Poisson Process Research Articles

Regression to the mean for overdispersed count data

In repeated measurements, regression to the mean (RTM) is a tendency of subjects with observed extreme values to move closer to the mean when measured a second time. Not accounting for RTM could lead to incorrect decisions such as when observed natural variation is incorrectly attributed to the effect of a treatment/intervention. A strategy for addressing RTM is to decompose the total effect, the expected difference in paired random variables conditional on the first being in the tail of its distribution, into regression to the mean and unbiased treatment effects. The unbiased treatment effect can then be estimated by subtraction. Formulae are available in the literature to quantify RTM for Poisson distributed data which are constrained by mean–variance equivalence, although there are many real life examples of overdispersed count data that are not well approximated by the Poisson. The negative binomial can be considered an explicit overdispersed Poisson process where the Poisson intensity is chosen from a gamma distribution. In this study, the truncated bivariate negative binomial distribution is used to decompose the total effect formulae into RTM and treatment effects. Maximum likelihood estimators (MLE) and method of moments estimators are developed for the total, RTM, and treatment effects. A simulation study is carried out to investigate the properties of the estimators and compare them with those developed under the assumption of the Poisson process. Data on the incidence of dengue cases reported from 2007 to 2017 are used to estimate the total, RTM, and treatment effects.

Pricing Asian options under the mixed fractional Brownian motion with jumps

The mixed fractional Brownian motion (mfBm) has gained popularity in finance because it can effectively model long-range dependence, self-similarity, and is arbitrage-free. This paper focuses on mfBm with jumps modeled by the Poisson process and derives an analytical formula for valuing geometric Asian options. Additionally, approximate closed-form solutions for pricing arithmetic Asian options and arithmetic Asian power options are obtained. Numerical examples are provided to demonstrate the accuracy of these formulas, which rely on a convenient approximation of the option strike price. The proposed approximation demonstrates significantly higher computational efficiency compared to Monte Carlo simulation.

Critical Markov branching process with infinite variance allowing Poisson immigration with increasing intensity

The article studies a single-type critical Markov branching process with infinite variance of the offspring distribution. The process admits also an immigration component at the time points of a non-homogeneous Poisson process. If additionally the mean number of immigrants is infinite, then proper limit distributions are obtained, under suitable normalization of the sample paths, depending on the rate of change of the intensity of the Poisson process.

A spontaneous spread of black locust (Robinia pseudoacacia L.): the importance of seed and vegetative reproduction

Abstract The importance of seed and vegetative propagation for the spontaneous expansion of black locust on abandoned agricultural land was evaluated in the present study. The dynamics of expansion was reconstructed on the basis of spatial distribution data on the age of individuals of this species. A non-homogeneous Poisson process with a linear trend and a Widom-Rowlinson model were the most appropriate in explaining the spatial distribution of R. pseudoacacia individuals. The negative linear trend was statistically significant for the vast majority of polygons in the area of spontaneous expansion of R. pseudoacacia, but insignificant in the artificial plantation. The model parameter η indicates the aggregated spatial distribution of R. pseudoacacia in the zone of spontaneous spread due to the vegetative mechanism of spread. The distribution of R. pseudoacacia in the artificial tree plantation was close to regular, or either random or aggregated. In the early stages, the seed reproduction is the most likely process of R. pseudoacacia spreading. The seed mechanism of spreading follows a spatial pattern, which is explained by a linear trend. The next stage is a combination of both seed and vegetative mechanisms of spread, which also lasts about 4–5 years. At the last stage, the spread occurs mainly through the vegetative mechanism, and the rate of spread of the community slows down significantly. The trigger for the spread is anthropogenic impact.

