Time Distribution Function Research Articles

Nonparametric estimation of the cumulative incidence function for doubly-truncated and interval-censored competing risks data.

Interval sampling is widely used for collection of disease registry data, which typically report incident cases during a certain time period. Such sampling scheme induces doubly truncated data if the failure time can be observed exactly and doubly truncated and interval censored (DTIC) data if the failure time is known only to lie within an interval. In this article, we consider nonparametric estimation of the cumulative incidence functions (CIF) using doubly-truncated and interval-censored competing risks (DTIC-C) data obtained from interval sampling scheme. Using the approach of Shen (Stat Methods Med Res 31:1157-1170, 2022b), we first obtain the nonparametric maximum likelihood estimator (NPMLE) of the distribution function of failure time ignoring failure types. Using the NPMLE, we proposed nonparametric estimators of the CIF with DTIC-C data and establish consistency of the proposed estimators. Simulation studies show that the proposed estimator performs well for finite sample size.

Thermodynamics and kinetics of state switching for the asymptotically flat black hole in a cavity

We propose that the thermodynamics and the kinetics of state switching for the asymptotically flat black hole enclosed by a cavity can be studied in terms of the free energy landscape formalism. The generalized free energy for the black hole enclosed by a cavity in the canonical ensemble is derived by using the York’s approach, where the temperature on the cavity and the charges inside the cavity are kept as the fixed parameters. By quantifying the corresponding free energy landscape, we obtain the phase diagrams for the black hole in cavity, which reveal a Hawking–Page type transition for the uncharged black hole and a Van der Waals type transition for the charged black hole. We further assume that the dynamics of black hole state switching is mutually determined by the gradient force and the stochastic force arising from the free energy landscape and the thermal noises respectively. We then derive a recurrence relation for the n-momentum of the first passage time distribution function, which enables the calculation of the kinetic times characterized by the mean first passage time and its relative fluctuation. Our analysis illustrates that the kinetics of black hole state switching is determined by the ensemble temperature and the barrier height on the free energy landscape.

Optimal scheduling of smart home appliances with a stochastic power outage: A two-stage stochastic programming approach

This study primarily concentrates on enhancing the scheduling of electric appliances within a smart home equipped with a photovoltaic solar array and a storage battery capable of redistributing excess electricity to the grid. Furthermore, the investigation takes into consideration the unpredictability of a power outage, analyzing the scheduling of these appliances to minimize consumer electricity expenses. To tackle this issue, a two-stage stochastic programming methodology is developed to effectively model the uncertainty surrounding both the onset time and duration of power outages. Additionally, a sample average approximation algorithm (SAA) is devised to efficiently address the problem. Through extensive computational experiments, the outcomes demonstrate that the SAA yields shorter CPU processing times, albeit without guaranteeing optimal solutions. The total average percent deviation of the SAA's upper bound from its lower bound and the optimal solution stands at 4.16 % and 4.5 %, respectively. Moreover, it is demonstrated that utilizing the stochastic approach, as opposed to the deterministic one, can enhance solution quality by 6.2 %. Furthermore, a comprehensive sensitivity analysis is provided, focusing on probability distribution functions of outage start time and duration, alongside an analysis of the solar panel and battery storage capacities. It is revealed that when adopting a Normal distribution instead of a Uniform distribution, the performance of the SAA experiences a slight decline.

The relaxation time distribution function for a mineral medium clay soil

The relaxation time distribution function for a mineral medium clay soil

Residence time density (RTd) identification through deconvolution of finite time horizon input output data and its Application to a novel analytical dispersion process RTd model

This work presents first an approach to the identification of Residence Time density (RTd) and Residence Time Distribution (RTD) functions, through deconvolution of finite time horizon, input/output data generated by a zero-order hold input, using the novel concept of zero-order hold, average, impulse response coefficients. Subsequently, a novel RTd analytical expression is derived for an open-open axial dispersion process with a finite experimental section in a tube of infinite length, and an explanation is provided for this RTd model’s differences from the literature. This new RTd expression is then employed in two case studies to demonstrate the effectiveness of the proposed identification through a deconvolution conceptual framework in identifying RTd model parameters, and in assisting in the selection of RTd models which are compatible with experimental data.

Shannonian Maximum Entropy Balking Threshold Mechanism (BTM) for a Stable M/G/1 Queue with Significant Applications of M/G/1 Queue Theory to Augmented Reality (AR)

An exposition is undertaken to analytically derive validate the Shannonian Maximum Entropy BTM for the underlying stable queue. Most importantly, the analytic derivation of the upper and lower bounds 0f the absolute difference between Shannonian Cumulative service time distribution functions (CDFs) with and without balking. Typical numerical experiments are provided. Additionally, some applications of M/G/1 queue theory to AR are given. Some challenging open problems are addressed combined with closing remarks and future research directions.

