Homogeneous Poisson Process Research Articles

Extreme Atlantic Hurricane Probability of Occurrence Through the Metastatistical Extreme Value Distribution

AbstractThe estimation of extreme hurricane probability is hampered by small samples and by limitations in our models of extreme tropical storms. Current best estimates of extreme hurricane probability based on the traditional extreme value theory assume hurricane arrivals to be a homogeneous Poisson process. We reformulate here, for application to Atlantic hurricanes, the Metastatistical Extreme Value Distribution (MEVD) that relaxes this key assumption in the traditional extreme value theory and uses all available observations. The MEVD application to the 1886–2017 observational record improves the accuracy of extreme Atlantic hurricane estimation by about 50% with respect to current best estimates. The MEVD allows trends to emerge above estimation uncertainty: A sliding‐window analysis shows a significant increase in extreme hurricane likelihood in the past century. This trend, previously uncertain and controversial, remains significant after correcting for hurricane underdetection in the presatellite era and when focusing only on the modern observational period (1970–2017).

Context-Aware Quantification for VANET Security: A Markov Chain-Based Scheme

Recently, the quantification of VANET security has drawn significant attention due to the lack of standard computational metrics. The salient features of VANET, such as highly dynamic connections, sensitive information sharing, and unreliable fading channels, make the security quantification challenging. Accurate measurement for VANET security depends on the sufficient understanding of “context”, or making sense of the states, environment, or situation. This article proposes a context-aware security quantification scheme for VANET based on the Markov chain. Firstly, a Homogeneous Continuous-Time Markov Chain (HCTMC)-based security state model is designed for VANET. The value of each state of the HCTMC is determined with a value function that incorporates the security strength of transmitted data, dynamic and randomness of the vehicular channel, and transmission delay of the current situated environment of VANETs. Finally, the state transition matrix is derived based on the Homogeneous Discrete-Time Markov Chain (HDTMC) and Homogeneous Poisson Process (HPP). Simulation results show that the security quantification method enables the VANET’s system to adopt context-aware defense strategies according to the situated environment.

Predict new cases of the coronavirus 19; in Michigan, U.S.A. or other countries using Crow-AMSAA method.

Statistical predictions are useful to predict events based on statistical models. The data is useful to determine outcomes based on inputs and calculations. The Crow-AMSAA method will be explored to predict new cases of Coronavirus 19 (COVID19). This method is currently used within engineering reliability design to predict failures and evaluate the reliability growth. The author intents to use this model to predict the COVID19 cases by using daily reported data from Michigan, New York City, U.S.A and other countries.The piece wise Crow-AMSAA (CA) model fits the data very well for the infected cases and deaths at different phases during the start of the COVID19 outbreak. The slope β of the Crow-AMSAA line indicates the speed of the transmission or death rate. The traditional epidemiological model is based on the exponential distribution, but the Crow-AMSAA is the Non Homogeneous Poisson Process (NHPP) which can be used to modeling the complex problem like COVID19, especially when the various mitigation strategies such as social distance, isolation and locking down were implemented by the government at different places.

Performance Analysis and Evaluation of Ternary Optical Computer Based on a Queueing System With Synchronous Multi-Vacations

The ternary optical computer (TOC) has attracted increasing attention from its providers and potential customers because of the advantages of its optical processor, such as low power consumption, numerous trits, parallelism, dynamical reconfigurability and bitwise allocability. The analysis of its performance has become an urgent problem to be solved in recent years. This paper builds a four-stage TOC service model by introducing synchronous multi-vacations and tandem queueing. Here, a vacation refers specifically to an optical processor vacation. Additionally, we propose a processor-divided-equally(PDE) strategy and a task scheduling and optical processor allocation algorithm under this strategy. Assuming that the intervals of task arrival follow a homogeneous Poisson process, we obtain some important performance indicators, such as the mean response time, mean number of tasks, and optical processor utilization under the PDE strategy, which are based on the M/M/1 and M/M/n queueing systems with exhaustive service and synchronous multi-vacations. The numerical results illustrate that the number of small optical processors has an important effect on the performance of TOC and that high vacation rate can improve the system performance to some extent.

