Poisson Process Research Articles

Sufficient Reduction Method for Bivariate Zero-Inflated Poisson Process

The sufficient reduction (SR) method was developed for detecting a mean shift in a bivariate zero-inflated Poisson process. The derived sequence of statistics from the reduction was monitored with the EWMA and EWMA-SN charts for monitoring a mean shift in a process. The detection performance was compared against other SR methods developed for a Poisson process and evaluated via the simulations under the different shift sizes and proportions of zero in the process. The results showed that the presence of zeros in the process influenced the performance of SR methods by delaying shift detection and reducing the detection accuracy, especially when shift size was small. The proposed method with the EWMA chart gave the shortest delay for detecting a small to moderate shift and gave the highest true alarm rate and the lowest non-detection rate for detecting a small shift compared to other methods.

Simulation-based variable neighborhood search for optimizing skill assignments in multi-server facilities with inventories

This paper addresses the joint optimization problem of skill assignments and inventory in a multi-skill, multi-server repair facility. Failures of different part types occur according to Poisson processes, and each part type requires a certain repair skill. The repair facility supplies ready-to-install spare parts when available in the inventory, according to the (S−1,S) inventory policy. The repair times follow exponential distributions, with rates dependent on the part type. After repair, the parts are returned to the inventory as ready-to-install spare parts. If the inventory is empty when a failed part arrives, the replacement part is backordered, and a penalty cost is incurred. The objective of the problem is to find an assignment of repair skills to servers and inventory levels that minimize the expected total cost of the system. That is, the costs for servers, the costs to upgrade the skills of servers, and the expected holding and backorder costs. We propose to solve this problem by a simulation-based Variable Neighborhood Search (VNS) approach, in which a Discrete Event Simulation is applied to evaluate the expected backorder and holding costs given the skill assignments.The proposed method is capable of significantly improving the results of a recently published Genetic Algorithm, achieving an average cost reduction of 5.1% in the same running time. Moreover, it is able to find comparable solutions in one fifth of the GA running time.

Properties related with conditional expectation for a non-homogeneous Poisson Process

Properties related with conditional expectation for a non-homogeneous Poisson Process

Abstract A034: Development of a novel class of genomic biomarkers through mathematical modeling of cancer genome contiguity

Abstract Introduction: Chromosomal instability (CIN) is a hallmark of cancer aggressiveness. However, in lung adenocarcinoma (LUAD) the most common methods for measuring genomic instability, such as the weighted Genome Instability Index (wGII), are not reliable predictors of patient outcomes. The wGII measures the portion of the genome altered by copy number variation (CNV). The wGII quantifies aneuploidy but ignores other properties of CNVs such as their length distribution. We hypothesized that quantification of genome contiguity may overcome the limitations of wGII and help identify novel prognostic biomarkers. Method: We develop a novel measure of CIN based on mathematical modeling of genome breakage patterns. Utilizing segmented CNV profiles we infer two properties of the genome breakage process – its intensity and dispersion. Intensity is proportional to the number of genome breakpoints and is estimated by the total number of CNV segments. Dispersion quantifies spatial dependence of breakpoints over a wide range of genomic scales. It is computed by Ripley’s K-function which compares the distribution of breakpoints with a random Poisson point process. Jointly, intensity and dispersion, which we term the weighted Genome Contiguity Index (wGCI), estimate the contiguity of the genome. We contrast the wGCI with a heuristic score based on quantiles of segment lengths. Results: We sought to evaluate the association between the wGCI and patients’ survival in a proteogenomic cohort of 215 primary LUAD tumors profiled using whole-genome sequencing (WGS). In support of our hypothesis, we observed that patients had significantly better outcomes if their tumor genomes had longer segments compared to those with a high degree of fragmentation (P=0.014). Utilizing the wGCI, we found that tumor genomes with dispersed breakage (and low contiguity) were associated with worst outcomes (P=0.01). To identify mechanisms associated with dispersed breakage patterns we compared genomic features of the two groups. We found that LUAD tumors with highly fragmented genomes have a higher rate of TERT amplifications (P=0.005) and increased TERT RNA expression (P=2e-6). These tumors are also enriched for TP53 mutations (P=2e-16) and MYC amplifications (P=0.01). Given the cost and practical challenges of clinical WGS we leveraged mass-spectrometry proteomics to identify protein biomarkers. We found IGF2BP3 to be significantly upregulated (P=0.0006) in highly fragmented tumors and to be associated with worse outcomes (P=0.011). We use immunohistochemistry (IHC) to develop and validate IGF2BP3 as a prognostic biomarker in independent lung cancer cohorts. Conclusion: We introduced a novel quantitative measure of CIN that uniquely captures the impact of chromosome breakage on LUAD patients' clinical outcomes. Further, we identified upregulation of IGF2BP3 as a proxy to identify fragmented genomes using IHC. Our work underscores the value of mathematical modeling of genome breakage patterns to develop complex genomic biomarkers. Citation Format: Noshad Hosseini, Rahul Mannan, Alexey Nesvizhskii, Saravana Dhanasekaran, Marcin Cieslik. Development of a novel class of genomic biomarkers through mathematical modeling of cancer genome contiguity [abstract]. In: Proceedings of the AACR Special Conference in Cancer Research: DNA Damage Repair: From Basic Science to Future Clinical Application; 2024 Jan 9-11; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2024;84(1 Suppl):Abstract nr A034.

