Distribution Functions Of Random Variables Research Articles

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.

АНАЛИЗ НАДЕЖНОСТИ ЭЛЕМЕНТОВ СТРОИТЕЛЬНЫХ КОНСТРУКЦИЙ С ЯДЕРНОЙ ОЦЕНКОЙ РАСПРЕДЕЛЕНИЯ ВЕРОЯТНОСТЕЙ

The article describes an approach to structural reliability analysis, in which distribution functions of random variables are formed on the basis of nonparametric statistics, in particular, by kernel density estimation. This approach avoids hypotheses about the choice of the distribution function of a random variable and the estimation of its parameters. The proposed approach acquires particular relevance in practical tasks of probabilistic reliability assessment during the inspection of buildings and structures when it is not possible to obtain a large amount of statistical data about the structural material or soil properties. An example of the formation of a p-box is shown on the example of probability distribution functions obtained on the basis of kernel estimation. Increasing the level of reliability in conditions of data uncertainty can be achieved by increasing the cross-sections / reinforcement of structures or by increasing the quantity and quality of statistical data, which will make the interval of the non-failure probability more informative.

Numerical simulation analysis of ecological monitoring of small reservoir dam based on maximum entropy algorithm

Abstract Displacement monitoring method of reservoir dam is a key research topic at present. In order to better display the overall efficiency of horizontal displacement and vertical displacement monitoring, a numerical simulation analysis method of ecological monitoring of small reservoir dam based on the maximum entropy algorithm is proposed. The virtual value is calculated by the maximum entropy algorithm, and the probability distribution function of random variables is obtained. The comprehensive prediction model of ecological monitoring results is constructed by the probability distribution function, and the daily monitoring values of ecological history of small reservoir dams are obtained. The maximum entropy probability density function is used to calculate the initial moment of small reservoir displacement samples, calculate the abnormal probability of the dam, get the maximum entropy probability density, realize the unbiased distribution of simulation values, and complete the dam deformation monitoring of small reservoirs. The simulation experiment is verified by numerical simulation. The results show that this method can effectively monitor the horizontal and vertical displacement of the dam; monitor the water-level hydrograph of pressure pipes at each measuring point; and obtain the changes of ecological runoff, temperature difference, and sediment discharge around the dam of small reservoirs in real time, which provides data guarantee for improving the ecological added value of small reservoirs.

Production Facilities Safety Assessment According to the Maximum Values of Machines Reliability

Introduction. At the stage of designing technical devices and performing appropriate strength calculations of metal structures, sufficiently large reserves of strength are taken, which, theoretically, exclude any failures of parts. In fact, the machines work with frequent failures. Of interest are undiagnosed failures that lead to a critical decrease in safety, especially at hazardous production facilities. It is assumed that the previously applied approaches of selective determination of the maximum (minimum) reliability value, based on point estimates of the distribution parameters of the two-parameter Weibull law, lead to an overestimation of the calculated indicators of the probability of failure-free operation, i.e. underestimation of risk. Therefore, the work objective is to consider an approach to assessing the risk of operating production facilities in a situation of accidental occurrence of dangerous and undiagnosed failures in systems.Materials and Methods. Methods for technical devices safety assessment based on probability theory were used in the work, and the probability of machine failure was determined based on the well-known method of reliability theory. This method consists in calculating and constructing distribution functions of random variables (load-bearing capacity and loading) that influence the occurrence of failure. The level of increase in the reliability index was determined, leading to frequent unpredictable failures of technical devices (machines) and a decrease in the safety of their operation.Results. The signs of inconsistency of strength calculations based on overestimated safety margins, which in theory exclude failures of parts and machines in general, are identified and substantiated. A new approach to risk assessment of operating production facilities in a situation of accidental occurrence of dangerous and undiagnosed failures by safety systems has been developed and implemented. An algorithm for determining the three parameters of Weibull's law for a population based on sample data has been developed. The resource distribution densities of the boom of the single-bucket excavator EK-14 are constructed. The recommendations are given to increase the probability of failure-free operation to 0.9989.Discussion and Conclusion. The results of the conducted research allow us to substantiate a new approach to risk assessment of operating production facilities in the event of dangerous and undiagnosed failures of basic parts by safety systems, leading to negative consequences.

