Probability Density Function Of Time Research Articles

Investigating the influence of sub-mesoscale current structures on Baltic Sea connectivity through a Lagrangian analysis

This study explores the impact of sub-mesoscale structures and vertical advection on the connectivity properties of the Baltic Sea using a Lagrangian approach. High-resolution flow fields from the General Estuarine Transport Model (GETM) were employed to compute Lagrangian trajectories, focusing on the influence of fine-scale structures on connectivity estimates. Six river mouths in the Baltic Sea served as initial positions for numerical particles, and trajectories were generated using flow fields with varying horizontal resolutions: 3D trajectories with 250m resolution as well as 2D trajectories with 250m and 1km resolutions. Several Lagrangian indices, such as mean transit time, arrival depths, and probability density functions of transit times, were analyzed to unravel the complex circulation of the Baltic Sea and highlight the substantial impact of sub-mesoscale structures on numerical trajectories. Results indicate that in 2D simulations, particles exhibit faster movement on the eastern side of the Gotland Basin in high-resolution compared to coarse-resolution simulations. This difference is attributed to the stronger coastal current in high-resolution compared to coarse-resolution simulations. Additionally, the study investigates the influence of vertical advection on numerical particle motion within the Baltic Sea, considering the difference between 3D and 2D trajectories. Findings reveal that denser water in the eastern and south-eastern areas significantly affects particle dispersion in 3D simulations, resulting in increased transit times. Conversely, regions in the North-western part of the basin accelerate particle movement in 3D compared to the 2D simulations. Finally, we calculated the average residence time of numerical particles exiting the Baltic Sea through the Danish strait. Results show an average surface layer residence time of approximately 790 days over an eight-year integration period, highlighting the relatively slow water circulation in the semi-enclosed Baltic Sea basin. This prolonged residence time emphasizes the potential for the accumulation of pollutants. Overall, the study underscores the pivotal role of fine-scale structures in shaping the connectivity of the Baltic Sea, with implications for understanding and managing environmental challenges in this unique marine ecosystem.

Estimating Hazard Function through Reliability Function and Empirical Methods

In this research, the reliability functions are applied to estimate the hazard function of four used car components (tires, brakes, lights, engine), which are inspected by aperiodic vehicles inspection (PVI) established in Erbil city, as a specialized company that conducts the annual technical inspection of vehicles to detect the failure component, that either require repair or replace it with a new one. For our purpose, the data about the failure components of a sample of size (50000) cars are obtained from the Erbil traffic directorate, which are annually inspected for 11 years (2010 – 2020) by a (PVI) company. From the available data the reliability function, hazard function, and probability density function of failure time of each component are found by the non-parametric method and the estimated Rayleigh distribution since the failure rates of the components are the linear functions of time, also the comparison between their reliability values have made by the mean absolute error method.

Adjoint subordination to calculate backward travel time probability of pollutants in water with various velocity resolutions

Abstract. Backward probabilities, such as the backward travel time probability density function for pollutants in natural aquifers/rivers, have been used by hydrologists for decades in water quality applications. Calculating these backward probabilities, however, is challenging due to non-Fickian pollutant transport dynamics and velocity resolution variability at study sites. To address these issues, we built an adjoint model by deriving a backward-in-time fractional-derivative transport equation subordinated to regional flow, developed a Lagrangian solver, and applied the model/solver to trace pollutant transport in diverse flow systems. The adjoint model subordinates to a reversed regional flow field, transforms forward-in-time boundaries into either absorbing or reflective boundaries, and reverses the tempered stable density to define backward mechanical dispersion. The corresponding Lagrangian solver efficiently projects backward super-diffusive mechanical dispersion along streamlines. Field applications demonstrate the adjoint subordination model's success with respect to recovering release history, groundwater age, and pollutant source locations for various flow systems. These include systems with upscaled constant velocity, nonuniform divergent flow fields, or fine-resolution velocities in a nonstationary, regional-scale aquifer, where non-Fickian transport significantly affects pollutant dynamics and backward probabilities. Caution is needed when identifying the phase-sensitive (aqueous vs. absorbed) pollutant source in natural media. The study also explores possible extensions of the adjoint subordination model for quantifying backward probabilities of pollutants in more complex media, such as discrete fracture networks.