Ontophylo: Reconstructing the evolutionary dynamics of phenomes using new ontology‐informed phylogenetic methods

Abstract Reconstructing ancestral states for discrete characters is essential for understanding trait evolution in organisms. However, most existing methods are limited to individual characters and often overlook the hierarchical and interactive nature of traits. Recent advances in phylogenetics now offer the possibility of integrating knowledge from anatomy ontologies to reconstruct multiple discrete character histories. Nonetheless, practical applications that fully harness the potential of these new approaches are still lacking. This paper introduces ontophylo, an R package that extends the PARAMO pipeline to address these limitations. Ontophylo enables the reconstruction of phenotypic entities composed of amalgamated characters, such as anatomical regions or entire phenomes. It offers three new applications: (1) reconstruction of evolutionary rates of amalgamated characters using phylogenetic non‐homogeneous Poisson process (pNHPP) that allows modelling rate variation across tree branches and time; (2) reconstruction of morphospace dynamics; and (3) visualization of evolutionary rates on vector images of organisms. Ontophylo incorporates ontological knowledge to facilitate these applications. Benchmarking confirms the accuracy of pNHPP in estimating character rates under different evolutionary scenarios, and example applications demonstrate the utility of ontophylo in studying morphological evolution in Hymenoptera using simulated data. Ontophylo can be easily integrated with other ontology‐oriented and general‐purpose R packages and offers new opportunities to examine morphological evolution on a phenomic scale using new and legacy data.

Clustered jamming and antenna beam-width fluctuations for UAV-assisted cellular networks

Jamming devices have the capability to deliberately disrupt legitimate communications. Clustered jamming pertains to the deployment of jammers that are concentrated within specific clusters to amplify the interference. This paper models the impact of clustered jamming interference on unmanned aerial vehicles (UAV)-assisted cellular networks, taking into account fluctuations of the 3-dimensional (3D) beam-width of low altitude platform (LAP) antennas. The distribution of UAVs and macro base station (MBS) is modeled by independent 3D and 2D homogeneous Poisson point processes, respectively, while jammers are modeled using an independent Matern cluster process. Antenna 3D beam-width fluctuations are modeled according to Normal distribution. The modeling centers on the probability of association as well as coverage and spectral efficiency for line-of-sight LAP, non-line-of-sight LAP, and MBS links. This modeling investigation is conducted under multiple jamming clusters and the antenna 3D beam-width fluctuations of UAV. Analytical expressions are derived to characterize association, coverage, and SE performance in the context of clustered jamming and UAV beam-width variation. The obtained results indicate that UAV-assisted cellular networks experience a notable decline in the proximity of clustered jammers. Furthermore, this performance degradation is significantly exacerbated when the antenna 3D beam-width exhibits fluctuations. Consequently, to enhance the performance of UAV-assisted cellular networks, it is imperative for system designers to consider the implementation of robust anti-jamming techniques.

