsystems within the framework of competing and complementary risks. Subsequently, we construct the overall reliability of the entire repairable system, utilizing day‐to‐day group‐censored maintenance data from two identical aircraft systems. Assuming that the lifetimes of subsystems follow non‐identical exponential distributions, it is theoretically justified that the system reliability can be modeled by homogeneous Poisson processes even though the number of subsystems of any particular type is unknown and the temporal order of multiple subsystem failures within a given time interval is uncertain due to interval censoring. Using the proposed model, we formulate the likelihood function for the mean time between failures of subsystems with different causes, and subsequently establish an inferential procedure for the model parameters. Given a considerable number of parameters to estimate, we explore the efficacy of a Bayesian approach, treating the contractor‐supplied estimates as the hyperparameters of prior distributions. This approach mitigates potential model uncertainty as well as the practical limitation of a frequentist‐based approach. It also facilitates continuous updates of the estimates as new maintenance data become available. Finally, the entire inferential procedures were implemented in Microsoft Excel so that it is easy for any reliability practitioner to use without the need to learn sophisticated programming languages. Thus, this research supports an ongoing, real‐time assessment of the overall mission reliability and helps early detection of any subsystem whose reliability is below the threshold level.

Reinforcing resilience for integrated design of green and grey infrastructure with real-time control rules by considering system failures

The mutual benefits of real-time control and green and grey infrastructure for urban drainage system (UDS) renovations have drawn great attention. However, risks emerged due to equipment and structural failures, and no studies considered such failures while optimizing the renovation scheme. To address this issue, this study proposed a multi-criteria optimization method for the integrated design of green and grey infrastructure with real-time control rules, where the economic cost and system performance under normal and exceptional conditions were optimized simultaneously. Failure probabilities of equipment and structure were quantified using homogeneous Poisson process models, and failure losses were estimated with the aid of a machine learning-based surrogate model. This approach was tested with a combined UDS in China. Results indicate that: (1) More investment does not necessarily increase system resilience to failures. Equipment and structural failures can significantly lower the effectiveness of grey infrastructure and real-time control. Therefore, solutions with more investments in grey infrastructure, which indicate higher costs, experience greater failure losses. (2) System resilience to failures can be significantly improved while maintaining other objectives when compared with the traditional design scheme. The proposed method allows for a new perspective in addition to the cost-and-resilience two-objectives design of UDS renovations, especially for systems threatened by various failures.

The Robust Classical MTBF Test

For the reliability demonstration of a repairable system, testers expediently assume that the rate of occurrence of failures (ROCOF) can be modeled by a homogenous Poisson process (HPP), also called an ordinary renewal process (ORP), and therefore the times between failure, or interarrival times, are exponentially distributed with a constant mean time between failures (MTBF). This often-used test is colloquially called the “classical MTBF test” because it has been widely used by both government and industry for the past 50 years. The test is simple; operate a system for a period and the sample estimate for MTBF is the operating time divided by the number of failures. A confidence interval is applied to the sample estimate to determine the test result. This paper considers the validity of the test result if the repairable system has an increasing or decreasing MTBF, that is, a nonhomogeneous Poisson process (NHPP). Specifically, the change in consumer’s risk is analyzed if the repairable system has a decreasing MTBF (increasing ROCOF). A simple risk mitigation strategy consisting of a burn-in period equal to the threshold MTBF is also considered and found to be effective. The classical MTBF test is shown to be a robust test and a burn-in period prior to testing makes the classical MTBF test even more robust.

Geometric random intersection graphs with general connection probabilities

Abstract Let $\mathcal{V}$ and $\mathcal{U}$ be the point sets of two independent homogeneous Poisson processes on $\mathbb{R}^d$ . A graph $\mathcal{G}_\mathcal{V}$ with vertex set $\mathcal{V}$ is constructed by first connecting pairs of points (v, u) with $v\in\mathcal{V}$ and $u\in\mathcal{U}$ independently with probability $g(v-u)$ , where g is a non-increasing radial function, and then connecting two points $v_1,v_2\in\mathcal{V}$ if and only if they have a joint neighbor $u\in\mathcal{U}$ . This gives rise to a random intersection graph on $\mathbb{R}^d$ . Local properties of the graph, including the degree distribution, are investigated and quantified in terms of the intensities of the underlying Poisson processes and the function g. Furthermore, the percolation properties of the graph are characterized and shown to differ depending on whether g has bounded or unbounded support.

