Poisson Process Research Articles

NHPP Software Reliability Model with Rayleigh Fault Detection Rate and Optimal Release Time for Operating Environment Uncertainty

Software is used in diverse settings and depends on development and testing environments. Software development should improve the reliability, quality, cost, and stability of software, making the software testing period crucial. We proposed a software reliability model (SRM) that considers the uncertainty of software environments and the fault detection rate function as a Rayleigh distribution, with an explicit mean value function solution in the model. The goodness-of-fit of the proposed model relative to several existing nonhomogeneous Poisson process (NHPP) SRMs is presented based on three software application failure datasets. Further, a cost model is also presented that addresses the error-removal risk level and required time. The optimal testing release policy for minimizing the expected total cost (ETC) is also determined for NHPP SRMs. The impact of the software environment is studied by varying it, and the optimal release times and minimum ETCs are compared. The goodness-of-fit comparison confirmed that the proposed model has more accurate prediction values than other models. Further, whereas the existing models applied to the cost model do not change after a certain operation period, the proposed model yields changes in release time even for long operating periods.

Joint Distribution of the Supremum and the Moment of Its Attainment by a Poisson Process with Linear Drift

Joint Distribution of the Supremum and the Moment of Its Attainment by a Poisson Process with Linear Drift

Growing integer partitions with uniform marginals and the equivalence of partition ensembles

We present an explicit construction of a Markovian random growth process on integer partitions such that given it visits some level n, it passes through any partition λ of n with equal probabilities. The construction has continuous time, but we also investigate its discrete time jump chain. The jump probabilities are given by explicit but complicated expressions, so we find their asymptotic behavior as the partition becomes large. This allows us to explain how the limit shape is formed.Using the known connection of the considered probabilistic objects to Poisson point processes, we give an alternative description of the partition growth process in these terms. Then we apply the constructed growth process to find sufficient conditions for a phenomenon known as equivalence of two ensembles of random partitions for a finite number of partition characteristics. This result allows to show that counts of odd and even parts in a random partition of n are asymptotically independent as n→∞ and to find their limiting distributions, which are, somewhat surprisingly, different.

Longitudinal Fluvial Dispersion of Coarse Particles: Insights From Field Observations and Model Simulations

AbstractIn this study, we use field observations augmented with model simulations to examine gravel dispersion over nine years (2007–2015) in Halfmoon Creek. The observations of flow, entrainment, and dispersion were used to develop a forward model utilizing the Einstein‐Hubbell‐Sayre (EHS) compound Poisson process. The observed mean virtual velocity of the tracer population slows down with cumulative excess energy after the 2010 large event. The forward model deviates from the observations in representation of tails, overpredicts mean displacements, and shows a narrower spatial distribution. The heavy‐tailed resting times indicate prolonged immobilization of some grains, suggesting the preferential movement of other most mobile grains. As such, 34% of most mobile grains constitute 50% of the total entrainments. The consideration of preferential movement explains the longitudinal spread but still overpredicts the displacement after the 2010 event. The model was then explored to consider additional transport‐related mechanisms causing deviations, such as reduction in virtual velocity, entrainment probability, and morphological trapping of meander bends, which helps to adequately recreate the observed dispersive behavior. The available historical flow records used for simulating dispersive behavior over multiple decades reveal an abrupt increase in displacements for exceptionally large events, suggesting the exhumation of deeply buried grains back in transport. The simulation results highlight the need for tracer studies with large sample sizes and improved recovery rates for longer time frames experiencing floods of widely varying magnitudes. Such models, inspired by Einstein's stochastic theory can be valuable for various river research applications.

Statistical Mapping of PFOA and PFOS in Groundwater throughout the Contiguous United States.

Per-and polyfluoroalkyl substances (PFAS) are synthetic chemicals that are increasingly being detected in groundwater. The negative health consequences associated with human exposure to PFAS make it essential to quantify the distribution of PFAS in groundwater systems. Mapping PFAS distributions is particularly challenging because a national patchwork of testing and reporting requirements has resulted in sparse and spatially biased data. In this analysis, an inhomogeneous Poisson process (IPP) modeling approach is adopted from ecological statistics to continuously map PFAS distributions in groundwater across the contiguous United States. The model is trained on a unique data set of 8910 PFAS groundwater measurements, using combined concentrations of two PFAS analytes. The IPP model predictions are compared with results from random forest models to highlight the robustness of this statistical modeling approach on sparse data sets. This analysis provides a new approach to not only map PFAS contamination in groundwater but also prioritize future sampling efforts.

