Poisson Process Research Articles

A Stochastic Model for Traffic Incidents and Free Flow Recovery in Road Networks

This study addresses the disruptive impact of incidents on road networks, which often lead to traffic congestion. If not promptly managed, congestion can propagate and intensify over time, significantly delaying the recovery of free-flow conditions. We propose an enhanced model based on an exponential decay of the time required for free flow recovery between incident occurrences. Our approach integrates a shot noise process, assuming that incidents follow a non-homogeneous Poisson process. The increases in recovery time following incidents are modeled using exponential and gamma distributions. We derive key performance metrics, providing insights into congestion risk and the unlocking phenomenon, including the probability of the first passage time for our process to exceed a predefined congestion threshold. This probability is analyzed using two methods: (1) an exact simulation approach and (2) an analytical approximation technique. Utilizing the analytical approximation, we estimate critical extreme quantities, such as the minimum incident clearance rate, the minimum intensity of recovery time increases, and the maximum intensity of incident occurrences required to avoid exceeding a specified congestion threshold with a given probability. These findings offer valuable tools for managing and mitigating congestion risks in road networks.

Face and cycle percolation

Abstract We consider face and cycle percolation as models for continuum percolation based on random simplicial complexes in Euclidean space. More precisely, we consider the Vietoris–Rips with connection radius 1, with respect to Euclidean norm, on a stationary Poisson process in $${\mathbb {R}}^D$$ R D of intensity $$\lambda $$ λ . Face percolation is defined through infinite sequences of d-simplices sharing a $$(d-1)$$ ( d - 1 ) -dimensional face. In contrast, cycle percolation demands the existence of infinite d-cycles, thereby generalizing the lattice notion of plaquette percolation. We discuss the sharp phase transition for face percolation and derive comparison results between the critical intensities for face and cycle percolation. Finally, we consider an alternate version of simplex percolation, by declaring simplices to be neighbors whenever they are sufficiently close to each other, and prove a strict inequality involving the critical intensity of this alternate version and that of face percolation.

On the use and misuse of time-rescaling to assess the goodness-of-fit of self-exciting temporal point processes

The paper first highlights important drawbacks and biases related to the common use of time-rescaling to assess the goodness-of-fit (Gof) of self-exciting temporal point process (SETPP) models. Then it presents a new predictive time-rescaling approach leading to an asymptotically unbiased Gof framework for general SETPPs in the case of single observed trajectories. The predictive approach focuses on forecasting accuracy and addresses the bias problem resulting from the plugged-in estimated parameters. Dawid's prequential approach is used and the models' checking is mainly based on the forecasting accuracy of arrival times. These times are transformed, using sequentially estimated parameters, into random vectors which are proved to converge in probability under the null hypothesis and standard regulatory conditions to vectors of iid Exponential(1) rv's. Numerical experiments are used to compare the performances of the standard and predictive time-rescaling for Gof assessment of non-homogeneous Poisson and Hawkes self-exciting temporal processes. Data of Japanese seismic events are also used to illustrate the dynamic aspect of the proposed model-checking approach.

Modelling the stochastic importation dynamics and establishment of novel pathogenic strains using a general branching processes framework.

The importation and subsequent establishment of novel pathogenic strains in a population is subject to a large degree of uncertainty due to the stochastic nature of the disease dynamics. Mathematical models need to take this stochasticity in the early phase of an outbreak into account in order to adequately capture the uncertainty in disease forecasts. We propose a general branching process model of disease spread that includes host-level heterogeneity, and that can be straightforwardly tailored to capture the salient aspects of a particular disease outbreak. We combine this with a model of case importation that occurs via an independent marked Poisson process. We use this framework to investigate the impact of different control strategies, particularly on the time to establishment of an invading, exogenous strain, using parameters taken from the literature for COVID-19 as an example. We also demonstrate how to combine our model with a deterministic approximation, such that longer term projections can be generated that still incorporate the uncertainty from the early growth phase of the epidemic. Our approach produces meaningful short- and medium-term projections of the course of a disease outbreak when model parameters are still uncertain and when stochasticity still has a large effect on the population dynamics.