Flexible Bayesian reliability demonstration testing

AbstractThe aim is to demonstrate the reliability of a population at consecutive points in time, where a sample at each current point must prove that at least 100% of the devices function until the next point with a probability of at least . To test the reliability of the population, we flexibilise standard lifetime models by allowing the unknown parameter(s) of the corresponding counting process to vary in time. At the same time, we assign a prior distribution that assumes the parameters to be constant within a certain interval. This flexibilisation has several advantages: it can be applied for all parametric lifetimes; its Markov property allows the efficient derivation of the number of defective devices, even for a large number of testing times; and the inference is less certain and hence more realistic and leads to less frequent acceptance of poor quality populations. On the other hand, the inference is stabilised by the informative prior. Based on the flexibilisation of the homogeneous Poisson process (HPP), we derive acceptance sampling plans to test the future reliability of a population. Applying the zero failure sampling plans on simulations of Weibull processes shows their good frequentist properties and their robustness. In the case of utility meters subject to German regulations (Mess‐ und Eichverordnung (MessEV). 2014: 2010–2073.), application of the derived sequential sampling plans when the conditions of these plans are met can lead to an extension of the verification validity period. These sampling plans protect the consumer better than those from an HPP and are still cost efficient.

Joint optimization of maintenance and spares inventory policy for a series-parallel system considering dependent failure processes

This paper investigates the joint optimization of condition-based opportunistic maintenance and spare inventory policies for a series-parallel system consisting of software and components. Software failure follows a homogeneous Poisson process. Components are subject to mutually dependent competing failure processes. When the software failure occurs, an opportunistic inspection will be implemented for the components. Besides the maintenance policy, the (s, S) inventory policy is applied for components. The maintenance and order decisions will be made according to the conditions of the components and inventory level at each inspection. The state transition probabilities and expected sojourn times can be obtained by the formulated semi-Markov decision process. The optimal inspection interval, the preventive replacement threshold of the component, and the order point can be derived by minimizing the expected average cost rate. Finally, the effectiveness and superiority of the proposed joint optimization model are verified by an illustrative example.

Optimizing the maintenance threshold in presence of shocks: A numerical framework for systems with non-monotonic degradation

Shocks have attracted considerable interest in reliability and maintenance engineering because of their impact on vulnerable systems. Most industrial systems suffer from both internal degradation caused by fatigue and wear-out, and external shocks that often occur randomly due to harsh weather conditions, overloading, etc. Developing maintenance optimization models without taking these stochastic shocks into account is often ineffective. This paper develops a model to optimize the maintenance alarm threshold for a single-component continuously monitored system which is exposed to both fatal and non-fatal shocks in the presence of lead time for hard time maintenance. The shocks occur randomly according to a homogeneous Poisson process during the whole degradation process and have a stochastic impact on the degradation level, while the system resistance to shocks decreases as the system approaches failure. We propose a new numerical maintenance optimization model to find the solution without Monte-Carlo simulation and the model is compared to the Wiener process. A numerical example and a real-time experimental case study on roller bearings are used to demonstrate the effectiveness of the model. The results show that the model is capable of improving maintenance decision-making in terms of failure probability and risk perspective.

Time-Dependent Probabilistic Seismic Hazard Analysis for Seismic Sequences Based on Hybrid Renewal Process Models

