Geometric Probability Distribution Research Articles

Pipelines reliability assessment considering corrosion-related failure modes and probability distributions characteristic using subset simulation

A methodology using the subset simulation and the modified Metropolis-Hastings algorithm with delayed rejection (SS-MMHDR) is developed to evaluate the time-dependent reliability of corroded pipelines under multiple failure modes. This methodology considers the competition between the small leak and burst failures and the correlated non-normal distribution of the defect variables, and characterizes the defect geometric probability distribution from extensive inspection data. Results demonstrate that SS-MMHDR has high accuracy and low variability in evaluating the pipeline failures, outperforming the SS-MMH and Monte Carlo simulation algorithms, especially under low probability levels, but with slightly increased computation time. The Generalized Extreme Value Type II (GEVII) distribution is the optimal distribution for internal corrosion defect lengths from the Kolmogorov-Smirnov test. Lognormal, Weibull and Normal distributions provide more conservative estimates on burst and composite failure probabilities and limiting maintenance times than GEVII. Effects of operating pressure and its variability, corrosion rate on failure probability, and pipeline limiting maintenance time are also revealed through numerical cases.

Two-Stage Stochastic Optimization Model for Multi-Microgrid Planning

This paper presents a Two Stage stochastic Programming (TSSP) model for the planning of Multi-Microgrids (MMGs) in Active Distribution Networks (ADNs). The model aims to minimize the total costs while benefiting from interconnections of Microgrids (MGs), considering uncertainties associated with electricity demand and Renewable Energy Sources (RESs). The associated uncertainties are analyzed using Geometric Brownian Motion (GBM) and probability distribution functions (pdfs). The model includes long-term purchase decisions and short-term operational constraints, using Geographical information Systems (GIS) to realistically estimate rooftop solar limits. The planning model is used to study the feasibility of implementing an MMG system consisting of 4 individual Microgrids (MGs) at an ADN in a municipality in the state of São Paulo, Brazil. The results show that the TSSP model tends to be less conservative than the deterministic planning model, which is based on simple and pessimistic reserve constraints, while performing faster than a simple Stochastic Linear Programming (SLP) algorithm, with higher accuracy.

A Discrete Model for the Evolution of Infection Prior to Symptom Onset

We consider a between-host model for a single epidemic outbreak of an infectious disease. According to the progression of the disease, hosts are classified in regard to the pathogen load. Specifically, we are assuming four phases: non-infectious asymptomatic phase, infectious asymptomatic phase (key-feature of the model where individuals show up mild or no symptoms), infectious symptomatic phase and finally an immune phase. The system takes the form of a non-linear Markov chain in discrete time where linear transitions are based on geometric (main model) or negative-binomial (enhanced model) probability distributions. The whole system is reduced to a single non-linear renewal equation. Moreover, after linearization, at least two meaningful definitions of the basic reproduction number arise: firstly as the expected secondary asymptomatic cases produced by an asymptomatic primary case, and secondly as the expected number of symptomatic individuals that a symptomatic individual will produce. We study the evolution of infection transmission before and after symptom onset. Provided that individuals can develop symptoms and die from the disease, we take disease-induced mortality as a measure of virulence and it is assumed to be positively correlated with a weighted average transmission rate. According to our findings, transmission rate of the infection is always higher in the symptomatic phase yet under a suitable condition, most of the infections take place prior to symptom onset.