Formula omitted] control of singularly perturbed systems using deficient hidden semi-Markov model

This paper deals with the H∞ control of a class of stochastic multi-timescale systems, called Markov jump singularly perturbed systems. The hidden semi-Markov model is introduced to handle the situation when system modes are unavailable in semi-Markov systems. Such a model is assumed deficient, that is, it lacks knowledge about the emission probability, transition probability, and probability density function of the sojourn time. It is a more general case compared with works conducted with perfect transition information. Depending on whether a fast or slow sampling rate is used, the resulting discrete-time singularly perturbed system is modeled differently, for both of which the controller design is conducted. Furthermore, criteria expressed in terms of linear matrix inequalities (LMIs) are developed that guarantee the δ-error mean-square stability. An approach to estimate the upper bound on δ-error with incomplete information is provided, meanwhile, the relationship between system performance and the upper of singular perturbation parameter is also presented. Finally, two simulation examples using real-world systems are provided to corroborate the validity as well as the practical merits of the results.

Compositional Safe Approximation of Response Time Probability Density Function of Complex Workflows

We evaluate a stochastic upper bound on the response time Probability Density Function (PDF) of complex workflows through an efficient and accurate compositional approach. Workflows consist of activities having generally distributed stochastic durations with bounded supports, composed through sequence, choice/merge, and balanced/unbalanced split/join operators, possibly breaking the structure of well-formed nesting. Workflows are specified using a formalism defined in terms of Stochastic Time Petri Nets that permits decomposition into a hierarchy of subworkflows with positively correlated response times, guaranteeing that a stochastically larger end-to-end response time PDF is obtained when intermediate results are approximated by stochastically larger PDFs and when dependencies are simplified by replicating activities appearing in multiple subworkflows. In particular, an accurate stochastically larger PDF is obtained by combining shifted truncated Exponential terms with positive or negative rates. Experiments are performed on sets of manually and randomly generated models with increasing complexity, illustrating under which conditions different decomposition heuristics work well in terms of accuracy and complexity and showing that the proposed approach outperforms simulation having the same execution time.

Transient properties of grazing ecosystem driven by Lévy noise and Gaussian noise

This paper explores how Lévy noise and Gaussian noise affect the ecological grazing system by using the fourth-order Runge–Kutta method to simulate dynamic system and the Janicki-Weron algorithm to produce Lévy noise. Two deterministic quantities, the mean first passage time (MFPT) and the probability density function (PDF) of the first passage time (FPT), are utilized to explore the transient properties of grazing ecosystem. Our research results show that: (i) The transitions between two alternative stable states can be induced by the Gaussian noise intensity, Lévy noise intensity, Lévy stability index and Lévy skewness parameter. (ii) A higher Lévy noise intensity and a larger Lévy stability index or Lévy skewness parameter make the MFPT from desert state to sustainable vegetated state shorter; a higher Gaussian noise intensity makes it longer, which indicates that increased Lévy noise intensity can mitigate ecosystem degradation; increased Gaussian noise intensity will lead to desertification of vegetation. (iii) For larger Lévy noise intensity, the MFPT from sustainable vegetated state to desert state as a function of Gaussian noise intensity exhibits one maximum value when it transformation from the sustainable vegetated state to the desert state, the noise enhanced stability (NES) phenomena of grazing ecosystems are observed.

Sensitivity of gyre-scale marine connectivity estimates to fine-scale circulation

Abstract. We investigated the connectivity properties of an idealized western boundary current system separating two ocean gyres, where the flow is characterized by a well-defined mean circulation as well as energetic fine-scale features (i.e., mesoscale and submesoscale currents). We used a time-evolving 3D flow field from a high-resolution (HR-3D) ocean model of this system. In order to evaluate the role of the fine scales in connectivity estimates, we computed Lagrangian trajectories in three different ways: using the HR-3D flow, using the same flow but filtered on a coarse-resolution grid (CR-3D), and using the surface layer flow only (HR-SL). We examined connectivity between the two gyres along the western boundary current and across it by using and comparing different metrics, such as minimum and averaged values of transit time between 16 key sites, arrival depths, and probability density functions of transit times. We find that when the fine-scale flow is resolved, the numerical particles connect pairs of sites faster (between 100 to 300 d) than when it is absent. This is particularly true for sites that are along and near the jets separating the two gyres. Moreover, the connectivity is facilitated when 3D instead of surface currents are resolved. Finally, our results emphasize that ocean connectivity is 3D and not 2D and that assessing connectivity properties using climatologies or low-resolution velocity fields yields strongly biased estimates.

