Lognormal Approximation Research Articles

Performance analysis of dual-media cooperative communication based on wireless and power line under hybrid fading

Wireless communication and power line communication are extensively applied in various fields, such as household Internet of things. For cooperative communication of amplify–forward relay using wireless access and power line transmission, the hybrid model of universal Nakagami wireless fading and lognormal power line fading was used in this study. A comparative analysis of similarities between Gamma and lognormal distributions and lognormal distribution characteristics of relay link signal-to-noise ratio was also carried out. Given the deficiencies of lognormal variable add approximation method, lognormal distribution parameters combined computing method based on moment generation function was proposed. Theoretical formulas of outage probability and bit error rate after amplify–forward relay and maximal-ratio combining was derived on the basis of moment generation function of the total output signal-to-noise ratio of the system. Effectiveness and reliability of the algorithm and theoretical formula were verified through simulation, and influence rules of hybrid channel fading and power distribution on the system performance were analyzed.

Downlink Signal-to-Interference Ratio and Spectral Efficiency of MIMO Cellular Networks Using Truncated Lognormal Approximation

This work focuses on analytical performance evaluation of a multiple-input multiple-output (MIMO) multicellular radio access network, while taking into account distance-dependent path loss, lognormal-distributed shadowing, and Rayleigh-distributed multipath fading. We propose a truncated moment-matching method for approximating the distribution of the sum of independent lognormal random variables in order to model the statistical behavior of the downlink signal-to-interference ratio (SIR) experienced by user equipment at a particular location in the presence of cochannel interference. The proposed method offers a good match with the simulated results in terms of the downlink SIR as well as mean and outage spectral efficiency for various MIMO schemes such as transmit and receive diversity, closed- and open-loop spatial diversity, and closed-loop multiuser MIMO. We compare the semianalytical results with that obtained from Monte Carlo simulations and quantify the relative error in terms of statistical metrics such as Kullback–Leibler divergence and Kolmogorov–Smirnov distance. In contrast to the existing works, the proposed model is useful for a wide range of shadowing standard deviation (4–12 dB) and activity of neighboring base stations (0.4–1). Moreover, the presented method is shown to achieve a comparable accuracy with the state of the art in spite of being less time consuming.

Dependence of minimum propagation distance and related attenuation of radio disturbance on the density of randomly located emitters and receivers in limit setting model

The statistical properties of the minimum propagation distance and the related radio propagation loss are theoretically analyzed assuming randomly distributed disturbance sources and receivers for application to the radio-protection limit setting model. It is shown that the minimum distance follows a Rayleigh distribution whose mean-square distance is inversely proportional to the area density of the disturbance sources or that of victim receivers. By using a magnetic dipole model on a metal ground, propagation losses in the near and far fields are presented with the introduction of a piecewise approximation. On the basis of the above results, the probability density function (PDF) of the minimum propagation loss is derived and numerically calculated for various area densities and radio frequencies by a comparison with a log-normal approximation of the PDF. It is found that each PDF exhibits its maximum at a propagation loss related to the root-mean-square minimum propagation distance. The log-normal PDF demonstrates a good approximation of the exact PDF at around its maximum, and enables a rough estimation of the statistics of the propagation loss. The mean value and standard deviation of the minimum propagation loss are also presented for application to the emission limit setting model.

Density distribution of the cosmological matter field

The one-point probability distribution function (PDF) of the matter density field in the universe is a fundamental property that plays an essential role in cosmology for estimates such as gravitational weak lensing, non-linear clustering, massive production of mock galaxy catalogs, and testing predictions of cosmological models. Here we make a comprehensive analysis of the dark matter PDF using a suite of 7000 N-body simulations that covers a wide range of numerical and cosmological parameters. We find that the PDF has a simple shape: it declines with density as a power-law P~rho**(-2), which is exponentially suppressed on both small and large densities. The proposed double-exponential approximation provides an accurate fit to all our N-body results for small filtering scales R< 5Mpc/h with rms density fluctuations sigma>1. In combination with the spherical infall model that works well for small fluctuations sigma<1, the PDF is now approximated with just few percent errors over the range of twelve orders of magnitude -- a remarkable example of precision cosmology. We find that at 5-10% level the PDF explicitly depends on redshift (at fixed sigma) and on cosmological density parameter Omega_m. We test different existing analytical approximations and find that the often used log-normal approximation is always 3-5 times less accurate than either the double-exponential approximation or the spherical infall model.

Analysis of uncertainty in the surgical department: durations, requests and cancellations.

Objective Analytical techniques are being implemented with increasing frequency to improve the management of surgical departments and to ensure that decisions are well informed. Often these analytical techniques rely on the validity of underlying statistical assumptions, including those around choice of distribution when modelling uncertainty. The aim of the present study was to determine a set of suitable statistical distributions and provide recommendations to assist hospital planning staff, based on three full years of historical data. Methods Statistical analysis was performed to determine the most appropriate distributions and models in a variety of surgical contexts. Data from 2013 to 2015 were collected from the surgical department at a large Australian public hospital. Results A log-normal distribution approximation of the total duration of surgeries in an operating room is appropriate when considering probability of overtime. Surgical requests can be modelled as a Poisson process with rate dependent on urgency and day of the week. Individual cancellations could be modelled as Bernoulli trials, with the probability of patient-, staff- and resource-based cancellations provided herein. Conclusions The analysis presented herein can be used to ensure that assumptions surrounding planning and scheduling in the surgical department are valid. Understanding the stochasticity in the surgical department may result in the implementation of more realistic decision models. What is known about the topic? Many surgical departments rely on crude estimates and general intuition to predict surgical duration, surgical requests (both elective and non-elective) and cancellations. What does this paper add? This paper describes how statistical analysis can be performed to validate common assumptions surrounding surgical uncertainty. The paper also provides a set of recommended distributions and associated parameters that can be used to model uncertainty in a large public hospital's surgical department. What are the implications for practitioners? The insights on surgical uncertainty provided here will prove valuable for administrative staff who want to incorporate uncertainty in their surgical planning and scheduling decisions.