State Monitoring of Gas Regulator Station Based on Feature Selection of Improved Grey Relational Analysis

The in-depth application of the IoT technology in the gas industry has improved the intelligence of the gas system. As an important part of the gas system, the optimization of state monitoring of regulator stations is of great significance. The efficiency of monitoring can be improved by feature selection, but the reduction of important features will reduce the accuracy of identification. Therefore, a feature selection method based on gray relational analysis is proposed. First, the distance between the comparison matrix and the reference matrix is transformed using the geometric probability distribution, which solves the problem of data initial transformation. Then, an adaptive value method of resolution coefficient based on the cuckoo algorithm is proposed. Finally, a weight coefficient is constructed by combining the redundancy and the correlation to improve the gray relational degree applied to nonsequential systems. The maximum classification accuracy and the number of selected features are used to compare the performance of feature selection methods for several different types of data sets. Simulation analysis shows that the proposed method has higher maximum classification accuracy and smaller selected feature set. Finally, the proposed method is applied to the state monitoring of gas regulator stations. Seven feature selection methods, including the proposed method, the classical methods, and the advanced methods are combined with a support vector machine, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -nearest neighbor, and decision tree, respectively. The experimental results indicate that the comprehensive performance of the proposed method combined with the three classifiers is good. It enables the subset containing the most identifiable features to be obtained quickly, which is beneficial to improve the efficiency of state monitoring.

Development and verification of three-dimensional equivalent discrete fracture network modelling based on the finite element method

The discrete fracture network (DFN) method is essential for estimating the mechanical properties of naturally fractured rock masses. A three-dimensional (3D) equivalent DFN model was proposed and thoroughly validated for fractured rock masses. In the equivalent DFN modelling, the Monte Carlo method and 3D rock failure process analysis (RFPA3D) software were used. Additionally, the fracture and its spatial distribution were described explicitly, and the strength and deformability of complex fractured rock masses were investigated. A case study was conducted at the dam site area of Lianghekou Hydropower Station's the left bank slope. The ShapeMetriX3D system was used to acquire the geometric parameters and probability distribution of fractures via digital photogrammetry. Additionally, a series of numerical models were created by combining the proposed equivalent DFN method with the fracture geometry information from the case study. Based on the concept of scale effect and anisotropy, the representative element volume (REV) and mechanical parameters of fractured rock masses were determined. The mechanical parameters of slope rock masses obtained by the proposed DFN modelling and generalized Hoek Brown strength criterion were compared. Furthermore, the applicability and reliability of the proposed model were validated using site data and numerical verifications. The proposed 3D equivalent DFN modelling can analyze the scale effect and anisotropy of mechanical properties of fractured rock masses effectively. Finally, the influence of water head on the hydraulic characteristics of fractured rock masses was analyzed based on the proposed model. Results showed that fractures were the main pathway of water.

Estimating Call-Drop Probability Rates among Wireless Cellular Networks in Ghana

The increasing rate of call-drop among wireless cellular networks has become a headache for subscribers of mobile voice networks in Ghana. Despite the many studies that have been carried out on call-drop, most of them have failed to examine call-drop probability rates, a key metric of tackling the issue of call-drop. This study combines the probability distribution function of call-drop and maximum likelihood estimator to estimate call-drop probability rates among the wireless cellular networks in Ghana. The call-drop of the networks was, first of all, fitted to a probability distribution. The call-drop probability rate was then estimated using the maximum likelihood estimator of the fitted distribution. The call-drop probability distribution of Airtel-Tigo, Vodafone, and Glo follow the Geometric probability distribution, while that of MNT follows the Negative Binomial probability distribution. The call-drop probability rate of MTN is 7.6%, Vodafone is 3.3%, Airtel-Tigo, and Glo is 3.0%. These call-drop rates are found to be outside the prescribed NCA's acceptable limit of less than 3 %.

A randomized response model based on blank card and geometric distribution to estimate sensitive proportion

Using geometric probability (GP) distribution as a randomization device, a new randomized response (RR) model has been proposed which has an immense potential to estimate the human population proportion that possess a stigmatized character. Privacy protection measure of proposed model and some of its properties have been investigated. In addition, empirical experiments are conducted to validate the theoretical results, which demonstrate the better performance of the suggested estimators over their direct competitors. Finally, results are analyzed and appropriate suggestions are made available to survey practitioners when dealing with sensitive aspects.