Random vibration and reliability analysis of fluid-conveying pipe under white noise excitations

Due to the random excitation in the external environment, the fluid-conveying pipe would inevitably undergo random vibration. This paper investigates the random vibration and reliability of simply-supported fluid-conveying pipes under white noise excitations. Dynamic equations of the nonlinear pipe conveying fluid under white noise excitations are modeled by the generalized Hamilton principle. Then probability density functions (PDFs) of the amplitude and energy of the system are determined by using the modified stochastic averaging method. The PDFs of displacement and velocity of the system are also discussed. The accuracy of results is verified by the Monte Carlo method. In addition, the first-passage problem of the pipe system is investigated for the first time. In order to obtain the PDFs of reliability and first failure time of the system, the forward finite difference method is adopted to solve the backward Kolmogorov equation of the system. The effects of different parameters on the amplitude, energy, reliability, and first failure time are investigated. The results show that the amplitude of the system becomes large and the first-passage time is advanced with the increase of excitation intensity or fluid speed. The amplitude of the system is decreased and the first-passage time is delayed with the increase of the damping coefficient.

The Lévy walk with rests under stochastic resetting

The Lévy walk with rests (LWR) model is a typical two-state stochastic process that has been widely and successfully adopted in the study of intermittent stochastic phenomena in physical and biological systems. Stochastic processes under resetting provide treatable and interesting schemes to study foraging and search strategies. In this manuscript, we focus on the anomalous diffusive behavior of the LWR under stochastic resetting. We consider both the case of instantaneous resetting, in which the particle stochastically returns to a given position immediately, and the case of noninstantaneous resetting, in which the particle returns to a given position with a finite velocity. The anomalous diffusive behaviors are analyzed and discussed by calculating the mean squared displacement analytically and numerically. Results reveal that the stochastic resetting can not only hinder the diffusion, where the diffusion evolves toward a saturation state, but also enhances it, where as compared with the LWR without resetting, the diffusion exponent surprisingly increases. As far as we know, the enhancement effect caused by stochastic resetting has not yet been reported. In addition, the resetting time probability density function (PDF) of the instantaneous resetting and the return time PDF of the noninstantaneous resetting are studied. Results reveal that the resetting time PDF could follow a power law provided that the sojourn time PDF is power-law distributed and the sojourn time with a heavier tail plays a decisive role in determining the resetting time PDF, whereas the shape of the return time PDF is determined by not only by the sojourn time PDF, but also by the return manner.

An Age-Dependent Prognostic Model for Nonlinear Degrading Devices Based on Diffusion Process

Degradation modeling based remaining useful life (RUL) predicting has become a significant basis of prognotics and system health management (PHM), and has drawn much favor of both scholars and engineers in the field of reliability. Based on a purely time-dependent diffusion process, this paper presents a nonlinear degradation model with special focus on the proportional relationship between the age-dependent expectation and variance of the degradation processes of the concerned devices. Exact probability density function (PDF) and cumulative distribution function (CDF) of life time and RUL in explicit forms are derived under the concept of first hitting time (FHT), incorporating the item-to-item variability. An framework of maximizing the likelihood function has been proposed to estimate the unknown parameters utilizing condition monitoring data of the devices. Case studies of fatigue crack length data of alloy and capacity data of electrolytic capacitors have been presented to illustrate the proposed prognostics model. The results show that the proposed degradation model can be well fitted to the nonlinear degradation data and provide an accurate prediction of RUL, and thus demonstrate the effectiveness of the proposed model.

Four-compartment epidemic model with retarded transition rates.

We study an epidemic model for a constant population by taking into account four compartments of the individuals characterizing their states of health. Each individual is in one of the following compartments: susceptible S; incubated, i.e., infected yet not infectious, C; infected and infectious I; and recovered, i.e., immune, R. An infection is visible only when an individual is in state I. Upon infection, an individual performs the transition pathway S→C→I→R→S, remaining in compartments C, I, and R for a certain random waiting time t_{C}, t_{I}, and t_{R}, respectively. The waiting times for each compartment are independent and drawn from specific probability density functions (PDFs) introducing memory into the model. The first part of the paper is devoted to the macroscopic S-C-I-R-S model. We derive memory evolution equationsinvolving convolutions (time derivatives of general fractional type). We consider several cases. The memoryless case is represented by exponentially distributed waiting times. Cases of long waiting times with fat-tailed waiting-time distributions are considered as well where the S-C-I-R-S evolution equationstake the form of time-fractional ordinary differential equations. We obtain formulas for the endemic equilibrium and a condition of its existence for cases when the waiting-time PDFs have existing means. We analyze the stability of healthy and endemic equilibria and derive conditions for which the endemic state becomes oscillatory (Hopf) unstable. In the second part, we implement a simple multiple-random-walker approach (microscopic model of Brownian motion of Z independent walkers) with random S-C-I-R-S waiting times in computer simulations. Infections occur with a certain probability by collisions of walkers in compartments I and S. We compare the endemic states predicted in the macroscopic model with the numerical results of the simulations and find accordance of high accuracy. We conclude that a simple random-walker approach offers an appropriate microscopic description for the macroscopic model. The S-C-I-R-S-type models open a wide field of applications allowing the identification of pertinent parameters governing the phenomenology of epidemic dynamics such as extinction, convergence to a stable endemic equilibrium, or persistent oscillatory behavior.