An Approach to Obsolescence Forecasting based on Hidden Markov Model and Compound Poisson Process

The popularity of electronic devices has sparked research to implement components that can achieve better performance and scalability. However, companies face significant challenges when they use systems with a long-life cycle, such as in avionics, which leads to obsolescence problems. Obsolescence can be driven by many factors, primary among which could be the rapid development of technologies that lead to a short life cycle of parts. Moreover, obsolescence problems can prove costly in terms of intermittent stock availability and unmet demand. Therefore, obsolescence forecasting appears to be one of the most efficient solutions. This paper presents a review of gaps in the actual approaches and proposes a method that can better forecast the product life cycle. The proposed approach will help companies to improve obsolescence forecasting and reduce its impact in the supply chain. The method introduces a stochastic approach to estimate the obsolescence life cycle through simulation of demand data using Markov chain and homogeneous compound Poisson process. This approach uses multiple states of the life cycle curve based on the change in demand rate and introduces hidden Markov theory to estimate the model parameters. Numerical results are provided to validate the proposed method. To examine the accuracy of this approach, the standard deviation (STD) of obsolescence time is calculated. The results showed that the life cycle curves of parts can be predicted with high accuracy.

Software Reliability-Combination of Models

Three software reliability growth models (SRGMs) - one proposed by the authors and two existing in literature based on non homogeneous Poisson Process (NHPP) are considered. Combinations of the three models are suggested as super models to measure software reliability. A comparative study of the suggested models is made with reference to three criteria as applied to eight different data sets and is noticed that the suggested model has a good contribution in the combination to come out as a better SRGM model.

Distribution of Distances between Elements in a Compact Set

In this article, we propose a review of studies evaluating the distribution of distances between elements of a random set independently and uniformly distributed over a region of space in a normed R -vector space (for example, point events generated by a homogeneous Poisson process in a compact set). The distribution of distances between individuals is present in many situations when interaction depends on distance and concerns many disciplines, such as statistical physics, biology, ecology, geography, networking, etc. After reviewing the solutions proposed in the literature, we present a modern, general and unified resolution method using convolution of random vectors. We apply this method to typical compact sets: segments, rectangles, disks, spheres and hyperspheres. We show, for example, that in a hypersphere the distribution of distances has a typical shape and is polynomial for odd dimensions. We also present various applications of these results and we show, for example, that variance of distances in a hypersphere tends to zero when space dimension increases.

A predictive Bayesian approach to sequential time‐between‐events monitoring

AbstractA fundamental problem with all process monitoring techniques is the requirement of a large Phase‐I data set to establish control limits and overcome estimation error. This assumption of having a large Phase‐I data set is very restrictive and often problematic, especially when the sampling is expensive or not available, eg, time‐between‐events (TBE) settings. Moreover, with the advancement in technology, quality practitioners are now more interested in online process monitoring. Therefore, the Bayesian methodology not only provides a natural solution for sequential and adaptive learning but also addresses the problem of a large Phase‐I data set for setting up a monitoring structure. In this study, we propose Bayesian control charts for TBE assuming homogenous Poisson process. In particular, a predictive approach is adopted to introduce predictive limit control charts. Beside the Bayesian predictive Shewhart charts with dynamic control limits, a comparison of the frequentist sequential charts, designed by using unbiased and biased estimator of the process parameter, is also a part of the present study. To assess the predictive TBE chart performance in the presence of practitioner‐to‐practitioner variability, we use the average of the average run length (AARL) and the standard deviation of the in‐control run length (SDARL).

Random forests for homogeneous and non-homogeneous Poisson processes with excess zeros.

We propose a general hurdle methodology to model a response from a homogeneous or a non-homogeneous Poisson process with excess zeros, based on two forests. The first forest in the two parts model is used to estimate the probability of having a zero. The second forest is used to estimate the Poisson parameter(s), using only the observations with at least one event. To build the trees in the second forest, we propose specialized splitting criteria derived from the zero truncated homogeneous and non-homogeneous Poisson likelihood. The particular case of a homogeneous process is investigated in details to stress out the advantages of the proposed method over the existing ones. Simulation studies show that the proposed methods perform well in hurdle (zero-altered) and zero-inflated settings, for both homogeneous and non-homogeneous processes. We illustrate the use of the new method with real data on the demand for medical care by the elderly.