The Probability Distribution of Time to Extinction: A simulation study

The time to extinction (TTE) is very important topic in some fields of studies; ecology, economics, corporate competition, bacterial sciences and epidemiology. The aim of the present paper is to investigate about the empirical parametric and empirical nonparametric probability distribution of the time to extinction for two related stochastic models; Rosenzweig and Macarthur model and May model. First, we assume the amplitude of r is a random variable with a continuous uniform probability distribution on a closed interval [r 1, r 2]. Second, we assume the number of consecutive years’ during which the amplitude of r remains constant is a random variable with a discrete geometric probability distribution with parameter p which is supported on a positive integer ℤ+ and p ∈ (0,1).

Study on the Probability Model of Ship-Bridge Collision

Based on the comprehensive analysis of the existing major ship collision probability models, this paper combined the advantages of the AASHTO model and the KUNZI model and improved AASHTO model by using the ship simulator to obtain the ship’s track probability distribution curve to replace the original geometric probability distribution curve. It also introduced ship stopping probability function to make up for the shortcomings of the AASHTO model in which the influence of human factors is not considered. Finally, this paper predicts the collision probability of the Longjiang Bridge in 2025 with the improved model.

A randomized response model based on blank card and geometric distribution to estimate sensitive proportion

Using geometric probability (GP) distribution as a randomization device, a new randomized response (RR) model has been proposed which has an immense potential to estimate the human population proportion that possess a stigmatized character. Privacy protection measure of proposed model and some of its properties have been investigated. In addition, empirical experiments are conducted to validate the theoretical results, which demonstrate the better performance of the suggested estimators over their direct competitors. Finally, results are analyzed and appropriate suggestions are made available to survey practitioners when dealing with sensitive aspects.

Automated Design of Realistic Contingencies for Big Data Generation

The letter proposes an algorithm for big data generation based on realistic selection of a set of contingencies for power systems described by undirected graphs. Every contingency is created by eliminating a certain number of elements in the system represented by graph edges. The number of elements as well as the distance between elements of the contingency is randomly selected according to a geometric probability distributions based on historical data. The duration of a fault that starts the contingency as well as the time intervals between elements of the contingency are chosen by sampling from a gamma distribution. In addition, the absence of islands in the system is assessed by analyzing the connectedness of the graph with deleted edges, which is quantified by computing the number of zero eigenvalues of the Laplacian matrix of the resulting graphs. The algorithm is validated on the Nordic 44-bus power system.

Local distributions of the 1D dilute Ising model

The local distributions of the one-dimensional dilute annealed Ising model with charged impurities are studied. Explicit expressions are obtained for the pair distribution functions and correlation lengths, and their low-temperature asymptotic behavior is explored depending on the concentration of impurities. For a more detailed consideration of the ordering processes, we study local distributions. Based on the Markov property of the dilute Ising chain, we obtain an explicit expression for the probability of any finite sequence and find a geometric probability distribution for the lengths of sequences consisting of repeating blocks. An analysis of distributions shows that the critical behavior of the spin correlation length is defined by ferromagnetic or antiferromagnetic sequences, while the critical behavior of the impurity correlation length is defined by the sequences of impurities or by the charge-ordered sequences. For the dilute Ising chain, there are no other repeating sequences whose mean length diverges at zero temperature. While both the spin correlation and the impurity correlation lengths can diverge only at zero temperature, the ordering processes result in a maximum of the specific heat at finite temperature defined by the maximum rate of change of the impurity-spin pairs concentration. A simple approximate equation is found for this temperature. We show that the non-ordered dilute Ising chains correspond to the regular Markov chains, while various orderings generate the irregular Markov chains of different types.

Geometric branching reproduction Markov processes

We present a model of a continuous-time Markov branching process with the infinitesimal generating function defined by the geometric probability distribution. It is proved that the solution of the backward Kolmogorov equation is expressed by the composition of special functions – Wright function in the subcritical case and Lambert-W function in the critical case. We found the explicit form of conditional limit distribution in the subcritical branching reproduction. In the critical case, the extinction probability and probability mass function are expressed as a series containing Bell polynomial, Stirling numbers, and Lah numbers.