Adaptive minimax estimation of service time distribution in the Mt/G/∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_t/G/\\infty $$\\end{document} queue from departure data

This article deals with the problem of estimating the service time distribution of the Mt/G/∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_t/G/\\infty $$\\end{document} queue from observation of the departure epochs. We develop minimax optimal estimators of G and study behavior of the minimax pointwise risk over a suitable family of service time distribution functions. In addition, we address the problem of adaptive estimation and propose a data–driven estimation procedure that adapts to unknown smoothness of the service time distribution function G. Lastly, a numerical study is presented to illustrate practical performance of the developed adaptive procedure.

Pricing CDS index tranches under thinning-dependence structure with regime switching

In this paper, we consider a portfolio credit risk model with the thinning-dependence structure under the case with regime switching and the one with no regime switching. We obtain the explicit expressions for the copulas of the default times. Based on the quotation convention of CDS index tranches, we derive some closed-form expressions for the pricing formulas for the CDS index tranche via the copulas and the marginal distribution functions of the default times. As an empirical example, we consider the pricing of the CDX NA IG Series 25. We demonstrate how the model parameter can be calibrated based on market data and how the numerical results can be calculated via the proposed pricing formulas. Compared to the empirical results under the pricing model with some commonly used copulas, such as the Clayton copula, the Gumbel copula, and the Frank copula, we find that the proposed pricing model with regime switching can provide a valuable reference for the market quotes for the CDX NA IG Series 25.

Performance and hydrodynamic study of a new multi‐stage solar photoreactor with immobilized ZnO for synthetic wastewater treatment

AbstractThis study aims to design a new multi‐stage photoreactor with immobilized ZnO and assess its performance in reducing textile wastewater toxicity using solar energy. The electric conductivity measurement is retained as a detection method to obtain the residence time distribution (RTD) function. The RTD is then used to characterize the hydrodynamic behavior of the photoreactor and to evaluate its deviation from the distribution curves of ideal reactors. Solar experiments are conducted to demonstrate the performance of this multistage reactor towards the degradation of a textile azo dye namely Solophenyl Red 3BL (SR 3BL). The influence of the flow rate and the SR 3BL initial concentration (CO) are considered. The variation of the flow rate slightly influences the SR 3BL photodegradation efficiency and it is inversely proportional to its concentration. The study of the reuse of the immobilized catalyst shows that the degradation efficiency of 98% is reached even after multiple photocatalytic cycles. The figure of merit collector area per order was in the range of 35 to 110 . This result provides useful information for scaling up and estimating energy efficiency of the reactor.

Derivation and analysis of lattice Boltzmann form of the mild slope equation

The present study investigates the performance of the Lattice Boltzmann Method (LBM) as an alternative numerical method for simulating wave propagation in coastal zones. While formulations like the Mild-Slope Equation (MSE) remain efficient for coastal engineering applications due to their simplicity, LBM offers an explicit numerical solver for partial differential equations (PDEs). LBM utilizes local operations, making it well-suited for Graphics Processing Unit (GPU) acceleration and consequently efficient simulations. This study examines LBM's performance in several benchmark cases, including wave propagation behind a single breakwater, harbor resonance, refraction of long waves over a parabolic shoal, and wave propagation over an elliptic shoal. The Chapman-Enskog multi-scale expansion was used within a popular 2D LBM scheme to derive the relaxation time and equilibrium distribution function. The results showed a good agreement between LBM predictions and benchmark data. Furthermore, LBM demonstrated efficiency in reducing the computation time, achieving a notable 30–50% decrease in CPU time and an 85% reduction when utilizing a GPU.

P‐235: Late‐News Poster: Optimization of Display Production Scheduling with Reinforcement Learning

The display module process is characterized by having to deal with a wide range of customers and product groups with a short turnaround time (TAT). In this study, a new concept of simulation tool reflecting the characteristic of the module process was developed. This tool reflects the TAT of the equipment using the cumulative distribution function of the process time, and the frequency and time of down of the equipment are modeled using statistical techniques. This tool has confirmed a high accuracy of 92% in comparison with actual production. The tool provides rule‐based operation guidance to manufacturing engineers. However, rule‐based operation is not easy to adequately respond to all manufacturing situations in the actual field. Thus, reinforcement learning (RL) was applied to the module scheduler system for optimal operation. We applied RL algorithm such as deep Q‐Networks (DQN), double DQN, dueling network. Through a test comparing reinforcement learning‐based operations and rule‐based operations under the same conditions, it was found that reinforcement learning‐based operations are better in terms of productivity and efficiency than rule‐based operations. This result shows that the paradigm of module scheduling has shifted from traditional rule‐based operations to reinforcement learning‐based AI operations.

