Poisson Process Research Articles

Non-Poissonian bursts in the arrival of phenotypic variation can strongly affect the dynamics of adaptation.

Modelling the rate at which adaptive phenotypes appear in a population is a key to predicting evolutionary processes. Given random mutations, should this rate be modelled by a simple Poisson process, or is a more complex dynamics needed? Here we use analytic calculations and simulations of evolving populations on explicit genotype-phenotype maps to show that the introduction of novel phenotypes can be 'bursty' or overdispersed. In other words, a novel phenotype either appears multiple times in quick succession, or not at all for many generations. These bursts are fundamentally caused by statistical fluctuations and other structure in the map from genotypes to phenotypes. Their strength depends on population parameters, being highest for 'monomorphic' populations with low mutation rates. They can also be enhanced by additional inhomogeneities in the mapping from genotypes to phenotypes. We mainly investigate the effect of bursts using the well-studied genotype-phenotype map for RNA secondary structure, but find similar behaviour in a lattice protein model and in Richard Dawkins's biomorphs model of morphological development. Bursts can profoundly affect adaptive dynamics. Most notably, they imply that fitness differences play a smaller role in determining which phenotype fixes than would be the case for a Poisson process without bursts.

Pricing derivatives on foreign assets using Markov-modulated cojump-diffusion dynamics

In this study, cross-currency derivatives pricing under stochastic interest rates is analyzed when Markov-modulated cojump-diffusion dynamics with both idiosyncratic and simultaneous jumps drive the underlying foreign equity price and foreign exchange rate processes. More specifically, the noise dynamics of these two assets are captured by the correlated Wiener processes, and the two types of jump events are described as two independent compound Poisson processes with log-normal jump amplitudes. Cojumping addresses the correlation between these two asset price jumps. In an incomplete market setting, the dynamic Esscher transform approach is applied to determine a pricing kernel. According to the resulting dynamics, explicit analytical formulas for evaluating values of European foreign equity call options are provided. Numerical experiments are also described.

Remarks on a stochastic geometric model for interference-limited cellular communications

A plethora of stochastic geometry (SG) models have been developed for cellular communications, especially in the context of internet of things (IoT) applica tions. A typical assumption in such models is that base stations (BS) are de ployed in the Euclidean plane as a spatial poisson point process (PPP) of some density λ, with each communicating equipment transmitting at some power p. The usual objective of these models is to characterize the cellular coverage prob ability in both the downlink (DL) and uplink (UL) directions. In this article we expose, in the form of four remarks, the peculiar behavior of a baseline stochas tic geometric model of an interference-limited cellular system. Specifically, we reveal that under some assumptions, the coverage probability in both the UL and DL directions for this system is independent of both λ and p, flagrantly contra dicting intuition. The aim of the article is by no means to invalidate the use of SG in modeling communications systems, but rather to point out that such modeling may not be adequate all the time.

A resampling-based approach to share reference panels

For many genome-wide association studies, imputing genotypes from a haplotype reference panel is a necessary step. Over the past 15 years, reference panels have become larger and more diverse, leading to improvements in imputation accuracy. However, the latest generation of reference panels is subject to restrictions on data sharing due to concerns about privacy, limiting their usefulness for genotype imputation. In this context, here we propose RESHAPE, a method that employs a recombination Poisson process on a reference panel to simulate the genomes of hypothetical descendants after multiple generations. This data transformation helps to protect against re-identification threats and preserves data attributes, such as linkage disequilibrium patterns and, to some degree, identity-by-descent sharing, allowing for genotype imputation. Our experiments on gold-standard datasets show that simulated descendants up to eight generations can serve as reference panels without substantially reducing genotype imputation accuracy.

Inhomogeneous Poisson processes in the disk and interpolation

Abstract We investigate different geometrical properties, related to Carleson measures and pseudo-hyperbolic separation, of inhomogeneous Poisson point processes on the unit disk. In particular, we give conditions so that these random sequences are almost surely interpolating for the Hardy, Bloch or weighted Dirichlet spaces.

Shortage Policies for a Jump Process with Positive and Negative Batch Arrivals in a Random Environment

We study a continuous-review stock management of a retailer for a single item in a limited storage (buffer) in a random environment. The stock level fluctuates according to two independent compound Poisson processes with discrete amounts of items (batches) that enter and leave the storage facility. The storage facility is controlled by a three-parameter base-stock replenishment policy. All items exceeding the storage capacity are transferred to an unlimited foreign facility. In addition, a restricted backlogging possibility is permitted; additional demands for items are lost sales. We further assume a random shelf life, the possibility of total inventory collapse, and a random lead time. Applying Markov theory, we derive the optimal control parameters minimizing the long-run expected total cost. A sensitivity analysis is conducted focusing on the comparison between the pure lost-sales policy and a partial backordering policy. Accordingly, we identify cases where one policy is cost effective compared to the other, particularly with respect to the batch patterns (sign, rate, average, and variability), and the associated costs.