СТОХАСТИЧНЕ МОДЕЛЮВАННЯ ОПТИМАЛЬНИХ ЗВ’ЯЗКІВ МІЖ ТІНЬОВОЮ ТА ЛЕГАЛЬНОЮ ЕКОНОМІКАМИ

The article is devoted to the stochastic modelling of optimal links between the shadow and legal economies using Wiener and Poisson processes. The study focuses on identifying the best management strategies that will help reduce the size of the shadow economy. The paper first considers a deterministic economic and mathematical model of the relationship between the shadow and legal economies, which is presented in the form of differential equations with initial conditions. These differential equations describe the rate of change of real sector workers' savings, the dynamics of owners' savings, the rate of change of the price of aggregate output, etc. The assumptions about the dynamics of cash savings and the price of aggregate output are supplemented by initial conditions. Restrictions on the consumer shares of savings used for consumption of legal and shadow products for employees and entrepreneurs are formed. To build an optimal process, the author formalizes an objective function (objective criterion) that maximizes the average total (integral) savings of employees and entrepreneurs over a certain period of time and uses stochastic sufficient conditions for optimality, where the controls are the shares of savings used to purchase aggregate products of the legal and shadow economies of employees and business owners, and the phase trajectories are the monetary savings of employees and business owners of legal and shadow economies, the price of a unit of aggregate products, and the price of a unit of aggregate products. Thus, the main idea of the process of solving the optimal management problem takes into account both the management process itself (strategies, management methods) and the paths along which the system moves from the initial state to the final result (phase trajectories). The structure of the optimal process of the proposed stochastic model is described. It is established that the optimal controls are deterministic values and do not depend on the coefficients of the Wiener processes in the proposed stochastic dynamic model. It is also worth noting that the specific formulation of this problem of optimal control of the interaction between the shadow and legal economies depends on the context of the study and may vary depending on the tasks and assumptions set.

Poisson process with linear drift and related function series

Рассматривается случайный процесс $Y(t)=at-\nu_+(pt)+\nu_-(-qt)$, $t\in(-\infty,\infty)$, где $\nu_{\pm}(t)$ - независимые стандартные пуассоновские процессы при $t\geqslant 0$ и $\nu_{\pm}(t)=0$ при $t<0$. Параметры $a$, $p$ и $q$ таковы, что $\mathbf{E}Y(t)<0$, $t\neq 0$. В работе найдены суммы $\varphi_m(z,r)=\sum_{k\geqslant 0}(re^{-r})^{k}(z+k)^{m+k-1}/k!$, $m=1,2,…$, $z\geqslant 0$, функциональных рядов с параметром $ r\in(0,1) $, которые используются для рекуррентного вычисления моментов $\mathbf{E}(t^*)^m$, $m\geqslant 1$, времени $t^*$ достижения максимума траекторией процесса $Y(t)$. Результаты работы применимы к задаче об оценивании параметра $\theta$ по $n$ наблюдениям с плотностью $f(x,\theta)$, которая имеет скачок в точке $x=x(\theta)$, $x'(\theta)\neq 0$. Если $\widehat\theta_n$ - это оценка максимального правдоподобия истинного параметра $\theta_0$, то предельным при $n\to\infty$ распределением для нормированных оценок $n(\widehat\theta_n - \theta_0)$ будет распределение аргумента максимума $t^*_{\theta_0}$ траектории процесса $Y(t)$ с параметрами $a$, $p$ и $q$, зависящими от односторонних пределов плотности в точке $x(\theta_0)$ и от производной $x'(\theta_0)$. Вычисление моментов $\mathbf{E}(t^*_{\theta_0})^m$, $m=1, 2$, в этом случае позволяет оценить величину асимптотического смещения оценки максимального правдоподобия и ее эффективность.