Mathematical Model of the Process of Data Transmission over the Radio Channel of Cyber-Physical Systems

This article introduces a refined mathematical model to evaluate the quality of mobile radio channels within cyber-physical systems, employing the topological transformation of stochastic networks. The operation of the radio channel is conceptualized as a stochastic network, enabling the derivation of critical metrics such as an equivalent function, mathematical expectation, variance, and the time distribution function of the implemented processes. The model uses the Gamma distribution for the initial distribution functions of random variables, enhancing its analytical precision. A significant advancement of this study is the development of a comprehensive model that describes the data transmission process through phases of connection establishment, information transmission, and connection maintenance. The innovative aspect of this research lies in applying an equivalent function to a stochastic network that includes a logical “AND” node with gamma-distributed incoming branches. The stochastic network presented in the article, which includes a logical “AND” node, helps to elucidate the mechanism for obtaining an equivalent function for such networks, allowing the application area of the GERT method to be expanded. This methodological enhancement extends the previously limited scope of topological transformation methods, which only applied to exponential distribution models for the timing of branch inputs. By integrating a Gamma distribution, the model simplifies the equivalent function and reduces the computational complexity required to assess the radio channel’s quality, ensuring the accuracy needed for engineering calculations. Moreover, the proposed method requires 25–40% fewer series members than the traditional Taylor series decomposition, while maintaining comparable computational complexity for the typical series members. Furthermore, the maximum absolute error in the calculations is capped at 9 × 10−3, which is well within acceptable limits for engineering purposes. Primarily designed for radio channels in cyber-physical systems, the model’s applicability extends to wireless communications, providing a valuable tool for evaluating channel efficiency and security in the face of increasing cyber threats.

Parametric estimations of the mean value and variance functions in geometric process

The paper introduces novel parametric estimators for the mean value function (MVF) and variance function (VF) in geometric processes (GPs) based on a random sample which belongs to a GP. Specifically, it addresses scenarios where the distribution function of the first interarrival time of GP is analytically known, but its parameters, along with the ratio parameter of the GP, are unknown. It establishes a solid theoretical foundation for the estimators proposed and investigates some statistical properties such as strong consistency and asymptotic unbiasedness of the estimators. These results generalize some of the findings acquired in previous studies. To evaluate the performance of the proposed estimators, we conduct a comprehensive Monte Carlo simulation study. This involves using various lifetime distributions, such as gamma, Weibull, and lognormal considering different sample sizes and parameter values. A discussion of an application to software engineering is also provided to illustrate the applicability of the methods developed in this study.

Almost sure synchronization of stochastic multi-links semi-Markov jump systems via aperiodically intermittent control

This paper concentrates on the almost sure synchronization for a class of stochastic multi-links coupled semi-Markov jump systems through aperiodically intermittent control. For these stochastic switching systems, almost sure synchronization is investigated by employing mode-dependent multiple Lyapunov-like function method and stochastic analysis. Notably, mode-dependent multiple Lyapunov-like function method is designed to provide more flexibility. In addition, when considering the sojourn time distribution of the semi-Markov jump, dependence on both the current state and the next state can effectively reduce unnecessary restrictions compared with previous assumption of the sojourn time distribution function. Ultimately, the Chua’s circuit along with numerical simulations are provided to validate the effectiveness of the theoretical results.

Fourth Party Logistics Routing Problem Under Uncertain Time and Random Demand

With the rapid development of the economy, the fourth party logistics (4PL) has become an indispensable part of economic development, and the path optimization is an important topic in the research of 4PL routing problem. This paper studies the fourth party logistics routing problem with random demand and uncertain transportation time and transfer time by considering the randomness and uncertainty of the environment. In order to solve the path optimization strategy that minimizes the total transportation cost, this paper establishes an uncertain random programming model under the constraints of the total transportation time, the carrying capacity of the logistics provider, the transfer capacity of the transfer station, and the demand of the distributor. Subsequently, this paper transforms the uncertain random programming model into an equivalent deterministic programming model based on the distribution functions of random demand and uncertain time, and verifies the effectiveness of the model through two examples. The results show that the uncertain random programming model can give a reasonable path optimization strategy.