On geometric probability distributions on the torus with applications to molecular biology

In this paper, we study a family of probability distributions, alternative to the von Mises family, called Inverse Stereographic Normal Distributions. These distributions are counterparts of the Gaussian Distribution on $\mathbb{S}^{1}$ (univariate) and $\mathbb{T}^{n}$ (multivariate). We discuss some key properties of the models, such as unimodality and closure with respect to marginalizing and conditioning. We compare this family of distributions to the von Mises’ family and the Wrapped Normal Distribution. Then, we discuss some inferential problems, introduce a notion of moments which is natural for inverse stereographic distributions and revisit a version of the CLT in this context. We construct point estimators, confidence intervals and hypothesis tests and discuss briefly sampling methods. Finally, we conclude with some applications to molecular biology and some illustrative examples. This study is motivated by the Protein Folding Problem and by the fact that a large number of proteins involved in the DNA-metabolism assume a toroidal shape with some amorphous regions.

Hugo Steinhaus problem of estimation of the war casualities on the base of contemporary press obituaries

In the memoirs of Professor Hugo Steinhaus the problem of estimation of the war casualties of German Army during the SecondWorldWar on the base of contemporary press obituaries is mentioned.A ssuming the inflated geometric probability distribution function for the size of German family, introducing some transformations of this random variable and taking into account the probability distribution function of the contens of obituaries we propose in the note a solution of the problem.

Hybrid Slotted-CSMA/CA-TDMA for Efficient Massive Registration of IoT Devices

Recently, the Wi-Fi alliance announced a new Wi-Fi standard known as IEEE 802.11ah (or Wi-Fi HaLow) to efficiently support Internet of Things (IoT) applications. However, the existing registration method under IEEE 802.11ah, based on carrier-sense multiple access with collision avoidance (CSMA/CA), was analyzed as not efficient enough for registration of large-scale machine-to-machine (M2M) communications where a massive number of devices try to access a single, centralized access point (AP). In this paper, we propose a hybrid slotted-CSMA/CA–time-division multiple access (TDMA) (HSCT) medium access control (MAC) protocol for efficient massive registration of IoT devices (up to 8000) in M2M networks. We focus on situations, where a large number of M2M devices simultaneously try to register at a single, centralized AP. In the proposed HSCT, contention-based slotted-CSMA/CA allows devices to send an authentication request via randomly selected backoff slots, whereas contention-free TDMA permits those devices to send/receive the subsequent association request/association response via an individually allocated TDMA slot. In addition, a centralized authentication control (CAC)-based mechanism with modified algorithms for optimal selection of CAC parameters and the slotted fixed-window CSMA protocol with Sift geometric probability distribution are used to mitigate severe contention between massive registrations upon network (re-)initialization from an AP reboot. This paper also analyzes the performance of the proposed scheme and determines the optimal configuration to enhance registration performance. Simulation results demonstrate that the proposed HSCT MAC protocol achieves substantial improvement, compared with the contention-free transmission, a combined authentication/association scheme, and the conventional IEEE 802.11ah with CSMA/CA.

Gaussian States Minimize the Output Entropy of the One-Mode Quantum Attenuator

We prove that Gaussian thermal input states minimize the output von Neumann entropy of the one-mode Gaussian quantum-limited attenuator for fixed input entropy. The Gaussian quantum-limited attenuator models the attenuation of an electromagnetic signal in the quantum regime. The Shannon entropy of an attenuated real-valued classical signal is a simple function of the entropy of the original signal. A striking consequence of energy quantization is that the output von Neumann entropy of the quantum-limited attenuator is no more a function of the input entropy alone. The proof starts from the majorization result of De Palma et al., IEEE Trans. Inf. Theory 62, 2895 (2016), and is based on a new isoperimetric inequality. Our result implies that geometric input probability distributions minimize the output Shannon entropy of the thinning for fixed input entropy. Moreover, our result opens the way to the multimode generalization, that permits to determine both the triple trade-off region of the Gaussian quantum-limited attenuator and the classical capacity region of the Gaussian degraded quantum broadcast channel.