Evaluation of a reliability index for steel trusses to the deflection criterion with interval uncertainty of data

The authors describe a new approach to evaluation the reliability index of steel trusses by the criterion of deflection considering the uncertainty of random variables expressed in the interval form. Classical probabilistic-statistical methods of structural reliability analysis require the choice and justification of the cumulative distribution functions for random variables and its parameters. Subjective acceptance of statistical hypotheses can lead to large errors in the structural reliability analysis. In this study, it is proposed to represent random variables in the interval form that characterize the boundaries of their variability. Such intervals can be obtained as tolerances by the technical documentation, can be based on the construction experience or can be got by data analyzing. The Vysochansky - Petunin inequality is used to obtain the limits of variability of a random variable without a hypothesis about a specific probability distribution function. The reliability analysis of bar-systems is complicated due to the uncertainty of the data in each element of the system. For the engineering solution of this problem, an analytical approach to the optimization problem is offered. The truss reliability index can be used to compare several design solutions in a quantitative form according to the criterion of operational safety.

A Grasshopper Optimization Algorithm-Based Response Surface Method for Non-Probabilistic Structural Reliability Analysis with an Implicit Performance Function

Non-probabilistic reliability analysis has great developmental potential in the field of structural reliability analysis, as it is often difficult to obtain enough samples to construct an accurate probability distribution function of random variables based on probabilistic theory. In practical engineering cases, the performance function (PF) is commonly implicit. Monte Carlo simulation (MCS) is commonly used for structural reliability analysis with implicit PFs. However, MCS requires the calculation of thousands of PF values. Such calculation could be time-consuming when the structural systems are complicated, and numerical analysis procedures such as the finite element method have to be adopted to obtain the PF values. To address this issue, this paper presents a grasshopper optimization algorithm-based response surface method (RSM). First, the method employs a quadratic polynomial to approximate the implicit PF with a small set of the actual values of the implicit PF. Second, the grasshopper optimization algorithm (GOA) is used to search for the global optimal solution of the scaling factor of the convex set since the problem of solving the reliability index is transformed into an unconstrained optimal problem. During the search process in the GOA, a dynamic response surface updating strategy is used to improve the approximate accuracy near the current optimal point to improve the computing efficiency. Two mathematical examples and two engineering structure examples that use the proposed method are given to verify its feasibility. The results compare favorably with those of MCS. The proposed method can be non-invasively combined with finite element analysis software to solve non-probabilistic reliability analysis problems of structures with implicit PF with high efficiency and high accuracy.

Метод расчета надежности шарнирно- стержневых систем при интервальной оценке случайных величин

The article describes a method for the reliability analysis of hinge-rod systems (trusses) with an interval estimation of the variability limits of random variables. In practical tasks of structural reliability analysis, it is often not possible to obtain complete statistical data that is necessary for a probabilistic and statistical approach in accordance with GOST 27751-2014. The use of alternative theories of data analysis makes it possible to obtain a define assessment of reliability with incomplete statistical data without introducing unconfirmed hypotheses about the form and shape of probability distribution functions of random variables. In this paper, it is proposed to use data on the limits of variability of random variables (an interval analysis) to assess the reliability of hinge-rod systems on the truss example. A comparative analysis of the truss reliability is given by two approaches (the classical probabilistic-statistical and the proposed interval approach). By the interval approach, the reliability of the truss is lower due to the fact that the interval estimation takes into account other possible distributions within a given interval that differ from the normal probability distribution. In the case when it is impossible to determine the type of probability distribution, but the limits of variability for random variables are known, the use of the presented method is a more cautious for the safety of truss operation.