Exploring non-Markovian dynamics: Stochastic analysis of a single-server retrial queue with recurrent clients and balking behavior under extended Bernoulli vacations

The present study delves into the dynamics of a specific form of queueing system described as an M/G/1 retrial queue. Here, the queue comprises two distinct categories of clients: transit clients and recurrent clients. Transit clients are those who appear at the queue following a Poisson process, reflecting a random arrival pattern commonly seen in queueing scenarios. On the other hand, recurrent clients are predefined entities who immediately rejoin the queue once they've been served, demonstrating repetitive behavior in their interactions with the system. Once the server completes servicing a client, it initiates a vacation period. Moreover, in this approach, an optional extended vacation is also taken into account, i.e., the server may opt to indulge in an extended vacation following the initial essential Bernoulli vacation. Also, the consumers are allowed to balk. Further, the ergodicity requirement for the system's stability and then analytical findings for the stationary distribution are obtained. Additionally, various performance metrics for the system are also established. Furthermore, a comprehensive cost function is formulated and further optimized by incorporating a particle swarm optimization (PSO) approach. The convergence analysis is conducted as well, which is supported by illustrative figures. As a result, this work provides a beneficial understanding of enhancing the efficiency of such intricate queueing systems.

On the Expansion of Resolvents and the Integrated Density of States for Poisson Distributed Schrödinger Operators

We consider a Schrödinger operator with random potential distributed according to a Poisson process. We show that under a uniform moment bound expectations of matrix elements of the resolvent as well as the integrated density of states can be approximated to arbitrary precision in powers of the coupling constant. The expansion coefficients are given in terms of expectations obtained by Neumann expanding the potential around the free Laplacian. Our results are valid for arbitrary strength of the disorder parameter, including the small disorder regime.

Waiting Time Distribution of Giant Pulses from the Crab Pulsar Modeled with a Non-stationary Poisson Process

We study the waiting time distribution of giant pulses from the Crab pulsar observed with the Nanshan 26-m radio telescope at a center frequency of 1556 MHz with 512 MHz bandwidth. The observations were performed over a duration of 12.6 hours with 32 μs sampling interval. Our analysis has led to the detection of 2097 giant pulses above a threshold of 10 σ, with flux density > 100 Jy. The occurrence rate of giant pulses is characterized by a highly intermittent giant pulse productivity in short clusters with high rates, separated by relatively long quiescent intervals with low occurrence rates, especially for the giant pulses associated with the interpulse emission. The distribution of waiting times between two subsequent giant pulses displays a power-law tail that can be modeled with a non-stationary Poisson process, which indicates that giant pulses are independent and random events. Distinct waiting time distributions between giant pulses in the main pulse and interpulse phases are presented, which implies that the giant pulse emission mechanisms maybe different in the opposite magnetic poles. The ramification for our understanding of the radio emission mechanisms is discussed.

Fuzzy Fractional Brownian Motion: Review and Extension

In traditional finance, option prices are typically calculated using crisp sets of variables. However, as reported in the literature novel, these parameters possess a degree of fuzziness or uncertainty. This allows participants to estimate option prices based on their risk preferences and beliefs, considering a range of possible values for the parameters. This paper presents a comprehensive review of existing work on fuzzy fractional Brownian motion and proposes an extension in the context of financial option pricing. In this paper, we define a unified framework combining fractional Brownian motion with fuzzy processes, creating a joint product measure space that captures both randomness and fuzziness. The approach allows for the consideration of individual risk preferences and beliefs about parameter uncertainties. By extending Merton’s jump-diffusion model to include fuzzy fractional Brownian motion, this paper addresses the modelling needs of hybrid systems with uncertain variables. The proposed model, which includes fuzzy Poisson processes and fuzzy volatility, demonstrates advantageous properties such as long-range dependence and self-similarity, providing a robust tool for modelling financial markets. By incorporating fuzzy numbers and the belief degree, this approach provides a more flexible framework for practitioners to make their investment decisions.

Statistical Divergence and Paths Thereof to Socioeconomic Inequality and to Renewal Processes.