Ecological drivers of tree assemblage in tropical, subtropical and subalpine forests

AbstractAimsIt is widely accepted that deterministic and neutral dispersal processes are two of the principal mechanisms driving community assembly. However, the relative importance of these ecological drivers between distinct life stages both at community and individual species levels remains poorly understood in different vegetation types. Here, we address the following questions: (a) is there a ubiquitous process shaping tree assemblage; (b) how does the relative importance of environmental and dispersal processes vary through ontogeny in different vegetation types; and (c) will the spatial patterns of trees at community and individual species levels reflect similar contribution between ecological processes?LocationYunnan province, southwest China.MethodsWe used the Homogeneous Poisson, Homogeneous Thomas, Inhomogeneous Poisson, and Inhomogeneous Thomas point process models to predict the effect of stochastic, dispersal and/or environmental processes, respectively, on the distribution of trees across ontogeny in tropical, subtropical and subalpine forests. To evaluate the relative importance of models, we compared the observed and simulated patterns of the species–area relationship and the pair correlation function g(r) at community and species level, respectively.ResultsThe Homogeneous Thomas model was the model with the lowest Akaike information criterion (AIC) scores across ontogeny and forest types at community level. At the species level, however, Homogeneous Thomas, and Inhomogeneous Thomas predicted the distribution of large and small trees, respectively, with lower AIC values in two of our three plots. In the third plot, the Homogeneous Thomas model is dominant across ontogeny at the species level.ConclusionsTree communities are assembled mainly by the dispersal process at community level. The relative importance of ecological processes for species distribution varies across life stages at the species level, suggesting that there is an ontogenetic shift of ecological processes in shaping tree distribution. Using different summary statistics at both community and species levels helps to discern the different spatial structures of species and increases our appreciation of the processes underpinning community assembly.

Pulsar Glitch Activity as a State-dependent Poisson Process: Parameter Estimation and Epoch Prediction

Abstract Rotational glitches in some rotation-powered pulsars display power-law size and exponential waiting-time distributions. These statistics are consistent with a state-dependent Poisson process, where the glitch rate is an increasing function of a global stress variable (e.g., crust-superfluid angular velocity lag), diverges at a threshold stress, increases smoothly while the star spins down, and decreases step-wise at each glitch. A minimal, seven-parameter, maximum likelihood model is calculated for PSR J1740−3015, PSR J0534+2200, and PSR J0631+1036, the three objects with the largest samples whose glitch activity is Poisson-like. The estimated parameters have theoretically reasonable values and contain useful information about the glitch microphysics. It is shown that the maximum likelihood, state-dependent Poisson model is a marginally (23%–27%) better post factum “predictor” of historical glitch epochs than a homogeneous Poisson process for PSR J1740−3015 and PSR J0631+1036 and a comparable predictor for PSR J0534+2200. Monte Carlo simulations imply that ≳50 glitches are needed to test reliably whether one model outperforms the other. It is predicted that the next glitch will occur at Modified Julian Date (MJD) 57,784 ± 256.8, 60,713 ± 1935, and 57,406 ± 1444 for the above three objects, respectively. The analysis does not apply to quasiperiodic glitchers like PSR J0537−6910 and PSR J0835−4510, which are not described accurately by the state-dependent Poisson model in its original form.

Construction of optimal reliability test plans for binary type multi-state strongly coherent systems

The concept of Binary Type Multi-State Strongly Coherent System (BTMSCS) or Multi-State Coherent System of Type 2 (MCS of type 2) has been defined by Natvig (1982). In this research article component based optimal reliability test plans for BTMSCS have been proposed. Here it is assumed that these systems are made up of n components, and the random variables representing lifetimes of systems and their components take values in set The construction of reliability test plans for BTMSCS is based on the use of (i) multivariate counting processes with marginal counting processes being homogeneous Poisson processes and (ii) positively upper orthant dependent property of the associated renewal processes. The applicability of the proposed reliability test plans has been demonstrated using all the three BTMSCSs made up of two components and one system consisting of three components with under the assumption that a bivariate random vector representing times spent by the ith component in states 1 and 2, follows Farlie-Gumbel-Morgenstern distribution with exponential marginals.