A fitting procedure for probability density functions of service restoration times. Application to underground cables in medium-voltage networks

A fitting procedure for probability density functions of service restoration times. Application to underground cables in medium-voltage networks

The transient characteristics of an underdamped periodic potential system excited by two different kinds of noise

In this paper, the transient characteristics of an underdamped periodic potential system excited by multiplicative Gaussian white noise and additive Lévy noise are studied in terms of the mean first-passage time(MFPT) and the probability density function(PDF) of the first-passage time. The second-order underdamped periodic potential system is equivalent to two first-order stochastic differential equations. The MFPT was obtained by averaging the response value of the first-passage time, and the PDF image of the first-passage time was drawn under different parameter values. It was found that the increase in damping coefficient and stability index will inhibit the particle crossing, while the increase of multiplicative noise intensity, additive noise intensity and skewness parameter will promote the particle crossing to a certain extent.

Electric vehicle scheduling based on stochastic trip time and energy consumption

The vehicle scheduling problem (VSP) is profound in public transit planning. Especially electric VSP (EVSP) occupies a hot topic, as electric vehicles have enjoyed a fast-expanding market share in the bus market in recent years. It used to be solved regardless of the dreadful variability of traffic by setting the fixed trip times and further deterministic energy consumption, then the robustness of resulting schedules is compromised. Therefore, EVSP based on stochastic trip time and energy consumption is studied. We propose a probabilistic model for EVSP based on the probability density function (PDF) of trip time to minimize the fleet size and operating cost and maximize on-time performance. For modeling, we define the time compatibility and energy compatibility of trips by PDF. Based on the time compatibility, the lower bound of the fleet size is derived. As energy compatibility is a non-inherent attribute of trips, we next develop an adaptive large neighborhood search (ALNS) heuristic for EVSP. Experiments demonstrate that ALNS may considerably surpass the large neighborhood search algorithm for solving EVSP, and the probabilistic model may lead to more robust schedules without increasing fleet size.

Nonlinear random vibration of the slender deep-water pier under seismic excitation

The present study investigates the nonlinear random vibration of the deep-water pier exposed to horizontal seismic excitation. First of all, the stochastic dynamic model of the pier is formulated. During the process, the pier is simplified as a cantilever beam fixed on the rock foundation, the seismic excitation is treated as Gaussian white noise, the hydrodynamic pressure is described with the radiation wave theory, and the equation for the nonlinear kinematic of the pier is deduced by the means of Kane’s method. Then, with the application of the stochastic averaging (SA) technique, the Fokker–Plank–Kolmogorov (FPK) equation governing the transient probability density function (PDF) of the amplitude envelope and the backward Kolmogorov (BK) equation for the conditional reliability function (CRF) are derived, respectively. The closed-form stationary PDF can be yielded directly from the reduced FPK equation, while the CRF and the conditional PDF of first-passage time are given after solving the BK equation numerically. Numerical discussions are performed to illustrate the trend of excitation intensity, mass ratio, immersion ratio, and inner and outer hydrodynamic effect on the stationary response and first-passage failure are examined, respectively. It has been shown that increases in the excitation intensity, mass ratio, and immersion ratio can amplify the response and reduce the reliability of the deep-water pier system. The hydrodynamic effect also leads to an amplification of the system response and a reduction in the reliability of the system. Similarly, the presence of inner water can also exacerbate these effects, and this phenomenon becomes more pronounced with increasing immersed ratios. Additionally, the analytical solution is validated by the result obtained by pertained Monte Carlo simulations (MCS). It is noted that this work will be helpful for the optimal seismic design of deep-water piers.