On Multivariate Distribution of Stochastically Escaping Tumor Cells from the Effect of Chemotherapy

Chemotherapy is used to treat cancer through a cytotoxic therapeutic approach. Certain conditions may allow some cells to evade the treatment and spread again. Doctors resort to completing the treatment with radiation, which increases the patient’s stress during the treatment journey. In this work, we study the multivariate probability distribution of these stochastically escaped cells to estimate their number in an attempt to reduce the doses of chemotherapy and radiation together. We consider these escaping tumor cells increased independently according to a Poisson process. We obtain not just this distribution but also the variance and expected value of these escaped cells at any given time. This time is depending on the total number of infected cells in the system. Furthermore, we may calculate the expected value of the treatment cells by knowing the expected number of escaping cells. This information aids in determining the right dosage of chemotherapy, which in turn helps to reduce the patients’ pain.

Properties of the vertex of a convex hull generated by a Poisson point process inside a parabola

A convex hull generated by the implementation of a Poisson point process inside a parabola is considered in the article. At that, the measure of intensity of the Poisson law is related to regularly varying functions near the boundary of the support. It is proven that the domain bounded by the perimeters of the convex hull and the boundary of the support - a parabola, can be represented as a sum of independent identically distributed random variables. Moreover, this value does not depend on the vertices of the convex hull itself. It is worth noting that having approximated the binomial point process by a Poisson one, P.Groeneboom [6], A.J.Cabo and P.Groeneboom [3], I.Hueter [9], T.Hsing [8] and others, using the martingale properties of stationary vertex processes, proved various options of the central limit theorem for functionals of a random convex hull in the case when the original distribution is uniformly concentrated in a convex polygon or ellipse. In this paper, the exact distribution and conditional distribution of vertex processes are found when the convex hull is generated by a inhomogeneous Poisson point process inside a parabola. In some special cases, it is shown that the area between the perimeter of the convex hull and the support of the distribution is expressed by the sum of independent random variables.

The role of oscillations in grid cells' toroidal topology.

Persistent homology applied to the activity of grid cells in the Medial Entorhinal Cortex suggests that this activity lies on a toroidal manifold. By analyzing real data and a simple model, we show that neural oscillations play a key role in the appearance of this toroidal topology. To quantitatively monitor how changes in spike trains influence the topology of the data, we first define a robust measure for the degree of toroidality of a dataset. Using this measure, we find that small perturbations (~100 ms) of spike times have little influence on both the toroidality and the hexagonality of the ratemaps. Jittering spikes by ~100-500 ms, however, destroys the toroidal topology, while still having little impact on grid scores. These critical jittering time scales fall in the range of the periods of oscillations between the theta and eta bands. We thus hypothesized that these oscillatory modulations of neuronal spiking play a key role in the appearance and robustness of toroidal topology and the hexagonal spatial selectivity is not sufficient. We confirmed this hypothesis using a simple model for the activity of grid cells, consisting of an ensemble of independent rate-modulated Poisson processes. When these rates were modulated by oscillations, the network behaved similarly to the real data in exhibiting toroidal topology, even when the position of the fields were perturbed. In the absence of oscillations, this similarity was substantially lower. Furthermore, we find that the experimentally recorded spike trains indeed exhibit temporal modulations at the eta and theta bands, and that the ratio of the power in the eta band to that of the theta band, [Formula: see text], correlates with the critical jittering time at which the toroidal topology disappears.

Optimizing Wait Times: Enhancing Server Accessibility with Two-Phase Imperative Services and Backup Server

This article analyses a bulk arrival retrial queueing system with a single server and two-stage service coupled with a backup server. Client arrivals occur in batches following a compound Poisson process. If the server is idle, one of the clients from the batch receives the first-stage service immediately, and the rest join the orbit. Otherwise, the entire arriving batch joins the orbit. Once the first stage of service is completed, the client moves to the second stage of the service facility. Unsatisfied clients after receiving two stages of service may become feedback clients. After every two-stage service completion, the server may take a vacation of random length. The server has the option to extend its vacation. Breakdowns of the main server can occur at any point during the busy state, and the server has the option to choose its repair type with probabilities. During the vacation and repair period of the main server, clients receive service from a standby (backup) server. Performance measures are derived, stochastic decomposition laws are also verified and special cases are analysed. Numerical results are presented to validate the impact of parameters on the system. Cost analysis and optimization have been carried out for the proposed model. AMS Subject Classification: 60K25, 68M20, 90B22, 90B25