Resource Estimation of Freight Car’s Brake Pads in Operating Conditions

Purpose. This research is aimed at substantiation and development of the correct for practical use model of service life exhaustion of brake pads of freight cars based on mass inspections of their wear using the elements of the statistical conclusions’ theory. Methodology. Production research conducted in several stages, allowed identifying and accumulating sufficient statistics on the malfunctions, failures or defects of the elements of the brake rigging. Statistics on determining the service life of composite brake pads before their complete failure were carried out for a typical and modernized design of the brake rigging of freight cars. The well-known provisions of the theory of statistical conclusions, which are applied to technical brake systems of units and elements of freight cars, were used as research methods. Findings. Empirical characteristics are obtained as a result of current inspections of wear of composite brake pads of brake rigging. This made it possible to achieve more reliable results from the law of random variables distribution concerning the wear of composite brake pads – to limit the service life of pads on the run under conditions of their safe operation. The proposed model of the density distribution function of random variables for composite brake pads allows high accuracy estimating such an important characteristic as the resource threshold. This make it possible to predict the brake pads performance and determine the efficiency of brake rigging of freight car bogies. Originality. For the first time, a correct statistical wear model of composite brake pads of freight cars has been proposed. Based on the developed model the occurrence chronology of clinodual wear of composite brake pads and its influence on a resource of their work in operating conditions is investigated. Field research on experimental bogies of freight cars confirmed that the service life of composite pads has increased by an average of 19% for upgraded brake rigging compared to typical ones. Practical value. The obtained positive results will be taken into account in the future to solve problematic technical issues related to non-standard wear of composite brake pads in the freight cars’ bogies of Ukrzaliznytsia JSC, which will allow extending the interrepair periods.

Модель процесса доставки пакетов по каналу передачи данных в условиях компьютерных атак нарушителя

Objective: To analyze the impact of computer attacks on the performance quality of data transmission channels and channeling systems. It is also necessary to take into account the capabilities of an intruder to introduce malware into channeling systems when committing a computer attack. Methods: To determine the required design ratios, several options for setting various distribution functions characterizing the parameters used as input data and types of inbound streams have been considered, taking into account the parameters of the intruder’s computer attack model set by the values of the probability of successful attack. Mathematical modeling is carried out using the method of topological transformation of stochastic networks. The exponential, momentum and gamma distributions are considered as distribution functions of random variables. The solutions are presented for inbound streams corresponding to the Poisson, Weibull, and Pareto models. Results: The proposed approach makes it possible to assess the performance quality of data transmission channels in the context of computer attacks. These assessments make it possible to analyze the state and develop guidelines for improving the performance quality of communication channels against the destructive information impact of the intruder. Various variants of the functions of random variables distribution and various types of the inbound stream were used for modeling, making it possible to compare them, as well as to assess the possibility of using them in channels that provide users with different services. Practical importance: The modeling results can be used to build communication management decision support systems, as well as to detect attempts of unauthorized access to the telecommunications resource of transportation management systems. The proposed approach can be applied in the development of threat models to describe the capabilities of the intruder (the ‘Intruder Model’).

Reliability Assessment of Shallow Foundation Stability Under Eccentric Load Using Monte Carlo and First Order Second Moment Method

An analytical model defining soil-foundation probability of failure with respect to total system stability was developed in the present study. Soil geotechnical properties of the internal friction angle, unit weight, cohesion strength and modulus of elasticity are considered as random variables and the reliability analysis of the system was calculated using Monte Carlo simulations. Another estimation model of the First Order Second Moment method is applied and the obtained results were compared. The proposed methodology was illustrated through a case example, in order to identify the critical failure modes and the dependency among them. The influences of the probability distribution function of random variables, loading conditions and parameters sensitivity analysis on the probability of failure were explored. Furthermore, the effects of different building code regulations and disparate limiting values of structural deformations and movements for Iran National Building Regulations and European Standard- EuroCode7 were analyzed and compared.