The last-success stopping problem with random observation times

AbstractSuppose N independent Bernoulli trials with success probabilities $$p_1, p_2,\ldots $$ p 1 , p 2 , … are observed sequentially at times of a mixed binomial process. The task is to maximise, by using a nonanticipating stopping strategy, the probability of stopping at the last success. The case $$p_k=1/k$$ p k = 1 / k has been studied by many authors as a version of the familiar best choice problem, where both N and the observation times are random. We consider a more general profile $$p_k=\theta /(\theta +k-1)$$ p k = θ / ( θ + k - 1 ) and assume that the prior distribution of N is negative binomial with shape parameter $$\nu $$ ν , so the arrivals occur at times of a mixed Poisson process. The setting with two parameters offers a high flexibility in understanding the nature of the optimal strategy, which we show is intrinsically related to monotonicity properties of the Gaussian hypergeometric function. Using this connection, we find that the myopic stopping strategy is optimal if and only if $$\nu \ge \theta $$ ν ≥ θ . Furthermore, we derive formulas to assess the winning probability and discuss limit forms of the problem for large N.

Generalized Itô’s lemma and the stochastic thermodynamics of diffusion with resetting

Abstract Methods from the theory of stochastic processes are increasingly being used to extend classical thermodynamics to mesoscopic non-equilibrium systems. One characteristic feature of these systems is that averaging the stochastic entropy with respect to an ensemble of stochastic trajectories leads to a second law of thermodynamics that quantifies the degree of departure from thermodynamic equilibrium. A well known mechanism for maintaining a diffusing particle out of thermodynamic equilibrium is stochastic resetting. In its simplest form, the position of the particle instantaneously resets to a fixed position x 0 at a sequence of times generated from a Poisson process of constant rate r. Within the context of stochastic thermodynamics, instantaneous resetting to a single point is a unidirectional process that has no time-reversed equivalent. Hence, the average rate of entropy production calculated using the Gibbs–Shannon entropy cannot be related to the degree of time-reversal symmetry breaking. The problem of unidirectionality can be avoided by considering resetting to a random position or diffusion in an intermittent confining potential. In this paper we show how stochastic entropy production along sample paths of diffusion processes with resetting can be analyzed in terms of extensions of Itô’s formula for stochastic differential equations (SDEs) that include both continuous and discrete processes. First, we use the stochastic calculus of jump-diffusion processes to calculate the rate of stochastic entropy production for instantaneous resetting, and show how previous results are recovered upon averaging over sample trajectories. Second, we formulate single-particle diffusion in a switching potential as a hybrid SDE and develop a hybrid extension of Itô’s stochastic calculus to derive a general expression for the rate of stochastic entropy production. We illustrate the theory by considering overdamped Brownian motion in an intermittent harmonic potential. Finally, we calculate the average rate of entropy production for a population of non-interacting Brownian particles moving in a common switching potential. In particular, we show that the latter induces statistical correlations between the particles, which means that the total entropy is not given by the sum of the 1-particle entropies.

Transient and Steady-State Analysis of an M/PH2/1 Queue with Catastrophes

In the paper, we consider the PH2-distribution, which is a particular case of the PH-distribution. In other words, The first service phase is exponentially distributed, and the service rate is μ. After the first service phase, the customer can to go away with probability p or continue the service with probability (1−p) and service rate μ′. We study an analysis of an M/PH2/1 queue model with catastrophes, which is regarded as a generalization of an M/M/1 queue model with catastrophes. Whenever a catastrophe happens, all customers will be cleaned up immediately, and the queuing system is empty. The customers arrive at the queuing system based on a Poisson process, and the total service duration has two phases. Transient probabilities and steady-state probabilities of this queuing system are considered using practical applications of the modified Bessel function of the first kind, the Laplace transform, and probability-generating function techniques. Moreover, some important performance measures are obtained in the system. Finally, numerical illustrations are used to discuss the system’s behavior, and conclusions and future directions of the model are given.