Analyzing dependence in incidence of diabetes and heart problem using generalized bivariate geometric models with covariates

ABSTRACTFor analyzing incidence data on diabetes and health problems, the bivariate geometric probability distribution is a natural choice but remained unexplored largely due to lack of models linking covariates with the probabilities of bivariate incidence of correlated outcomes. In this paper, bivariate geometric models are proposed for two correlated incidence outcomes. The extended generalized linear models are developed to take into account covariate dependence of the bivariate probabilities of correlated incidence outcomes for diabetes and heart diseases for the elderly population. The estimation and test procedures are illustrated using the Health and Retirement Study data. Two models are shown in this paper, one based on conditional-marginal approach and the other one based on the joint probability distribution with an association parameter. The joint model with association parameter appears to be a very good choice for analyzing the covariate dependence of the joint incidence of diabetes and heart diseases. Bootstrapping is performed to measure the accuracy of estimates and the results indicate very small bias.

Design and Analysis of an Optimal Inventory Model for Perishable Goods with Fixed Life Time

Background/Objectives: The main purpose of this paper is to plan and implement an optimum inventory model for the deteriorating things that minimizes the entire expected cost over a finite horizon. Methods/Statistical Analysis: The model presents a projected price operates for the inventory system with fixed life time. This paper accords with the hassle of ciphering optimum ordering policies for one item for successive intervals. Hyper geometric probability distribution is taken into consideration to examine the expected outdated cost. Findings: The optimum inventory model is solved analytically to review the impact of modification within the worth of the parameters. The solution methodology provided within the model helps the choice maker to enforce the first in first out policy by exposing customers to merchandise of a similar age. The results have been explained with numerical examples and graphical illustration. Applications/Improvements: First in First out issuing policy is designed to reveal the fact that it minimizes the expected outdates.

Steps and Gaps in Ground Ruptures: Empirical Bounds on Rupture Propagation

Abstract We analyze a set of 76 mapped surface ruptures for relationships between geometrical discontinuities in fault traces and earthquake rupture extent. The combined set includes 46 strike‐slip, 16 normal, and 14 reverse mechanism events. The survey shows ∼90% of ruptures have at least one end at a mappable discontinuity, either a fault end or a step of 1 km or greater. Dip‐slip ruptures cross larger steps than strike‐slip earthquakes, with maxima of ∼12 versus ∼5 km, respectively. Large steps inside strike‐slip ruptures are rare; only 8% (5 of 62) are ≥4 km. A geometric probability distribution model of steps as “challenges” to rupture propagation predicts that steps of 1 km or greater will be effective in stopping rupture about 46% of the time. The rate is similar for dip‐slip earthquakes, but, within this set, steps are relatively more effective in stopping reverse ruptures and less effective in stopping normal ruptures. By comparing steps at rupture terminations to the set of steps broken in rupture, we can estimate the importance of step size for stopping rupture. We define the passing ratio for a given step size as the fraction of steps broken divided by the corresponding fraction that stop rupture. A linear model for steps from 1 to 6 km in strike‐slip ruptures leads to the passing ratio =1.89–0.31× step width. Steps of ∼3 km are equally likely to be broken or to terminate rupture, and steps ≥6 km should almost always stop rupture. A similar comparison suggests that extensional steps are somewhat more effective than compressional steps in stopping ruptures. We also compiled the incidence of gaps of 1 km and longer in surface ruptures. Gaps occur in ∼43% of ruptures and occur more frequently in dip‐slip than strike‐slip ruptures. Online Material: Figures of annotated surface rupture maps for 40 earthquakes.