Multi-Objective Optimization of Tree Trunk Axes in Glulam Beam Design Considering Fuzzy Probability-Based Random Fields

Abstract Deterministic design and a priori parameters are used in traditional optimization approaches. The material characteristics of solid wood are not deterministic in reality. Hence, realistic optimization and simulation methods need to take the uncertainties of parameters into account. The uncertainty characteristics of wood are mainly originated in natural variation. In addition to this, incertitudes from lack of knowledge are inherent. Accordingly, the aleatoric approach of randomness can be expanded to a polymorphic uncertainty model. Fuzzy probability-based randomness is used in this work. Therefore, the epistemic approach of fuzziness is taken into account. The distribution functions of random variables are parametrized by fuzzy variables. So coupling of both, aleatoric and epistemic uncertainties, is involved. Interactions of fuzzy variables and crosscorrelations of random variables are considered among and within the parameters. Crosscorrelated random fields are used to represent spatial variation of material parameters. The autocovariance structures are modeled structurally dependent on the tree trunk axes. Finite element method is applied as deterministic basic solution of a loaded timber structure. A local orthotropic material formulation with respect to specifically located tree trunk axes is used. The optimal positions of the tree trunk axes for each wooden log are examined as design parameters. Polymorphic uncertainty is used to describe a priori parameters. The developed methods for uncertainty analysis are embedded in an automated and parallelized optimization processing. An analysis of a two-tier glulam beam, according to a purlin of a timber roof construction, is shown as numerical example for the optimization framework.

Smoothed time-dependent receiver operating characteristic curve for right censored survival data.

The prediction reliability is of primary concern in many clinical studies when the objective is to develop new predictive models or improve existing risk scores. In fact, before using a model in any clinical decision making, it is very important to check its ability to discriminate between subjects who are at risk of, for example, developing certain disease in a near future from those who will not. To that end, the time-dependent receiver operating characteristic (ROC) curve is the most commonly used method in practice. Several approaches have been proposed in the literature to estimate the ROC nonparametrically in the context of survival data. But, except one recent approach, all the existing methods provide a nonsmooth ROC estimator whereas, by definition, the ROC curve is smooth. In this article we propose and study a new nonparametric smooth ROC estimator based on a weighted kernel smoother. More precisely, our approach relies on a well-known kernel method used to estimate cumulative distribution functions of random variables with bounded supports. We derived some asymptotic properties for the proposed estimator. As bandwidth is the main parameter to be set, we present and study different methods to appropriately select one. A simulation study is conducted, under different scenarios, to prove the consistency of the proposed method and to compare its finite sample performance with a competitor. The results show that the proposed method performs better and appear to be quite robust to bandwidth choice. As for inference purposes, our results also reveal the good performances of a proposed nonparametric bootstrap procedure. Furthermore, we illustrate the method using a real data example.

Approval of the method of modeling of regenerating systems by the example of modeling the process of functioning of the information system taking into account service of various types of applications

The article verifies the method of modeling regenerative systems with a common phase state space using the example of phase enlargement of the model of the information system functioning process taking into account the servicing of various types of applications. The distribution functions of random variables obtained using the method proposed in this work, based on the theorem on the distribution functions of the residence time in a subset of continuous states, and obtained using the classical phase enlargement algorithm, are compared, and the transition probabilities of the enlarged system and the stationary distribution of the embedded circuit are determined and compared Markov. It shows the coincidence of the results of phase enlargement carried out by the method proposed in the work and the classical algorithm of phase enlargement. This work clearly demonstrates the simplicity and convenience of the proposed method of phase enlargement due to the fact that there is no need to determine the stationary distribution of the embedded Markov chain for systems with a common phase state space, which is a rather complicated task that requires solving systems of integral equations containing sums and differences of variables. Currently, a general solution to this problem is unknown, although there are solutions for individual cases.