This paper establishes a general framework for measuring statistical divergence. Namely, with regard to a pair of random variables that share a common range of values: quantifying the distance of the statistical distribution of one random variable from that of the other. The general framework is then applied to the topics of socioeconomic inequality and renewal processes. The general framework and its applications are shown to yield and to relate to the following: f-divergence, Hellinger divergence, Renyi divergence, and Kullback-Leibler divergence (also known as relative entropy); the Lorenz curve and socioeconomic inequality indices; the Gini index and its generalizations; the divergence of renewal processes from the Poisson process; and the divergence of anomalous relaxation from regular relaxation. Presenting a 'fresh' perspective on statistical divergence, this paper offers its readers a simple and transparent construction of statistical-divergence gauges, as well as novel paths that lead from statistical divergence to the aforementioned topics.

Analysis of incorporating modified Weibull model fault detection rate function into software reliability modeling

When software systems are introduced, they are typically deployed in field environments similar to those used during development and testing. However, these systems may also be used in various other locations with different environmental conditions, making it challenging to improve software reliability. Factors such as the specific operating environment and the location of bugs in the code contribute to this difficulty. In this paper, we propose a new software reliability model that accounts for the uncertainty of operating environments. We present the explicit closed-form mean value function solution for the proposed model. The model's goodness of fit is demonstrated by comparing it to the nonhomogeneous Poisson process (NHPP) model based on Weibull model, using four sets of failure data sets from software applications. The proposed model performs well under various estimation techniques, making it a versatile tool for practitioners and researchers alike. The proposed model outperforms other existing NHPP Weibull based in terms of fitting accuracy under two different methods of estimation and provides a more detailed and precise evaluation of software reliability. Additionally, sensitivity analysis shows that the parameters of the suggested distribution significantly impact the mean value function.

Modeling Non-Homogenous Poisson Process and Estimating the Intensity Function for Earthquake Occurrences in Iraq using Simulation for data from January 1st 2018 to April 30th 2023

Modeling Non-Homogenous Poisson Process and Estimating the Intensity Function for Earthquake Occurrences in Iraq using Simulation for data from January 1st 2018 to April 30th 2023

Markovian phase-type single working vacation queue with breakdown and standby server

We consider a single server queuing model with three service types, each with three phases. In which the customers arrive according to a Poisson process and the service time of each customer follows Hyper-Erlang distribution. We analyzed the model with interruptions, such as breakdown, repair and single working vacation. The standby server works whenever the main server under breakdown. Using the matrix geometric method, the steady-state probability vector of the model is examined and finally, some numerical examples are presented.

Performance on massive MIMO enabled mobile edge computing networks: Parallel computing modeling

With the development of communication and computation technologies, mobile edge computing (MEC) has become an emerging technology being deployed in 5G/B5G mobile networks to improve the mobile user experience. In this paper, we study the asymptotic performance of parallel computing in MEC networks with massive multi-input and multi-output (MIMO) in the millimeter wave band. First, a massive MIMO enabled MEC network with a parallel-limited computing model is constructed, and the task completed probability is defined as a new network performance metric. In the proposed model, we derive the task offloading and computing probabilities via the probability generation functional of the Poisson point process, from which, the task completed probability is obtained. Then we propose a computing resource allocation optimization algorithm to maximize the task completed probability within a given delay constraint. The numerical simulation results prove the accuracy of the obtained theoretical results and verify the performance of the proposed algorithm.

Interacting many-particle systems in the random Kac–Luttinger model and proof of Bose–Einstein condensation

Following a model originally considered by Kac and Luttinger, we study interacting many-particle systems in a random background. The background consists of hard spherical obstacles with fixed radius and that are distributed via a Poisson point process with constant intensity on Rd, 2≤d∈N. Interactions among the (bosonic) particles are described through repulsive pair potentials of mean-field type. As a main result, we prove (complete) Bose–Einstein condensation (BEC) in the thermodynamic limit and into the minimizer of a Hartree-type functional, in probability or with probability almost one depending on the strength of the interaction. As an important ingredient, we use very recent results obtained by Alain-Sol Sznitman regarding the spectral gap of the Dirichlet Laplacian in a Poissonian cloud of hard spherical obstacles in large boxes. To the best of our knowledge, our paper provides the first proof of BEC for systems of interacting particles in the Kac–Luttinger model, or in fact for some higher-dimensional interacting random continuum model.