Sensitivity analysis of a non-Markovian feedback retrial queue, reneging, delayed repair with working vacation subject to server breakdown

This study investigated the steady-state characteristics of a non-Markovian feedback retrial queue with reneging, delayed repair, and working vacation. In this scenario, we assumed that consumers arrive through Poisson processes and the server provides service to consumers during both regular and working vacation periods. However, it is subject to breakdowns at any moment, resulting in a service interruption for a random duration. Additionally, the concept of delay time was also presented. The consumer that is dissatisfied with the service may re-enter the orbit to receive another service; this individual is considered a feedback consumer. The server will go on a working vacation if the orbit is empty after successfully serving a satisfied consumer. By utilizing the supplementary variable technique (SVT), we examined the steady-state probability generating function of the system and orbit sizes. Finally, numerical outcomes and a sensitivity analysis were given to verify the analytical findings of important performance indicators.

A pertinent approach: Two class of uncertain priority queuing models under the steady state condition

A pertinent approach: Two class of uncertain priority queuing models under the steady state condition

Analysis of a renewal arrival process subject to geometric catastrophe with random batch killing

This paper considers a population model (system) which is prone to catastrophe that kills individuals in batches. Individuals enter the system in accordance with the renewal process and catastrophe occurs as per the Poisson process. The catastrophe attacks the population in a successive order in batches of random size, each batch of individuals dies with probability ξ. This successive process ends when the whole population is wiped out or a batch of individuals survives with probability 1 − ξ. This type of killing pattern is known as geometric catastrophe. The supplementary variable technique is used to develop the steady-state governing equations. Further using the difference operator, the distributions of population size are evaluated at arbitrary, pre-arrival, and post-catastrophe epochs. In addition to that, a few different measurements of the system’s performance are derived. In order to demonstrate the applicability of the model, a number of numerical and graphical outcomes are presented in the form of tables and graphs.

Bayesian regression analysis of panel count data under frailty nonhomogeneous Poisson process model with an unknown frailty distribution

Bayesian regression analysis of panel count data under frailty nonhomogeneous Poisson process model with an unknown frailty distribution

Modeling and identifiability of non-homogenous Poisson process cure rate model

Modeling and identifiability of non-homogenous Poisson process cure rate model

On the Construction of Conditional Probability Densities in the Brownian and Compound Poisson Filtrations

In this paper, we construct supermartingales valued in [0,1] as solutions of an appropriate stochastic differential equation on a given reference filtration generated by either a Brownian motion or a compound Poisson process. Then, by means of the results contained in [M. Jeanblanc and S. Song, Stochastic Processes Appl. 121 (2011) 1389–1410], it is possible to construct an associated random time on some extended probability space admitting such a given supermartingale as conditional survival process and we shall check that this construction (with a particular choice of supermartingale) implies that Jacod’s equivalence hypothesis, that is, the existence of a family of strictly positive conditional probability densities for the random times with respect to the reference filtration, is satisfied. We use the components of the multiplicative decomposition of the constructed supermartingales to provide explicit expressions for the conditional probability densities of the random times on the Brownian and compound Poisson filtrations.

Timely Cache Updating in Parallel Multi-Relay Networks

We consider a system consisting of a server, which receives updates for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> files according to independent Poisson processes. The goal of the server is to deliver the latest version of the files to a user through a parallel network of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> caches. We consider an update received by the user successful, if the user receives the same file version that is currently prevailing at the server. We derive an analytical expression for information freshness at the user. We observe that freshness for a file increases with increase in consolidation of rates across caches. To solve the multi-cache problem, we first solve the auxiliary problem of a single-cache system. We then rework this auxiliary solution to our parallel-cache network by consolidating rates to single routes as much as possible. This yields an approximate (sub-optimal) solution for the original problem. We provide an upper bound on the gap between the sub-optimal solution and the optimal solution. We present counterpart expressions and policies for version age of information by employing a stochastic hybrid system approach. Numerical results for both timeliness metrics show that the proposed sub-optimal policy closely follows the optimal policy.