Modeling the Impact of Community First Responders

In community first responder (CFR) systems, traditional emergency service response is augmented by a network of trained volunteers who are dispatched via an app. A central application of such systems is out-of-hospital cardiac arrest (OHCA), where a very fast response is crucial. For a target performance level, how many volunteers are needed, and from which locations should they be recruited? We model the presence of volunteers throughout a region as a Poisson point process, which permits the computation of the response-time distribution of the first-arriving volunteer. Combining this with known survival-rate functions, we deduce survival probabilities in the cardiac arrest setting. We then use convex optimization to compute a location distribution of volunteers across the region that optimizes either the fraction of incidents with a fast response (a common measure in the industry) or patient survival in the case of OHCA. The optimal location distribution provides a bound on the best possible performance with a given number of volunteers. This can be used to determine whether introducing a CFR system in a new region is worthwhile or can serve as a guide for additional recruitment in existing systems. Effective target areas for recruitment are not always obvious because volunteers recruited from one area may be found in various areas across the city depending on the time of day; we explicitly capture this issue. We demonstrate these methods through an extended case study of Auckland, New Zealand. This paper was accepted by Carri Chan, healthcare management. Funding: This research was financed in part by the Netherlands Organization for Scientific Research (NWO) in the form of a Rubicon grant (019.172EN.016) to C. J. Jagtenberg and a Veni grant (VI.Veni.191E.005) to P. L. van den Berg. S. G. Henderson and H. Li were supported in part by National Science Foundation [Grant CMMI-2035086]. Vrije Universiteit Amsterdam and Erasmus University received funding from TKI Dinalog [Grant 2023-1-307TKI]. Some of this work has been executed under the TKI Dinalog [Grant 2023-1-307TKI]. Part of this work was completed during a research visit made possible by a [Distinguished Visitor Award] from the University of Auckland. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.04024 .

Short-Time Infrequent Metadynamics for Improved Kinetics Inference.

Infrequent Metadynamics is a popular method to obtain the rates of long time-scale processes from accelerated simulations. The inference procedure is based on rescaling the first-passage times of the Metadynamics trajectories using a bias-dependent acceleration factor. While useful in many cases, it is limited to Poisson kinetics, and a reliable estimation of the unbiased rate requires slow bias deposition and prior knowledge of efficient collective variables. Here, we propose an improved inference scheme, which is based on two key observations: (1) the time-independent rate of Poisson processes can be estimated using short trajectories only. (2) Short trajectories experience minimal bias, and their rescaled first-passage times follow the unbiased distribution even for relatively high deposition rates and suboptimal collective variables. Therefore, by basing the inference procedure on short time scales, we obtain an improved trade-off between speedup and accuracy at no additional computational cost, especially when employing suboptimal collective variables. We demonstrate the improved inference scheme for a model system and two molecular systems.

Flexible Bayesian reliability demonstration testing

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

A Genuinely Hybrid, Multiscale 3D Cancer Invasion and Metastasis Modelling Framework

We introduce in this paper substantial enhancements to a previously proposed hybrid multiscale cancer invasion modelling framework to better reflect the biological reality and dynamics of cancer. These model updates contribute to a more accurate representation of cancer dynamics, they provide deeper insights and enhance our predictive capabilities. Key updates include the integration of porous medium-like diffusion for the evolution of Epithelial-like Cancer Cells and other essential cellular constituents of the system, more realistic modelling of Epithelial–Mesenchymal Transition and Mesenchymal–Epithelial Transition models with the inclusion of Transforming Growth Factor beta within the tumour microenvironment, and the introduction of Compound Poisson Process in the Stochastic Differential Equations that describe the migration behaviour of the Mesenchymal-like Cancer Cells. Another innovative feature of the model is its extension into a multi-organ metastatic framework. This framework connects various organs through a circulatory network, enabling the study of how cancer cells spread to secondary sites.

The contact process on scale-free geometric random graphs

We study the contact process on a class of geometric random graphs with scale-free degree distribution, defined on a Poisson point process on Rd. This class includes the age-dependent random connection model and the soft Boolean model. In the ultrasmall regime of these random graphs we provide exact asymptotics for the non-extinction probability when the rate of infection spread is small and show for a finite version of these graphs that the extinction time is of exponential order in the size of the graph.