Accessing power-law statistics under experimental constraints

Power-law distributions appear in a large variety of situations and influence our understanding of various physical phenomena. Their identification and characterization are notoriously difficult because of the large fluctuations inherent to empirical data and also because of the unknown range over which the power-law behavior holds. Furthermore, the data on which one is trying to detect power laws are affected by technical constraints and experimental limitations. Here, we show how a power-law distribution is modified by two fundamental limitations: the spatiotemporal resolution and the time window. We consider a time series of events or states and investigate the interevent time probability density function (PDF) or the PDF of the duration of a state. We present in detail how each limitation affects the PDF and derive mathematical expressions that relate the observed distribution to the true one: the resolution globally affects the shape of PDF while preserving the asymptotic exponent and the time window introduces a nonexponential cutoff. We demonstrate that, instead of looking for a simple power law in experimental data, one should fit the data with the modified PDF that we derived for given experimental constraints. We apply our theory to data from an experimental study of the transport of mRNA-protein complexes along dendrites. The presented mathematical theory widens our understanding of the identification and characterization of power-law distributions in experimental data and can be used in a broad spectrum of science fields.

Statistical Analysis Approach of Single-sided Ventilation Based on Particle Track Technique Using Large Eddy Simulation

The main objective of this study is to investigate the airflow characteristics in single-sided ventilation. A CFD method of large eddy simulation (LES) using particle track technique was conducted in this study, and the simulation results are validated by experiment results. The mostly used ventilation efficiency indicators of natural ventilation are Air Flow Rate (AFR) and Purging Flow Rate (PFR). AFR is the volumetric airflow exchange rate through the openings and PFR is the effective airflow rate that removes indoor air pollutants. The ventilation efficiency (εν) of natural ventilation, which is defined as the ratio of PFR and AFR, is not well investigated in previous research, therefore, this study discusses the case of the εν of single-sided ventilation and tries to explain the distinction between AFR and PFR by statistical analysis approach based on particle movements. Particles are emitted from the opening and the particles’ movements are tracked by tracing their locations at different time steps. The residence time measures the time between the time that particles enter the room and the time that particles leave the room, and indoor travel distance indicates the sum of the displacement when the particle travels in the room. The Probability Density Function (PDF) of the residence time and indoor travel distance is used to depict airflow characteristics of the ventilation path of single-sided ventilation. The analysis method of study might be another useful approach to studying natural ventilation in detail.

Approximating the First Passage Time Density of Diffusion Processes with State-Dependent Jumps

We study the problem of the first passage time through a constant boundary for a jump diffusion process whose infinitesimal generator is a nonlocal Jacobi operator. Due to the lack of analytical results, we address the problem using a discretization scheme for simulating the trajectories of jump diffusion processes with state-dependent jumps in both frequency and amplitude. We obtain numerical approximations on their first passage time probability density functions and results for the qualitative behavior of other statistics of this random variable. Finally, we provide two examples of application of the method for different choices of the distribution involved in the mechanism of generation of the jumps.

Optimizing travel routes using temporal networks constructed from global positioning system data in kyoto tourism

Owing to the complexity of urban transportation networks and temporal changes in traffic conditions, the assessment of real-time traffic situations is a challenge. However, the development of mobile information devices using the global positioning system (GPS) has made it easier to obtain personal mobility information. In this study, we developed a method for evaluating the mobility of people in a city using GPS data. We applied two methods: evaluating human mobility using temporal networks constructed from GPS data, and searching for the shortest path by constructing and solving the time-dependent traveling salesman problem (TDTSP). The estimation is expected to be more realistic if transportation delays from congestion are considered. This study makes two major contributions. First, we propose a new method for estimating the time weights of edges in temporal networks using probability density functions for the travel time. Second, to apply ant colony optimization to the TDTSP, we propose a new method for estimating the congestion level from GPS data and calculating the transition probability using the estimated congestion level. As a case study, we conducted a human mobility analysis in Kyoto City.

A general theory for infectious disease dynamics

We present a general theory of infection spreading. It directly follows from conservation laws once known the probability density functions of latent times. The theory can deal with any distribution of compartments latent times. Real probability density function can be then employed, thus overcoming the limitations of standard SIR, SEIR and other similar models that implicitly make use of exponential or exponential-related distributions. SIR and SEIR-type models are, in fact, a subclass of the theory here presented. We show that beside the infection rate, the probability density functions of latent times in the exposed and infectious compartments govern the dynamics of infection spreading. We study the stability of such dynamical system and provide the general solution of the linearized equations in terms of the characteristic functions of latent times probability density functions. We exploit the theory to simulate the spreading of COVID-19 infection in Italy during the first 120 days.