Choosing between additive and conventional manufacturing of spare parts: On the impact of failure rate uncertainties and the tools to reduce them

Recently, there has been great interest in using additive manufacturing (AM) to produce spare parts: allowing to produce spare parts with short lead times and close to the point of use, AM reduces the need for large inventories otherwise required by conventional manufacturing (CM) techniques to deal with intermittent spare parts demand. However, using AM to produce spare parts is limited by two main drawbacks: high production costs and uncertain failure rates arising from AM still being a relatively new production technique. While the former can be counterbalanced by inventory cost reductions, it is unclear how the latter impacts the sourcing option decision (i.e. AM or CM). We aim to fill this gap by studying a periodic review model in which spare parts demands follow a Poisson process. To make our analysis accurate and reliable, we leverage a material science approach to obtain realistic values for the failure rate uncertainties. From the results, it emerges that AM is heavily penalized by failure rate uncertainties much higher than those of CM. Consequently, we then focus on some recent tools developed to reduce AM failure rate uncertainties: porosity assessment and in-situ monitoring. We find that the failure rate uncertainty should be reduced by five and six percentage points to make it worthwhile to invest in porosity assessment and in-situ monitoring, respectively. Finally, we determine under which circumstances each tool is preferred over the other and found that porosity assessment is typically the most competitive uncertainty reduction tool.

The poisson property of extreme events in optics

Statistics of extreme events in optics, defined as above-threshold counts of an optical signal, is shown to converge, in the large-sample-size limit, to a generalized Poisson distribution whose mean is found via the exponent of the respective extreme-value distribution. Specifically, extreme-event readouts from polynomial and exponential optical nonlinearities are shown to converge in their statistics to Poisson distributions whose means are, respectively, exponential and slower-than-exponential functions of the extreme-event-counter threshold. Extreme-event counts of a phase readout, on the other hand, converge to a Poisson process whose mean is a light-tailed function of the threshold. The Poisson-limit property of extreme events in optics suggests a powerful resource for a unified treatment of a vast variety of extreme-event phenomena, ranging from optical rogue waves to laser-induced damage.

Mixed Poisson Processes with Dropout for Consumer Studies

We adapt the classical mixed Poisson process models for investigation of consumer behaviour in a situation where after a random time we can no longer identify a customer despite the customer remaining in the panel and continuing to perform buying actions. We derive explicit expressions for the distribution of the number of purchases by a random customer observed at a random subinterval for a given interval. For the estimation of parameters in the gamma–Poisson scheme, we use the estimator minimizing the Hellinger distance between the sampling and model distributions, and demonstrate that this method is almost as efficient as the maximum likelihood being much simpler. The results can be used for modelling internet user behaviour where cookies and other user identifiers naturally expire after a random time.

Testing coverage based NHPP software reliability growth modeling with testing effort and change-point

This paper investigates a software reliability growth model based on non-homogeneous Poisson process that includes coverage testing and change-points. The model integrate effort spent into a model to evaluate software reliability based on testing coverage. The amount of testing effort spent follows the Weibull distribution, while the amount of coverage is modeled by logistic, delayed S-shaped and exponential functions. Additionally, we look into the cost requirement-based software release time for exponential functions with a reliability constraint. We introduce the genetic algorithm, which is a powerful tool for dealing with search and optimization issues. For the purpose of testing the model's goodness of fit, two real failure datasets are used, and the model’s performance is analyzed based on goodness-of-fit metrics. By comparing the proposed model to perfect debugging models that have been described in the literature, it is demonstrated that the proposed model demonstrates notable advancements in this regard. The results of our analysis indicate that a software reliability estimation model should include change-points in a testing process if they exist.

Changepoint detection on daily home activity pattern: a sliced Poisson process method.

The problem of health and care of people is being revolutionized. An important component of that revolution is disease prevention and health improvement from home. A natural approach to the health problem is monitoring changes in people's behavior or activities. These changes can be indicators of potential health problems. However, due to a person's daily pattern, changes will be observed throughout each day, with, eg, an increase of events around meal times and fewer events during the night. We do not wish to detect such within-day changes but rather changes in the daily behavior pattern from one day to the next. To this end, we assume the set of event times within a given day as a single observation. We model this observation as the realization of an inhomogeneous Poisson process where the rate function can vary with the time of day. Then, we propose to detect changes in the sequence of inhomogeneous Poisson processes. This approach is appropriate for many phenomena, particularly for home activity data. Our methodology is evaluated on simulated data. Overall, our approach uses local change information to detect changes across days. At the same time, it allows us to visualize and interpret the results, changes, and trends over time, allowing the detection of potential health decline.