ABSTRACT Probabilistic seismic hazard analysis (PSHA) is a methodology with a long history and has been widely implemented. However, in the conventional PSHA and sequence-based probabilistic seismic hazard analysis (SPSHA) approaches, the occurrence of mainshocks is modeled as the homogeneous Poisson process, which is unsuitable for large earthquakes. To account for the stationary occurrence of small-to-moderate (STM) mainshocks and the nonstationary behavior of large mainshocks, we propose a time-dependent sequence-based probabilistic seismic hazard analysis (TD-SPSHA) approach by combining the time-dependent mainshock probabilistic seismic hazard analysis (TD-PSHA) and aftershock probabilistic seismic hazard analysis, consisting of four components: (1) STM mainshocks, (2) aftershocks associated with STM mainshocks, (3) large mainshocks, and (4) aftershocks associated with large mainshocks. The approach incorporates an exponential-magnitude, exponential-time model for STM mainshocks, and a renewal-time, characteristic-magnitude model for large mainshocks to assess the time-dependent hazard for mainshocks. Then nonhomogeneous Poisson process is used to model the occurrence of associated aftershocks, in which the aftershock sequences can be modeled using the Reasenberg and Jones (RJ) model or the epidemic-type aftershock sequence (ETAS) model. To demonstrate the proposed TD-SPSHA approach, a representative site of the San Andreas fault is selected as a benchmark case, for which five time-dependent recurrence models, including normal, lognormal, gamma, Weibull, and Brownian passage time (BPT) distributions, are chosen to determine the occurrence of large mainshocks. Then sensitivity tests are presented to show the effects on TD-SPSHA, including (1) time-dependent recurrence models, (2) mainshock magnitude, (3) rupture distance, (4) aftershock duration, (5) escaped time since the last event, and (6) future time interval. Furthermore, the bimodal hybrid renewal model is utilized by TD-SPSHA for another case site. The comparison results illustrate that the sequence hazard analysis approach ignoring time-varying properties of large earthquakes for long periods and the effects of associated aftershocks will result in a significantly underestimated hazard. The TD-SPSHA-based hazard curves using the ETAS model are larger than those of the RJ model. The proposed TD-SPSHA approach may be of significant interest to the field of earthquake engineering, particularly in the context of structural design or seismic risk analysis for the long term.

Reliability Analysis and Optimal Replacement Policy for Systems with Generalized Pólya Censored δ Shock Model

A fresh censored δ shock model is investigated. The arrival of random shocks follows a generalized Pólya process, and the failure mechanism of the system occurs based on the censored δ shock model. The generalized Pólya process is used for modeling because the generalized Pólya process has excellent properties, including the homogeneous Poisson process, the non-homogeneous Poisson process, and the Pólya process. Thus far, the lifetime properties of the censored δ shock model under the generalized Pólya process have not been studied. Therefore, for the established generalized Pólya censored δ shock model, the corresponding reliability function, the upper bound of the reliability function, the mean lifetime, the failure rate, and the class of life distribution are obtained. In addition, a replacement strategy N, based on the number of failures of the system, is considered using a geometric process. We determined the optimal replacement policy N* by objective function minimization. Finally, a numerical example is presented to verify the rationality of the model.

Mixed Poisson process with Stacy mixing variable

Stacy distribution defined for the first time in 1961 provides a flexible framework for modeling of a wide range of real-life behaviors. It appears under different names in the scientific literature and contains many useful particular cases. Homogeneous Poisson processes are appropriate apriori models for the number of renewals up to a given time t > 0. This article mixes them and considers a mixed Poisson process with Stacy mixing variable. We call it a Poisson-Stacy process. The resulting counting process is one of the Generalized Negative Binomial processes, and the distribution of its time-intersections are very-well investigated in the scientific literature. Here we define and investigate their joint probability distributions. Then, the corresponding mixed renewal process is investigated and Exp-Stacy and Erlang-Stacy distributions are defined and partially studied. The article finishes with a simulation study of these stochastic processes. Some plots of the probability density functions, probability mass functions, mean square regressions, and sample paths are drawn together with the corresponding code for the simulations.

A new frailty-based GEE approach of the informatively case K interval-censored failure time data

Interval-censored failure time data are commonly encountered in many fields and some methods and models have been proposed for them in the literature. In the real situations, interval censoring is informative. For this problem, we propose a new frailty-based generalized estimating equation (GEE) method of the proportional hazards (PH) model with the informatively case K interval-censored failure time data. In this article, a frailty-based generalized estimating equation is developed that can yield an unbiased estimator of the cumulative hazards function. The observation process of the proposed method does not need a non homogeneous Poisson process used in many existing literatures. The asymptotic properties of the proposed method are established, and an extensive simulation study is conducted to assess the performance of the proposed procedure. In addition, an application to an AIDS clinical trial data set is presented.