Detecting Abrupt Changes in High-Dimensional Self-Exciting Poisson Processes

High-dimensional self-exciting point processes have been widely used in many application areas to model discrete event data in which past and current events affect the likelihood of future events. In this paper, we are concerned with detecting abrupt changes of the coefficient matrices in discrete-time high-dimensional self-exciting Poisson processes, which have yet to be studied in the existing literature due to both theoretical and computational challenges rooted in the non-stationary and high-dimensional nature of the underlying process. We propose a penalized dynamic programming approach which is supported by a theoretical rate analysis and numerical evidence.

Information divergences and likelihood ratios of Poisson processes and point patterns

Information divergences and likelihood ratios of Poisson processes and point patterns

Joint distribution of the supremum and the time of its attaining for a trajectory of the Poisson process with linear drift

Joint distribution of the supremum and the time of its attaining for a trajectory of the Poisson process with linear drift

Low-noise preamplifier for scanning tunneling microscope

The current preamplifier is one of the important components of the scanning tunneling microscope (STM), and its performance is crucial to the basic operations of the STM system, as well as for the development of demanding novel functionalities such as autonomous atomic fabrication. In this study, the factors that affect the performance of a current preamplifier, including its noise spectrum density and the bandwidth, are analyzed in depth, and a preamplifier is designed and fabricated specifically for the STM system. By using a carefully selected low-noise op amp chip, the optimized current preamplifier has a noise floor as low as 4 &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ {\mathrm{f}}{\mathrm{A}}/\sqrt{{\mathrm{H}}{\mathrm{z}}} $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240560_M2.jpg"/&gt;&lt;graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240560_M2.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and a bandwidth of 2.3 kHz, at its most sensitive transimpedance gain of 1 GΩ. It has three transimpedance gains, 10 MΩ, 100 MΩ, and 1 GΩ, that can be switched through digital control signals. A two-switch configuration is adopted to minimize the noise floor while maintaining the optimal bandwidth. The current detectable by this three-level preamplifier ranges from pA to μA, satisfying the requirements of most STM operations. Using this preamplifier, the fundamental functions of the STM system are successfully demonstrated, including surface topographic characterization, scanning tunneling spectroscopy, and single atom/molecule manipulation. The measurement of shot noise in tunneling current is also explored, and a linear relationship between shot noise and tunneling current is obtained by carefully analyzing noise. It is illustrated that the Fano factor of the shot noise in a normal metallic tunneling junction is approximately equal to 1, revealing the expected Poisson process for electron tunneling in such a scenario. The results are valuable for the high-resolution characterization of correlation systems in the future.

Performance approximations of Markov Modulated Poisson Process arrival queue with Markovian Service using matrix geometric approach

Queue analysis of correlated work of the arrival and the service process are not available much in literature. This research paper applies matrix geometric approach to study the MMPP/MSP/1 queuing model under the quasi-birth death(QBD) process. Customers admitted in the system are having two different intensity of arrival followed by Markov Modulated Poisson Process(MMPP). MMPP is a versatile class of MAP and it is necessary to adapt the first characterization of the queuing model. In service mechanism Markov Service Process(MSP) is adopted. As like MMPP, MSP is also a versatile and can capture the correlation of the modulated arrivals. The methodology presented in this research work calculates the metric performances not depending on the series of infinite state of the system. The assumed model is formulated as QBD process to obtain the steady state probability of the system. The complexity model formulated involves structures matrices with appropriate dimension. Neut?s pioneered the matrix geometric approach to such complex models which involves the rate computation of rate matrix R. The rate matrix R is approximated to derive the performance metrics of the system and then compute the cost for the underlying Markov chain. To attain this approximation the sensitivity analysis is done with numerical results.