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.

Model for random internalization of nanoparticles by cells

We propose a stochastic model for the internalization of nanoparticles by cells formulating cellular uptake as a compound Poisson process with a random probability of success. This is an alternative approach to the one presented by Rees [] who explained overdispersion in nanoparticle uptake and associated negative binomial distribution by considering a Poisson distribution for particle arrival and a gamma-distributed cell area. In our stochastic model, the formation of new pits is represented by the Poisson process, whereas the capturing process and the population heterogeneity are described by a random Bernoulli process with a beta-distributed probability of success. The random probability of success generates ensemble-averaged conditional transition probabilities that increase with the number of newly formed pits (self-reinforcement). As a result, an ensemble-averaged nanoparticle uptake can be represented as a Polya process. We derive an explicit formula for the distribution of the random number of pits containing nanoparticles. In the limit of the fast nucleation and low probability of nanoparticle capture, we find the negative binomial distribution. Published by the American Physical Society 2024

Spatial distribution of poultry farms using point pattern modelling: A method to address livestock environmental impacts and disease transmission risks.

The distribution of farm locations and sizes is paramount to characterize patterns of disease spread. With some regions undergoing rapid intensification of livestock production, resulting in increased clustering of farms in peri-urban areas, measuring changes in the spatial distribution of farms is crucial to design effective interventions. However, those data are not available in many countries, their generation being resource-intensive. Here, we develop a farm distribution model (FDM), which allows the prediction of locations and sizes of poultry farms in countries with scarce data. The model combines (i) a Log-Gaussian Cox process model to simulate the farm distribution as a spatial Poisson point process, and (ii) a random forest model to simulate farm sizes (i.e. the number of animals per farm). Spatial predictors were used to calibrate the FDM on intensive broiler and layer farm distributions in Bangladesh, Gujarat (Indian state) and Thailand. The FDM yielded realistic farm distributions in terms of spatial clustering, farm locations and sizes, while providing insights on the factors influencing these distributions. Finally, we illustrate the relevance of modelling realistic farm distributions in the context of epidemic spread by simulating pathogen transmission on an array of spatial distributions of farms. We found that farm distributions generated from the FDM yielded spreading patterns consistent with simulations using observed data, while random point patterns underestimated the probability of large outbreaks. Indeed, spatial clustering increases vulnerability to epidemics, highlighting the need to account for it in epidemiological modelling studies. As the FDM maintains a realistic distribution of farm location and sizes, its use to inform mathematical models of disease transmission is particularly relevant for regions where these data are not available.

An emergency supply policy for an inventory replenishment model with returns and partial backorders

AbstractThis paper studies a continuous-review inventory replenishment model with a limited storage capacity S in an uncertain environment. We assume that the demands and returns follow independent Poisson processes. We further assume a ra1079 shelf life, a random lead time, and early loss. The storage is managed according to the base-stock (S, s) policy for $$s<S,S>0.$$ s < S , S > 0 . In case of overstock, each returned item exceeding S is transferred to a foreign facility. If during the lead time a demand reaches zero stock, we consider two alternatives: either allow partial backordering up to $$L_{B}$$ L B items, beyond which the unsatisfied demand is lost, or call for an immediate and costly emergency supply up to level $$0<Q_{B}^{e}\le S$$ 0 < Q B e ≤ S . Our objective is to study how the thresholds s, S, $$L_{B},$$ L B , and $$Q_{B}^{e}$$ Q B e are impacted by the system’s parameters, such as returns, demands, and costs. Using a Markovian framework, we derive the steady-state probabilities for the inventory level, and construct closed-form expressions for the average cost functions. Then, we numerically investigate the impact of the different parameters on the best policy and on the threshold levels. We compare the two alternatives and identify situations in which calling for an emergency supply is economically profitable.