Asymptotics for sum-ruin probabilities of a bidimensional risk model with heavy-tailed claims and stochastic returns

This paper considers a bidimensional risk model with geometric Lévy price processes and dependent heavy-tailed claims, where the two claim-number processes generated by the two different lines of business are almost arbitrarily dependent. When the distributions of the claims are subexponential with a positive lower Matuszewska index, an asymptotic formula for the finite-time sum-ruin probability is derived, which has a more transparent form when the distributions of the claims are regularly-varying-tailed and the two claim-number processes are homogeneous Poisson processes. Some simulation studies are conducted to verify the accuracy and sensitivity of the asymptotic result by employing the crude Monte Carlo method.

Method for Evaluating the Reliability and Competitive Failure of Wind Turbine Gearbox Bearings Considering the Fault Incubation Point

Aiming to resolve the problem where the reliability of gearbox bearings of wind turbines is easily affected by random impact, this paper puts forward a reliability evaluation method for the competitive failure of gearbox bearings of wind turbines considering the fault incubation point. Firstly, we use Weibull distribution to simulate the fault latent process of wind turbine gearbox bearings. Secondly, a natural degradation model of gearbox bearing is established based on the Wiener process. Then, we model the random impact arrival frequency and impact intensity through a homogeneous Poisson process and normal distribution, respectively. Finally, based on considering the fault incubation point, the natural degradation of the bearing, the instantaneous degradation caused by impact, and the decline in the impact resistance of the bearing, a reliability evaluation model of gearbox bearings of wind turbines is established. A high-speed bearing of a gearbox from a wind farm in northern China is selected for simulation analysis. The results show that the proposed method can better describe the reliability decline process of the gearbox bearings of wind turbines, which has a specific guiding significance for the maintenance of wind turbines.

Intra-hour photovoltaic forecasting through a time-varying Markov switching model

This work presents a Markov switching model with a time-varying transition matrix to forecast intra-hour photovoltaic (PV) power output, aiming at providing forecasting flexibility. First, the proposed methodology captures images of the sky employing a self-made, low-cost all-sky imager. Second, the goal is to limit exposure problems in those images via the exposure fusion technique. Third, the proposed algorithm identifies groups of pixels forming clouds through a super paramagnetic clustering algorithm. Finally, we model the problem with a homogeneous Poisson process and forecast the cloud location and the shadowed area on a PV plant for the coming minutes. The shadowed area together with meteorological data are the inputs to this model. In the case study, our approach shows better performance than the persistence method, in particular for changing cloud conditions.

Random motions in [formula omitted] with orthogonal directions

This paper is devoted to the detailed analysis of three-dimensional motions in R3 with orthogonal directions switching at Poisson times and moving with constant speed c>0. The study of the random position at an arbitrary time t>0 on the surface of the support, forming an octahedron Sct, is completely carried out on the edges and faces (Fct). In particular, the motion on the faces Fct is analyzed by means of a transformation which reduces it to a three-directions planar random motion. This permits us to obtain an integral representation on Fct in terms of integral of products of first order Bessel functions. The investigation of the distribution of the position p=p(t,x,y,z) inside Sct implied the derivation of a sixth-order partial differential equation governing p (expressed in terms of the products of three D’Alembert operators). A number of results, also in explicit form, concern the time spent on each direction and the position reached by each coordinate as the motion develops. The analysis is carried out when the incoming direction is orthogonal to the ongoing one and also when all directions can be uniformly chosen at each Poisson event. If the switches are governed by a homogeneous Poisson process many explicit results are obtained.

Optimal condition‐based maintenance policy for systems with mutually dependent competing failure

AbstractIn this paper, a novel condition‐based maintenance (CBM) policy for systems with mutually dependent competing failure processes (DCFPs) is studied. The cumulative random shocks contribute to the increased rate of degradation, meanwhile, the hard failure threshold is reduced when the amount of degradation reaches a threshold. The degradation process is simulated by a non‐stationary Gamma process and random shocks are modeled as a homogeneous Poisson process. Preventive replacement (PR) and corrective replacement (CR) are accounted for in the developed maintenance strategy. The optimal periodic inspection interval, PR threshold, and the number of cumulative random shocks can be derived by minimizing the average long‐term maintenance cost rate. In the end, a numerical case is applied to demonstrate the availability of the developed models.