Spatial and Temporal Corrosion Degradation Modelling with A Lévy Process based on ILI (In-Line) Inspections

Corrosion defects affect the structural integrity of onshore pipelines making them prone to a Loss of Containment (LOC). LOCs may trigger significant consequences over the surrounding people and environment. Therefore, In-Line (ILI) inspections are commonly implemented to measure indicators of the metal loss along the pipe due to corrosion to support future intervention decisions. However, ILI tools are subject to detection uncertainties that hide the real number of defects because of the accuracy of the technique and the complexity of the spatially distributed corrosion degradation. This paper presents a framework in which a corroding pipeline is assessed spatially and temporally based on ILI measurements. This framework includes new defects generated over time, which are clustered with those already detected, degraded, and assessed regarding the reliability. To this end, the number of new defects is estimated with a Homogeneous Poisson Process. The defects are clustered with the DNV RP-F101 criterion, and their degradation is predicted using Levy processes. Finally, reliability is assessed temporally and spatially using Monte Carlo simulations based on a dynamic segmentation and a failure region. Following a real case study, the approach results allow us to identify critical segments of the pipeline.

LARGE DEVIATIONS FOR THE LONGEST GAP IN POISSON PROCESSES

The longest gap $L(t)$ up to time $t$ in a homogeneous Poisson process is the maximal time subinterval between epochs of arrival times up to time $t$; it has applications in the theory of reliability. We study the Laplace transform asymptotics for $L(t)$ as $t\rightarrow \infty$ and derive two natural and different large-deviation principles for $L(t)$ with two distinct rate functions and speeds.

First displacement time of a tagged particle in a stochastic cluster in a simple exclusion process with random slow bonds

AbstractWe consider the nearest-neighbour simple exclusion process on the one-dimensional discrete torus $\mathbb{T}_N=\mathbb{Z}/N\mathbb{Z}$ , with random rates $c_N=\{c_{x,N}\colon x \in \mathbb{T}_N\}$ defined in terms of a homogeneous Poisson process on $\mathbb{R}$ with intensity $\lambda$ . Given a realization of the Poisson process, the jump rate along the edge $\{x,x+1\}$ is 1 if there is not any Poisson mark in $ (x,x+1) $ ; otherwise, it is $\lambda/N,\, \lambda \in( 0,1]$ . The density profile of this process with initial measure associated to an initial profile $\rho_0\colon \mathbb{R} \rightarrow [0,1]$ , evolves as the solution of a bounded diffusion random equation. This result follows from an appropriate quenched hydrodynamic limit. If $\lambda=1$ then $\rho$ is discontinuous at each Poisson mark with passage through the slow bonds, otherwise the conductance at the slow bonds decreases meaning no passage through the slow bonds in the continuum. The main results are concerned with upper and lower quenched and annealed bounds of $T_j$ , where $T_j$ is the first displacement time of a tagged particle in a stochastic cluster of size j (the cluster is defined via specific macroscopic density profiles). It is possible to observe that when time t grows, then $\mathbb{P}\{T_j \geq t\}$ decays quadratically in both the upper and lower bounds, and falls as slow as the presence of more Poisson marks neighbouring the tagged particle, as expected.

On stochastic comparisons and ageing properties of multivariate proportional hazard rate mixtures

In this paper, we study mixtures of the multivariate proportional hazard model, where the frailty (an unobservable non negative random variable) acts on the hazard rates of lifetimes at the same time to model common risk factors. We investigate ageing properties and dependence between components for the mixture model under study. In addition, we carry out stochastic comparisons assuming dependence between components of the baseline random vector. More specifically, these stochastic comparisons refer to mixture models that share the same frailty random variable but different baseline random vectors. The results are applied for inter epoch times of a non homogeneous mixed Poisson process.