Two transitions in complex eigenvalue statistics: Hermiticity and integrability breaking

Open quantum systems have complex energy eigenvalues which are expected to follow non-Hermitian random matrix statistics, when chaotic, or two-dimensional (2d) Poisson statistics, when integrable. We investigate the spectral properties of a many-body quantum spin chain, i.e., the Hermitian Heisenberg model with imaginary disorder. Its rich complex eigenvalue statistics is found to separately break both Hermiticity and integrability at different scales of the disorder strength. With no disorder, the system is integrable and Hermitian, with spectral statistics corresponding to the 1d Poisson point process. At very small disorder, we find a transition from 1d Poisson statistics to an effective D-dimensional Poisson point process, showing Hermiticity breaking. At intermediate disorder, we find integrability breaking, as inferred from the statistics matching that of non-Hermitian complex symmetric random matrices in class AI†. For large disorder, as the spins align, we recover the expected integrability (now in the non-Hermitian setup), indicated by 2d Poisson statistics. These conclusions are based on fitting the spin-chain data of numerically generated nearest- and next-to-nearest-neighbor spacing distributions to an effective 2d Coulomb gas description at inverse temperature β. We confirm that such an effective description of random matrices also applies in classes AI† and AII† up to next-to-nearest-neighbor spacings. Published by the American Physical Society 2025

Nonparametric testing of first-order structure in point processes on linear networks

In this paper we address a two-sample problem in the context of point processes on linear networks. The aim is to determine whether two given point patterns defined over the same linear network and under the assumption of Poissonness, share the same spatial structure. To do so, a Kolmogorov–Smirnov and a Cramér von Mises type test statistics are developed and analysed through an extensive simulation study. We have included different types of networks, balanced and unbalanced sample sizes, and homogeneous and inhomogeneous Poisson point processes. The results show a good level adjustment and high power values, the latter increasing with the sample size and the discrepancy between the two generating intensities. Finally, these methods have also been applied to the analysis of traffic accidents in Rio de Janeiro (Brazil), studying their distribution at different rush hours.

Local time statistics and permeable barrier crossing: From Poisson to birth-death diffusion equations

Barrier crossing is a widespread phenomenon across natural and engineering systems. While abundant cross-disciplinary literature on the topic has emerged over the years, the stochastic underpinnings of the process have yet to be linked quantitatively to easily measurable observables. We bridge this gap by developing a microscopic representation of Brownian motion in the presence of permeable barriers that allows us to treat barriers with constant asymmetric permeabilities. Our approach relies upon reflected Brownian motion and on the crossing events being Poisson processes subordinated by the local time of the underlying motion at the barrier. Within this paradigm, we derive the exact expression for the distribution of the number of crossings and find an experimentally measurable statistical definition of permeability. We employ Feynman-Kac theory to derive and solve a set of governing birth-death diffusion equations and extend them to the case when barrier permeability is constant and asymmetric. As an application, we study a system of infinite, identical, and periodically placed asymmetric barriers for which we derive analytically effective transport parameters. This periodic arrangement induces an effective drift at long times whose magnitude depends on the difference in the permeability on either side of the barrier as well as on their absolute values. As the asymmetric permeabilities act akin to localized ratchet potentials that break spatial symmetry and detailed balance, the proposed arrangement of asymmetric barriers provides an example of a noise-induced drift without the need to time modulate any external force or create temporal correlations on the motion of a diffusing particle. By placing only one asymmetric barrier in a periodic domain, we also show the emergence of a nonequilibrium steady state. Published by the American Physical Society 2025