An analytical method for reliability evaluation of power distribution system with time-varying failure rates

Reliability evaluation is crucial in assessing the power supply capacity and guiding the planning and operation of the distribution system. Traditional analytical reliability evaluation usually assumes a constant failure rate for components, implying that the life distribution follows an exponential distribution. However, in reality, the failure rate of components varies over time due to external disturbances and aging. As a result, traditional analytical reliability evaluation methods are no longer applicable. To address this issue, this paper presents an analytical method for evaluating the reliability of the distribution system with time-varying failure rates. The method considers short-term and long-term time-varying failure rates using density evolution and non-homogeneous Poisson process techniques, respectively. Both steady-state and instantaneous indices are calculated to reflect the level of system reliability. Finally, the proposed method is applied to the IEEE RBTS Bus 6 test system, demonstrating its correctness and efficiency.

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.

A novel lifetime analysis of repairable systems via Daubechies wavelets

AbstractIt is well-known that minimally repaired systems are characterized by non-homogeneous Poisson processes. When the underlying lifetime distribution function or its associated intensity function is completely known, the statistical inference of the cumulative number of failures/repairs is made through some standard techniques. On the other hand, when the parametric form of the lifetime distribution or the intensity function is not known, non-parametric inference techniques such as non-parametric maximum likelihood estimate (NPMLE) and kernel-based estimates are employed. Since these techniques are based on the underlying lifetime data, the resulting estimates of the cumulative number of failures/repairs with respect to the system operation time are discontinuous and troublesome in assessing the reliability measures such as reliability function and mean time to failure, etc. In this paper, we propose a novel lifetime analysis of the repairable systems via Daubechies wavelets. More specifically, we generalize the seminal Daubechies wavelet estimate by Kuhl and Bhairgond (Winter Simul Conf Proc 1:562–571, 2000) by combining the ideas on NPMLE and KMBs. In both simulation experiments and real data analyses, we investigate applicability of our Daubechies wavelet-based approaches and compare them with the conventional non-parametric methods.

Quenched large deviations in renewal theory

In this paper we introduce and study renewal–reward processes in random environments where each renewal involves a reward taking values in a Banach space. We derive quenched large deviation principles and identify the associated rate functions in terms of variational formulas involving correctors. We illustrate the theory with three examples: compound Poisson processes in random environments, pinning of polymers at interfaces with disorder, and returns of Markov chains in dynamic random environments.

Long-term software fault prediction with wavelet shrinkage estimation

Wavelet shrinkage estimation received considerable attentions to estimate stochastic processes such as a non-homogeneous Poisson process in a non-parametric way, and was applied to software reliability estimation/prediction. However, it lacks the prediction ability for unknown future patterns in long term and penalizes assessing the software reliability in practice. In this paper, we focus on the long-term prediction of the number of software faults detected in the testing phase and propose many novel long-term prediction methods based on the wavelet shrinkage estimation. The fundamental idea is to adopt both the denoised fault-count data and prediction values, and to minimize several kinds of loss functions to make effective predictions. We also develop an automated wavelet-based software reliability assessment tool, W-SRAT2, which is a drastic extension of the existing tool, W-SRAT, by adding those prediction algorithms. In numerical experiments with 6 actual software development project data, we investigate the predictive performance of our long-term prediction approaches, which consist of 2,640 combinations, and compare them with the common software reliability growth models with the maximum likelihood estimation. It is shown that our wavelet shrinkage estimation/prediction methods outperform the existing software reliability growth models.