Sum of defects (SOD) chart for joint monitoring of the two parameters of a zero-inflated Poisson process

Rapid technological advancements and automation in today's world have led to many high quality manufacturing processes, where most of the products are free from defects. The defects data from high quality processes are commonly modeled by twoparameter zero- inflated Poisson (ZIP) distribution. Most of the existing control charting procedures for ZIP processes are developed assuming that all the manufactured units are inspected one by one, and most often separate control charts are proposed for monitoring the two parameters of a ZIP process. However, joint monitoring of the two parameters in a single chart can offer significant operational advantages because one needs to focus on a sole chart and a single charting statistic. Again, in many manufacturing set ups 100% inspection may not always be feasible. Recently, a few one-chart schemes with group inspection of small samples are reported in literature, and all these schemes require estimation of the two parameters based on accumulated samples till the current sampling stage for computation of the monitoring statistics. This accumulation of samples introduces several limitations in these schemes. In this paper, probability mass function and the cumulative distribution function (cdf) of the sum of defects (SOD) in a sample of size are defined, and a control chart for SOD is proposed. Both the parameters of a ZIP process can be jointly monitored using the SOD chart. Again, since no accumulation of the samples is required for application of the SOD chart, it is free from all limitations of the existing one-chart schemes with group inspection. Two case studies based on past data are presented. The performance of the SOD chart is found to be very encouraging.

ELLA: Modeling Subcellular Spatial Variation of Gene Expression within Cells in High-Resolution Spatial Transcriptomics.

Spatial transcriptomic technologies are becoming increasingly high-resolution, enabling precise measurement of gene expression at the subcellular level. Here, we introduce a computational method called subcellular expression localization analysis (ELLA), for modeling the subcellular localization of mRNAs and detecting genes that display spatial variation within cells in high-resolution spatial transcriptomics. ELLA creates a unified cellular coordinate system to anchor diverse cell shapes and morphologies, utilizes a nonhomogeneous Poisson process to model spatial count data, leverages an expression gradient function to characterize subcellular expression patterns, and produces effective control of type I error and high statistical power. We illustrate the benefits of ELLA through comprehensive simulations and applications to four spatial transcriptomics datasets from distinct technologies, where ELLA not only identifies genes with distinct subcellular localization patterns but also associates these patterns with unique mRNA characteristics. Specifically, ELLA shows that genes enriched in the nucleus exhibit an abundance of long noncoding RNAs or protein-coding mRNAs, often characterized by longer gene lengths. Conversely, genes containing signal recognition peptides, encoding ribosomal proteins, or involved in membrane related activities tend to enrich in the cytoplasm or near the cellular membrane. Furthermore, ELLA reveals dynamic subcellular localization patterns during the cell cycle, with certain genes showing decreased nuclear enrichment in the G1 phase while others maintain their enrichment patterns throughout the cell cycle. Overall, ELLA represents a calibrated, powerful, robust, scalable, and versatile tool for modeling subcellular spatial expression variation across diverse high-resolution spatial transcriptomic platforms.

More than one landslide per road kilometer – surveying and modeling mass movements along the Rishikesh–Joshimath (NH-7) highway, Uttarakhand, India

Abstract. The rapidly expanding Himalayan road network connects rural mountainous regions. However, the fragility of the landscape and poor road construction practices lead to frequent mass movements alongside roads. In this study, we investigate fully or partially road-blocking landslides along National Highway (NH-7) in Uttarakhand, India, between Rishikesh and Joshimath. Based on an inventory of >300 landslides along the ∼250 km long corridor following exceptionally high rainfall during September and October 2022, we identify the main controls on the spatial occurrence of mass-movement events. Our analysis and modeling approach conceptualizes landslides as a network-attached spatial point pattern. We evaluate different gridded rainfall products and infer the controls on landslide occurrence using Bayesian analysis of an inhomogeneous Poisson process model. Our results reveal that slope, rainfall amounts, lithology and road widening are the main controls on landslide occurrence. The individual effects of aggregated lithozones are consistent with previous assessments of landslide susceptibilities of rock types in the Himalayas. Our model spatially predicts landslide occurrences and can be adapted to other rainfall scenarios, thus having potential applications for efficiently allocating efforts for road maintenance. To this end, our results highlight the vulnerability of the Himalayan road network to landslides. Climate change and increasing exposure along this pilgrimage route will likely exacerbate landslide risk along the NH-7 in the future.