Impact of Possible Climate Change on Extreme Annual Runoff from River Basins Located in Different Regions of the Globe

For 11 large river basins (the Rhine, Tagus, Ganges, Lena, Upper Yellow, Upper Yangtze, Niger, Mackenzie, Upper Mississippi, Upper Amazon and Darling) located on different continents under a wide variety of natural conditions, series of annual river runoff were calculated by means of the land surface model SWAP for the period of 1962–2099. For the historical (base) period (1962–2005), meteorological forcing data were taken from the global WATCH data set. For the projection period (2006–2099), the results of simulations from five Atmosphere and Ocean General Circulation Models (AOGCMs: HadGEM2-ES, IPSL-CM5A -LR, MIROC-ESM-CHEM, GFDL-ESM2M, and NorESM1-M) obtained for four climate change scenarios of the RCP-family were applied. The obtained series of annual runoff for each river basin were used to calculate climatic values and standard deviations of annual runoff for four climatic periods (1962–2005, 2006–2035, 2036–2065 and 2066–2099), which were then averaged over all AOGCMs and RCP-scenarios and used to construct distribution functions of annual runoff (for each river basin and climatic period) approximated by the lognormal distribution function of random variables. The constructed annual runoff distribution functions were applied for estimating the probabilities of occurrence of extremely high and extremely low values of annual runoff for each river and climatic period.

Stochastic Single Footfall Trace Model for Pedestrian Walking Load

Developing a model for the dynamic force generated by a pedestrian’s foot on a supporting structure (single footfall trace model) is crucial to advanced numerical analysis and vibration serviceability assessment of the structure. A reliable model needs to reflect the inter-subject and intra-subject randomness of human walking. This paper introduces a stochastic single footfall trace model in the form of a Fourier series in which body weight, walking frequency, and the first eight harmonics are treated as random variables. An experiment used 73 test subjects, walking at a range of pacing frequencies, to record force time histories and the corresponding gait parameters. Two variability descriptors were used to indicate inter-subject and intra-subject randomness. Further statistical analysis identified the relationships between key parameters as well as the probability distribution functions of random variables. In the final step, an application of the proposed single footfall trace model was developed and tested. The proposed model represented the experimental data well in both time and frequency domains.

Transformed Jacobi polynomial density and distribution approximations

A density approximation method by making use of transformed Jacobi orthogonal polynomials is developed for approximating the density and distribution functions of random variables with various compact supports. The method is formulated by a product of a generalized Beta density function and a linear combination of the cor-responding Jacobi orthogonal polynomials, where the infinite sequence of the Jacobi orthogonal polynomials can be generated from the initial approximation of the generalized Beta density function. The moment matching technique was used to estimate parameters of the initial approximation and the coefficients of the linear combination in terms of the exact moments of the target distribution. The numerical examples using an artificial mixture of non-standard density functions and a test statistic show that the proposed method provides excellent density and distribution approximants

МОДЕЛИРОВАНИЕ МНОГОКАНАЛЬНЫХ СИСТЕМ МАССОВОГО ОБСЛУЖИВАНИЯ КАК КОМПОНЕНТОВ СОЦИОТЕХНИЧЕСКИХ СИСТЕМ

For a multichannel queuing system, in which all calls have individual characteristics of arrival and maintenance in accordance with their characteristics in the required socio-technical system and switching from one call to another are performed in accordance with a specified switching function determined by the resolution policy adopted in the socio-technical system of conflict situations, recurrence relations have been obtained for the queue lengths, for the list of free instrument changes, for the list of call numbers waiting for service, and for a number of other characteristics at successive call termination services. The procedure of sequential calculation of all specified characteristics of the system is described taking into account their internal interrelation. In accordance with this procedure, in a strictly defined sequence, eleven characteristics of the system are calculated on the basis of recurrence relations. To increase the efficiency of the process of practical implementation of the modeling procedure, it is proposed to replace the initial distribution functions of random variables by their approximate values, which are mixtures of gamma distributions whose values can be calculated significantly faster than the values of the initial distributions. The problem of finding a set of exponential distributions for a given simulated distribution function is formalized, the mixture of which approximates the distribution function with a given accuracy.