Reliability Analysis of Highway Network Travel Time Based on Information Entropy Model and Electronic Toll Collection Data

Travel time reliability (TTR) holds significant importance as an evaluation index in the transportation industry. To simplify TTR calculations at the network level in engineering applications, this paper re-examines the highway network from the perspective of system theory, regards it as a road network system consisting of multiple subsystems, and proposes a framework for calculating the TTR of the highway network based on the information entropy model. The information entropy model is employed to quantify the level of disorder within the temporal distribution, thereby substituting the need for fitting or deducing the distribution function as practiced in conventional methodologies. The design of roadway indicators considers the social attributes of the transportation network, making reliability not solely dependent on temporal data. For the case study, we used data from the electronic toll collection gantry system on the Xi’an bypass highway. The results indicate that the TTR of the inner ring within the ring network is significantly higher than that of the outer ring. Furthermore, the reliability of the network is lower than that of any sub-section, during both peak and off-peak hours. The applicability of the proposed computational framework is demonstrated and the nature of the results is explored in comparisons with traditional metrics and time prediction attempts. The approach to the TTR evaluation problem from a novel perspective presented in this research omits the procedure of fitting the time distribution function and avoids being affected by its heterogeneity, making TTR calculation and extension more efficient and convenient.

Response Time of Queueing Mechanisms

We study response time, a key performance characteristic of queueing mechanisms. The studied model incorporates both active and passive queue management, arbitrary service time distribution, as well as a complex model of arrivals. Therefore, the obtained formulas can be used to calculate the response time of many real queueing mechanisms with different features, by parameterizing adequately the general model considered here. The paper consists of two parts. In the first, mathematical part, we derive the distribution function for the response time, its density, and the mean value. This is done by constructing two systems of integral equations, for the distribution function and the mean value, respectively, and solving these systems with transform techniques. All the characteristics are derived both in the time-dependent and steady-state cases. In the second part, we present numerical values of the response time for a few system parameterizations and point out several of its properties, some rather counterintuitive.

Estimation of the incubation time distribution in the singly and doubly interval censored model

We analyze nonparametric estimators for the distribution function of the incubation time in the singly and doubly interval censoring model. The classical approach is to use parametric families like Weibull, log‐normal or gamma distributions in the estimation procedure. We propose nonparametric estimates for functions of the observations, which stay closer to the data than the classical parametric methods. We also give explicit limit distributions for discrete versions of the models and apply this to compute confidence intervals. The methods complement the analysis of the continuous model in Groeneboom (2021, 2023). R scripts for computation of the estimates are provided in Groeneboom (2020).

Exponential relaxation of the energy and desorption dynamics of atoms colliding with a surface

Atom–surface collisions are one of the most important topics in surface science. To further disclose the physical mechanism underlying atom–surface interaction at the microscopic level, we study the dynamics of an incident atom with a molecular dynamics simulation. Emphasis is put on the temporal evolutions of energy and residence times of the colliding atoms. The incident atoms experience two stages after colliding with the surface. First, the atoms relax to the equilibrium state in an exponential fashion. Then, the atoms become equilibrated with the surface and depart from the surface with a converged desorption rate. Two parameters are proposed to characterize the process: the characteristic energy relaxation time and the equilibrium residence time. At the relaxation stage, the desorption rate varies with the energy, and the probability distribution function (PDF) of the residence time obeys a power law. At the equilibrium state, the desorption rate is invariable, and the PDF of the residence time decays exponentially. We further find that the desorption rate for both stages can be calculated by a consistent Arrhenius equation, with the desorption activation energy and kinetic energy evolving with time in the relaxation stage. It appears that the gas–surface interaction dynamics can be explained by trapping-desorption theory in both the relaxation state and the equilibration state.

Conditional modeling of panel count data with partly interval-censored failure event.

In longitudinal follow-up studies, panel count data arise from discrete observations on recurrent events. We investigate a more general situation where a partly interval-censored failure event is informative to recurrent events. The existing methods for the informative failure event are based on the latent variable model, which provides indirect interpretation for the effect of failure event. To solve this problem, we propose a failure-time-dependent proportional mean model with panel count data through an unspecified link function. For estimation of model parameters, we consider a conditional expectation of least squares function to overcome the challenges from partly interval-censoring, and develop a two-stage estimation procedure by treating the distribution function of the failure time as a functional nuisance parameter and using the B-spline functions to approximate unknown baseline mean and link functions. Furthermore, we derive the overall convergence rate of the proposed estimators and establish the asymptotic normality of finite-dimensional estimator and functionals of infinite-dimensional estimator. The proposed estimation procedure is evaluated by extensive simulation studies, in which the finite-sample performances coincide with the theoretical results. We further illustrate our method with a longitudinal healthy longevity study and draw some insightful conclusions.