Joint optimization of inventory replenishment and rationing policies for an omnichannel store with both in-store and online demands

Boosted by the COVID-19 pandemic, omnichannel retail has become a popular retail model and is expected to further grow. We study inventory replenishment and rationing of a product in an omnichannel store that fulfils both in-store and online demands. Both demands follow Poisson process. As each in-store demand must be satisfied immediately whereas each online order has a delivery time window, the two demand classes have different behaviors facing stockout: offline demand is lost while online order is backlogged with backorder cost charged only if it is delivered late. The inventory replenishment and rationing between the two demand classes in this omnichannel inventory system are controlled by a continuous review (Q, r) policy and a critical level policy, respectively. The intertwined nature of inventory replenishment and rationing in this system makes its analytical cost evaluation and its joint optimization of the two policies challenging. After deriving an analytical formula for evaluating the expected cost of the system, two heuristic algorithms are proposed to jointly optimize the two policies. Numerical experiments on randomly generated instances demonstrate that the multi-start local search algorithm and the scatter search algorithm can obtain a near-optimal solution quickly with percentage cost gap no larger than 0.81% and 0.18%, respectively, and the critical level policy can yield cost savings up to 10.91% compared to the first-come, first-served policy. Finally, a sensitivity analysis is conducted to evaluate the influences of system parameters on the optimal critical level.

A probabilistic framework for describing a class of switching signals in continuous-time

We present an interpretation of switching signals with certain average dwell time in terms of Poisson process, which allows to build a probabilistic framework to accommodate all the switching signals of this kind. As a result, we can learn more knowledge about such signals. In particular, such a switching signal and the solution of the corresponding switching system jointly constitute a Markov process. And the Markov property will facilitate the asymptotic behaviour analysis of the switching dynamics. As a byproduct, we present an upper bound on the joint spectral radius of a family of Hurwitz or anti-Hurwitz matrices, which turns out to be quite sharp as compared with the existing results.

Burst traffic: Congestion management and performance optimization strategies in heterogeneous cognitive radio networks

AbstractIn order to solve the network congestion problem caused by burst service flows in heterogeneous cognitive radio networks (Het‐CRN), a two‐stage dual‐threshold random early detection (TD‐RED) control mechanism is proposed here. Meanwhile, a three‐state Markov modulated Poisson process (MMPP‐3) is used to model the burst service to analyze its arrival in depth. Moreover, a performance evaluation system (PES) is proposed to quantify the interference of burst services to different classes of services and their quality of service, which can comprehensively evaluate the performance enhancement of the TD‐RED control mechanism for each service. The system examines performance metrics such as packet loss rate, average delay, average captain etc. PES simulation results show that TD‐RED significantly reduces the packet rejection rate and transmission delay compared to the no congestion control (NCC) and random early detection (RED) methods, thus providing superior quality of service to all services. thereby providing a superior quality of service to cognitive users. Its lower average captain fully proves that TD‐RED can effectively alleviate congestion.

Reliability analysis for continuous degrading systems subject to multi-level failure dependence

This article explicitly models multi-level failure dependence existing in continuous degrading systems with multiple degrading components. Dependence within the component level is specified as that arrived shocks increase the vulnerability to inherent degradation. Dependence among components refers to the fact that the degradation of a particular component can be accelerated by others. The Gamma process is employed as a comprehensive framework for modeling the dependence above. A compound Poisson process and normal distribution are both introduced to describe the stochastic characteristics of random shocks. Procedures for obtaining analytical solutions and simulation results of reliability analysis are both presented. At last, an illustrative example of a hydraulic system is studied to demonstrate the effectiveness and advantages of the proposed method.

Some mathematical properties of the premium function and ruin probability of a generalized Cramér–Lundberg model driven by mixed poisson processes

Some mathematical properties of the premium function and ruin probability of a generalized Cramér–Lundberg model driven by mixed poisson processes

Photophysical image analysis: Unsupervised probabilistic thresholding for images from electron-multiplying charge-coupled devices.

We introduce the concept photophysical image analysis (PIA) and an associated pipeline for unsupervised probabilistic image thresholding for images recorded by electron-multiplying charge-coupled device (EMCCD) cameras. We base our approach on a closed-form analytic expression for the characteristic function (Fourier-transform of the probability mass function) for the image counts recorded in an EMCCD camera, which takes into account both stochasticity in the arrival of photons at the imaging camera and subsequent noise induced by the detection system of the camera. The only assumption in our method is that the background photon arrival to the imaging system is described by a stationary Poisson process (we make no assumption about the photon statistics for the signal). We estimate the background photon statistics parameter, λbg, from an image which contains both background and signal pixels by use of a novel truncated fit procedure with an automatically determined image count threshold. Prior to this, the camera noise model parameters are estimated using a calibration step. Utilizing the estimates for the camera parameters and λbg, we then introduce a probabilistic thresholding method, where, for the first time, the fraction of misclassified pixels can be determined a priori for a general image in an unsupervised way. We use synthetic images to validate our a priori estimates and to benchmark against the Otsu method, which is a popular unsupervised non-probabilistic image thresholding method (no a priori estimates for the error rates are provided). For completeness, we lastly present a simple heuristic general-purpose segmentation method based on the thresholding results, which we apply to segmentation of synthetic images and experimental images of fluorescent beads and lung cell nuclei. Our publicly available software opens up for fully automated, unsupervised, probabilistic photophysical image analysis.