Compound Poisson particle approximation for McKean–Vlasov SDEs

Abstract We present a comprehensive discretization scheme for linear and nonlinear stochastic differential equations (SDEs) driven by either Brownian motions or $\alpha $-stable processes. Our approach utilizes compound Poisson particle approximations, allowing for simultaneous discretization of both the time and space variables in McKean–Vlasov SDEs. Notably, the approximation processes can be represented as a Markov chain with values on a lattice. Importantly, we demonstrate the propagation of chaos under relatively mild assumptions on the coefficients, including those with polynomial growth. This result establishes the convergence of the particle approximations towards the true solutions of the McKean–Vlasov SDEs. By only imposing moment conditions on the intensity measure of compound Poisson processes our approximation exhibits universality. In the case of ordinary differential equations (ODEs) we investigate scenarios where the drift term satisfies the one-sided Lipschitz assumption. We prove the optimal convergence rate for Filippov solutions in this setting. Additionally, we establish a functional central limit theorem for the approximation of ODEs and show the convergence of invariant measures for linear SDEs. As a practical application we construct a compound Poisson approximation for two-dimensional Navier–Stokes equations on the torus and demonstrate the optimal convergence rate.

Directional Handover Analysis with Stochastic Petri Net and Poisson Point Process in Heterogeneous Networks

Handover is crucial for ensuring seamless connectivity in heterogeneous networks (HetNet) by enabling user equipment (UE) to switch its connection link between cells based on signal conditions. However, conventional analytical approaches ignored the distinctions between macro-cell to small-cell (M2S) and small-cell to macro-cell (S2M) scenarios during a handover decision-making process, which resulted in handover failures (HoF) or ping-pong handovers. Therefore, this paper proposes a novel framework, Do-SPN-PPP, that combines stochastic Petri net (SPN) and the Poisson point process (PPP) to quantitatively analyze M2S and S2M handover performance differences. The proposed framework also reveals and predicts how handover parameters affect UE residence time in a cell within the HetNet, and it exhibits a higher predictive accuracy compared with the traditional conventional analytical approach. In addition, the Monte Carlo simulation verified the Do-SPN-PPP framework, and the proposed framework exhibits a 96% reduction in computation time while maintaining a 95% confidence interval and 0.5% error tolerance compared with the simulation.

Ensuring representative sample volume predictions in microplastic monitoring

A large body of literature is available quantifying microplastic contamination in freshwater and marine systems across the globe. “Microplastics” do not represent a single analyte. Rather, they are usually operationally defined based on their size, polymer and shape, dependent on the sample collection method and the analytical range of the measurement technique. In the absence of standardised methods, significant variability and uncertainty remains as to how to compare data from different sources, and so consider exposure correctly. To examine this issue, a previously compiled database containing 1603 marine observations and 208 freshwater observations of microplastic concentrations from across the globe between 1971 and 2020 was analysed. Reported concentrations span nine orders of magnitude. Investigating the relationship between sampling methods and reported concentrations, a striking correlation between smaller sample unit volumes and higher microplastic concentrations was observed. Close to half of the studies reviewed scored poorly in quality scoring protocols according to the sample volume taken. It is critical that sufficient particles are measured in a sample to reduce the errors from random chance. Given the inverse relationship with particle size and abundance, the volume required for a representative sample should be calculated case-by-case, based on what size microplastics are under investigation and where they are being measured. We have developed the Representative Sample Volume Predictor (RSVP) tool, which standardises statistical prediction of sufficient sample volumes, to ensure microplastics are detected with a given level of confidence. Reviewing reports in freshwater, we found ~ 12% of observations reported sample volumes which would have a false negative error rate > 5%. Such sample volumes run the risk of wrongly concluding that microplastics are absent in samples and are not sufficient to be quantitative. The RSVP tool also provides a harmonised Poisson point process estimation of confidence intervals to test whether two observations are likely to be significantly different, even in the absence of replication. In this way, we demonstrate application of the tool to evaluate historic data, but also to assist in new study designs to ensure that environmental microplastic exposure data is relevant and reliable. The tool can also be applied to other data for randomly dispersed events in space or time, and so has potential for transdisciplinary use.Graphical

Controlled Filtered Poisson Processes

Filtered Poisson processes are used as models in various applications, in particular in statistical hydrology. In this paper, controlled filtered Poisson processes are considered. The aim is to minimize the expected time that the process will spend in the continuation region. The dynamic programming equation satisfied by the value function is derived. To obtain the value function, and hence the optimal control, a non-linear integro-differential equation must be solved, subject to the appropriate boundary conditions. Various cases for the size of the jumps are treated and explicit results are obtained.