Generalized multi‐release modelling of software reliability growth models from the perspective of two types of imperfect debugging and change point

AbstractOwing to release of software in multiple releases, code changes take place in software. Because of this added complexity in software, the testing team may be unable to correct the fault upon detection, leaving the actual fault to reside in the software, termed as imperfect debugging or there may be replacement of original fault by other fault, leading to error generation. Many other factors exist that affect the testing phase of software like strategies of testing, test cases, skill, efficiency, and learning of testing team. All these factors cannot be kept stable during the whole process of testing. They may change at any time moment causing the background processes to experience change, which is known as change‐point. Keeping all these critical testing environment factors under consideration, a new software reliability growth model has been proposed, which is derived from an non homogenous Poisson process (NHPP)based unified scheme for multi‐release two‐stage fault detection/observation and correction/removal software reliability models. The developed model is numerically illustrated on tandem data set for four releases.

Generalized software fault detection and correction modeling framework through imperfect debugging, error generation and change point

In this paper, a non homogeneous Poisson process based unified scheme for two stage fault detection and fault correction software reliability model is proposed under realistic environment of imperfect debugging, error generation and change-point. Due to complexity of faults, the testers may be unable to correct the fault upon detection leaving the actual fault to dwell in the software, termed as imperfect debugging or there may be replacement of original fault by other fault, leading to error generation. Also, the factors like test cases, skill, efficiency, learning etc. may not remain static during the whole phase of testing. They may change at any time moment causing the various parameters to change known as change-point. Here, fault detection and correction are taken as two stage processes because it is quite unrealistic to say that fault correction immediately follows detection without having a time lag between the two. The developed model is numerically illustrated on tandem data set and compared with existing model. It is concluded that the results obtained here are much better than the existing one.

Generalized Earthquake Counting Processes for Sequence‐Based Hazard

AbstractSequence‐based probabilistic seismic hazard analysis (SPSHA) allows us to account for the effect of aftershocks in the assessment of seismic structural‐design actions (Iervolino et al., 2014, 2018). In fact, it generalizes classical probabilistic seismic hazard analysis (PSHA; Cornell, 1968), combining it with aftershock‐PSHA (Yeo and Cornell, 2009). SPSHA associates in time aftershocks to mainshocks and, therefore, retains a desirable property of classical PSHA; that is, events (earthquakes in PSHA and mainshock–aftershock sequences in SPSHA) occur according to homogeneous Poisson processes (HPPs). Nevertheless, the number of earthquakes in SPSHA is not Poisson‐distributed. This is addressed herein, in which the probability distribution is formulated and discussed for the following random variables: (1) the count of all earthquakes pertaining to sequences originating in any time interval; (2) the count of all earthquakes occurring in any time interval; (3) the count of all earthquakes that cause exceedance of an arbitrary ground‐motion intensity threshold at the site of interest, generated by sequences originating in any time interval. An application referring to central Italy is also developed to help the discussion. The three main findings are that: (1) the formulated SPSHA counting processes further generalize PSHA; that is, they degenerate in the corresponding mainshock HPPs, if aftershocks are neglected; (2) to associate the aftershocks to the corresponding mainshocks in time is fit for hazard assessment purposes; and (3) the variance‐to‐mean ratio of the counting distributions is significantly larger than one; consequently, the occurrence processes cannot be approximated by Poisson processes. These results, which complete the SPSHA framework, can be a reference for model calibration exercises when SPSHA is computed via simulation and in those cases in which the probability of an exact number of exceedances is of interest, rather than that of observing at least one exceedance (e.g., for seismic damage accumulation studies).

An intermittent sensor versus a target that emits glimpses as a homogenous Poisson process

We consider a sensor that alternates randomly between working and broken versus a target that reluctantly gives away glimpses as a homogenous Poisson process. Over any interval of time, the sensor has a probability of detecting n glimpses, of detecting the kth glimpse, and of detecting the kth glimpse when there are n glimpses in that interval. The article provides closed-form approximations to the distributions for those probabilities, proves that the approximations become perfect as the time interval becomes infinitely long (asymptotic distributions, pointwise convergence), and compares the approximations with empirical results obtained from simulations.