STAR: Spread of innovations on graph structures with the Susceptible‐Tattler‐Adopter‐Removed model

Adoptions of a new innovation such as a product, service or idea are typically driven both by peer‐to‐peer social interactions and by external influence. Social graphs are usually used to efficiently model the peer‐to‐peer interactions, where new adopters influence their peers to also adopt the innovation. However, the influence to adopt may also spread through individuals close to the adopters, known as tattlers, who only share information regarding the innovation. We extend an inhomogeneous Poisson process model accounting for both external and peer‐to‐peer influence to include an optional tattling stage, and we term the extension the Susceptible‐Tattler‐Adopter‐Removed (STAR) model. In an extensive simulation study, the proposed model is shown to be stable and identifiable and to accurately identify tattling when present. Further, using simulations, we show that both inference and prediction of the STAR model are quite robust against missing edges in the social graph, a common situation in real‐world data. Simulations and theoretical considerations demonstrate that, when edges are missing, the STAR model is able to accurately estimate the shares attributed to the external and internal sources of influence. Furthermore, the STAR model may be used to improve the inference of the external and viral parameters and subsequent predictions even when tattling is not part of the real data‐generating mechanism.

Spatial distribution of woody plants in relation to mistletoe-infected Vachellia karroo trees in a semi-arid African savanna

By increasing resource heterogeneity, mistletoe-infected trees can restructure plant community processes and distribution patterns. No information is available on how mistletoe-infected Vachellia (Acacia) karroo trees within V. karroo dominated stands are spatially distributed, and on how they influence the spatial patterns of their surrounding conspecifics and heterospecifics. Each woody plant was stem mapped using a cartesian plane (x, y) within three 50×50 m plots located in V. karroo dominated stands in a semi-arid savanna, South West Zimbabwe. Pair correlations g(r) were used for the univariate analysis and Poisson process null models were applied to quantify and detect overall departure from randomness. For the bivariate analysis, pair correlations g12(r) under the null model of independence were used, whilst the mark correlation function (kmm(r)) was used to analyse the correlation of tree canopy area and mistletoe infection intensity. For each plot, size class distributions, based on tree height and basal stem diameter displayed negative J curves, with steep negative regression slopes across the size classes, clearly indicating a strongly recruiting population of V. karroo. The univariate patterns of all trees (infected and non-infected) were consistent with a random pattern, which is attributed to unsystematic mistletoe seed dispersal by birds. The univariate analysis of all woody plants (adults and juveniles) exhibited aggregation at small spatial scales due to the high abundance of clustered seedlings and saplings. At small spatial scales, understory woody plants (both conspecifics and non-conspecifics) were positively associated with mistletoe-infected trees due to mistletoephily which is the facilitation (or nurse protégé interactions) within the more resource-rich mistletoe-infected tree subcanopies. These results provide strong evidence suggesting that the variations in spatial pattern modification by mistletoe-infected trees could further increase spatial heterogeneity in this semi-arid savanna. As such, by increasing heterogeneity, mistletoe-infected trees can increase the resilience of semi-arid savannas in the face of perturbations and stochastic events.

Blending Different Latency Traffic With Beta Mixing

We analyze the anonymity provided by continuous mixnets (e.g., Loopix) when messages with different latency requirements are sent through the same network. The anonymity provided by existing mixnets that offer bounded latency guarantees has only been studied considering that all the traffic in the network follows the same latency distribution. In this work we evaluate whether it is beneficial to aggregate different types of traffic in the same network or to keep them separate, when the latency distributions are exponential and the traffic arrivals are a poisson process --- as is the case in Loopix and related designs. We present a novel evaluation method to analyze the leakage to the adversary when multiple different types of traffic are sent through the same network of continuous mixes. We apply the method to empirically evaluate the end-to-end anonymity (in terms of entropy) for each type of traffic in the presence of a global passive adversary that may additionally compromise a constant fraction of mixes or may have knowledge about the type of traffic of network output messages. Finally we show via empirical evaluation using our analytical framework that it is beneficial for anonymity to blend different types of traffic in the same mixnet.