A bathtub pattern for repairable systems: From better-than-minimal to minimal and worse-than-minimal repairs

We introduce a new combined repair process to describe repairs that are initially better-than-minimal, then become minimal, before finally becoming worse-than-minimal. The extended generalized Polya process (EGPP), non-homogeneous Poisson process (NHPP), and generalized Polya process (GPP) are used to describe this repair pattern, respectively. Several useful properties are derived for the combined process under two settings: change in repair type after a specified time and change in repair type after a specified number of failures/repairs. As an application, the optimal age replacement problem is defined and its optimal solution is analyzed. Detailed numerical examples support our findings.

Convergence of Poisson point processes and of optimal transport regularization with application in variational analysis of PET reconstruction

Abstract Poisson distributed measurements in inverse problems often stem from Poisson point processes that are observed through discretized or finite-resolution detectors, one of the most prominent examples being positron emission tomography (PET). These inverse problems are typically reconstructed via Bayesian methods. A natural question then is whether and how the reconstruction converges as the signal-to-noise ratio tends to infinity and how this convergence interacts with other parameters such as the detector size. In this article we carry out a corresponding variational analysis for the exemplary Bayesian reconstruction functional from [1,2], which considers dynamic PET imaging (i.e. the object to be reconstructed changes over time) and uses an optimal transport regularization.

On the one-chart schemes for joint monitoring of the two parameters of a zero-inflated Poisson (ZIP) process

One-chart scheme for joint monitoring of two parameters allows the practitioners to focus on a single chart and thus, it offers significant operational advantages. Recently, four one-chart schemes are reported in literature for joint monitoring of the two parameters (Φ, λ) of a zero-inflated Poisson (ZIP) process. One of these schemes, namely - Gamma chart, is developed assuming samples of large size will be inspected and other three charts, namely DS-chart, Max-chart and LR chart, are developed assuming samples of small size will be inspected after every certain interval. The plotting statistics of these three charts are computed using the estimated ZIP parameters. However, unless the sample size (n) is sufficiently large, the excessive number of zeros in the ZIP process may offer a sample of zeros only. Therefore, these three schemes require estimation of ZIP parameters and subsequent computation of the plotting statistic based on the accumulated samples till the sampling stage. Because of the accumulation of samples, these schemes suffer from several limitations. In this paper, these one-chart schemes are modified for large sample size to make them free from their limitations. Subsequently, the performances of the four one-chart schemes are evaluated extensively using simulation studies. The results reveal that among the three modified charts likelihood ratio (LR) chart is the most preferable with respect to in-control and out-of-control average run length (ARL) performances. However, all these three modified charts give false alarms when the process parameter(s) shift towards desirable direction(s) implying process improvement. On the other hand, the Gamma chart is slightly inferior with respect to out-of-control ARL performances. But it does not give false alarm when there is a process improvement. Further, since Gamma chart directly monitors the average number of defects, it offers significant advantages in terms of implementation and interpretations.

Pseudorandomness of primes at large scales

ABSTRACT Assuming a q-variant of the prime k-tuple conjecture uniformly, we compute mixed moments of the number of primes in disjoint short intervals and progressions. This involves estimating the mean of singular series along products of lattices, which is of independent interest. As a consequence, we establish the convergence of both sequences of suitably normalized primes to a standard Poisson point process.

Central limit theorems describing isolation by distance under various forms of power-law dispersal

In this paper, we uncover new asymptotic isolation by distance patterns occurring under long-range dispersal of offspring. We extend a recent work of the first author, in which this information was obtained from forwards-in-time dynamics using a novel stochastic partial differential equations approach for spatial $\Lambda$-Fleming-Viot models. The latter were introduced by Barton, Etheridge and V\'eber as a framework to model the evolution of the genetic composition of a spatially structured population. Reproduction takes place through extinction-recolonisation events driven by a Poisson point process. During an event, in certain ball-shaped areas, a parent is sampled and a proportion of the population is replaced. We generalize the previous approach of the first author by allowing the area from which a parent is sampled during events to differ from the area in which offspring are dispersed, and the radii of these regions follow power-law distributions. In particular, while in previous works the motion of ancestral lineages and coalescence behaviour were closely linked, we demonstrate that local and non-local coalescence is possible for ancestral lineages governed by both fractional and standard Laplacians.