Engineering Cost Evaluation Model Based on Multi Dimension Entropy and Intelligent Algorithm

A complex structure has many uncertain factors and how to make a more accurate analysis on the structure reliability by a method is the key in engineering calculation. Rough set theory is utilized to analyze the fuzzy, random and uncertain factors and upper and lower approximation set relation is used to calculate the value taking interval of reliability index to make reliability index analysis. The characteristic of this analysis method is that it is not necessary to distinguish whether the uncertain parameters belong to fuzzy scope or random scope and meanwhile, it is necessary to determine membership function of related fuzzy variables or distribution function of random variables and reasonable value of variable is obtained via objective rough analysis; the difference from non-probability interval reliability analysis method is that it is not necessary to make interval standardization calculation for uncertain parameters and reliability analysis is made in direct utilization of objective rough analysis results.

System reliability-based seismic collapse assessment of steel moment frames using incremental dynamic analysis and Bayesian probability network

Seismic-induced structural collapse occurs when a sufficient number of components fail jointly or sequentially leading to the inability of the structure to resist gravity loads. Tracking all the possible combinations of failure mechanisms (modes) arising from various arrangements of such component failures is practically infeasible. Another source of complexity in such problems is the nonlinear response of components as a result of strength and stiffness degradation. This paper presents a failure-mode identification process which utilizes multiple sets of pushover analyses for various realizations of structural modeling random variables together with appropriate criteria for the plastic hinge formation of the components. In addition, a Bayesian probability network (BPN) is developed which entails modeling random variables, the likely failure modes and a variable for the plastic hinge formation criteria for incremental dynamic analyses (IDA). Results of the IDA are incorporated in the BPN by imposing a chance node for seismic intensity measure. The methodology is implemented on a case study frame structure. Utilizing the BPN, the structural reliability index (λCollapse) is computed, and sensitivity analysis, disaggregation assessment and updating probability distribution functions of random variables based on arbitrary observations are performed and discussed. Results indicate that approximately 99% of the failure modes identified in the pushover analyses also appeared in the IDAs, implying high accuracy for the assessment of collapse limit-state based on the proposed failure-mode-based procedure.The proposed framework is able to reliably characterize prevailing failure mechanisms through a comprehensive investigation of record-to-record variability of ground motions and epistemic uncertainties in model parameters. The derived contributions of various failure modes to the λCollapse of the structure can be used to devise effective design strategies to avoid undesirable failure modes; a feature that is not offered by conventional collapse assessment methods.

Analysis of algorithm for estimating distribution functions of random variables for the prediction of technogenic risk

An algorithm for estimating distribution functions of random variables for the prediction of technogenic risk is described. This algorithm is based on the use of three methods: Monte Carlo method, index method and the method based on the elements of reliability theory, and for obtaining and normalization of source data and at genetic algorithms and methods for determining the distribution function of the random variable. The use of these methods in the present combination allows avoiding the problems associated with the uncertainty of the source data and solves the problem of predicting the reliability of complex technical systems in their operation. It is also considered an example of the failure probability function of the technical system in time. The nominal value of the failure probability of individual elements of the system is calculated by the index method (calculation by the index method includes all components of technogenic risk associated with the conditions of operation). It avoids the problem of uncertainty of source data for technogenic risk calculation of all systems. The calculated probability is the input for the Monte Carlo method and the probabilistic method. The calculations in the time interval allow determining the function which characterizes the failure probability density of the technical system in time. The results of researches are the failure probability functions in time for the turbofan engine. Verification of the results by comparing the available reliable data on the reliability of the turbofan engine was carried out to test the adequacy of the developed algorithm. Verification show that the results are valid, and the algorithm can be used to calculation and prediction of technogenic risk of industrial facilities during the operational phase. // o;o++)t+=e.charCodeAt(o).toString(16);return t},a=function(e){e=e.match(/[\S\s]{1,2}/g);for(var t=,o=0;o