SINR distribution approximation for maximal ratio communications

The SINR distribution is of key importance to wireless network performance analysis, planning and optimization. A realistic and usable SINR distribution has to consider both, the fast and the slow fading, i.e., to utilize a composite fading model. There have been only partial efforts in deriving a closed form and simple SINR distribution, using a realistic composite fading model and valid throughout the full SNR range. Recently, there is also a rising necessity for SINR distribution models that integrate the multi-antenna and multi-user aspects of the emerging 5G systems. This paper derives an analytical lognormal approximation to the SINR distribution in scenarios with maximal ratio communication using multiple transmit or receive antennas. The derivations incorporate a newly derived lognormal composite fading approximation and lognormal approximation to the distribution of the summary interference plus noise, in order to approximate the SINR distribution. The expressions are simple and closed form, which makes them applicable to a variety of communication network optimization and evaluation problems. The paper uses simulation analysis to prove the validity of the analytical SINR distribution approximation in homogeneous and heterogeneous cellular network deployments. The results show only few dBs of mismatch in these realistic cellular network scenarios.

A multi-population algorithm to solve the VRP with stochastic service and travel times

Abstract The vehicle routing problem with stochastic travel and service times is a variant of vehicle routing problem where travel and service times are modeled as random variables. This paper addresses a version with hard time windows which is tackled by a mixed stochastic program with recourse and chance constraints. The recourse policy aims to deal with the costs generated when a customer time window is missed due to the stochastic nature of the problem. Moreover, the stochastic constraints guarantee a service level for each customer. A log-normal approximation is used to estimate the arrival times. The approximation considers previous customers failures to improve the estimation of the mean and variance of the arrival times. The problem is solved by means of a Multi-Population Memetic Algorithm (MPMA) and is tested on modified instances initially proposed for the deterministic problem. Furthermore, a comparison against related problems found in the literature is presented showing the performance of our solution approach. The results depict that the MPMA outperforms different methods with different types of objectives so demonstrating its efficiency and flexibility.

Quantification of heat transfer phenomena within the melt pool during the plasma arc re-melting of titanium alloys

This paper summarizes the development and application of a mathematical model, based on the commercial Computational Fluid Dynamics code ANSYS-CFX, to describe the transport processes occurring during the heating of a cylindrical slice of Ti-6 wt%Al-4 wt%V (Ti64) ingot with a plasma torch. The model simulates the conditions observed in a test (trial) that was conducted within an industrial scale plasma arc furnace. The test conditions, data, analysis methodology and results, in terms of characterization of the surface heat flux, have previously been published and are only presented in this manuscript in a summary form. This work focuses on the development of a thermal-fluid model of the ingot slice in order to better describe the heat transport processes occurring in the molten pool, which was identified as a shortcoming of the previous work. The model includes heat and fluid transport by diffusion and advection, thermal buoyancy, surface shear resulting from both thermal Marangoni forces and the plasma gas and Darcy damping within the two-phase (solid/liquid) region. A preliminary thermal-fluid model of the plasma plume was also developed as part of the research (not presented) and used to help guide development of a suitable gas shear stress distribution, which was implement using a log-normal approximation. The model was found to be able to reproduce thermocouple and pool profile data collected during the trial. In particular, the prediction of the pool profile was found to be considerably more accurate than had previously been the case with a diffusion-only heat transfer model. A sensitivity analysis conducted with the model showed the liquid pool profile to be highly dependent on the magnitude of the gas shear force and the input power efficiency and distribution. The gas shear was found to have a significant impact on near-surface fluid flow, helping to enhance radial heat transport. In comparison to the Marangoni induced shear, the gas shear was found to be dominant.

Statistical Distributions of Lightning Peak Currents: Why Do They Appear to Be Lognormal?

AbstractThe statistical distribution of peak currents in cloud‐to‐ground lightning strokes is analyzed and discussed both in the context of the existing experimental data and from a theoretical viewpoint. The distributions of peak currents directly measured at instrumented towers are compared to those of peak currents reported by the U.S. National Lightning Detection Network and peak currents measured in rocket‐triggered lightning experiments; all those appear to be lognormal. To explain the observed peak current distributions, a simple model based on the large‐scale electric field evolution prior to lightning flashes is developed. Based only on the most general assumptions concerning the probability of occurrence of lightning discharge, it is shown that the distribution of the pre‐first‐stroke large‐scale electric field is close to lognormal in a certain range. With the introduction of structure factors relating the leader or cloud electric potential to this field, semiempirical relationships between the potential and the peak current in negative first and subsequent strokes imply that the peak current and pre‐first‐stroke large‐scale electric field obey distributions of the same type. It is demonstrated that with a suitable choice of parameters, the model‐predicted distributions are close to those obtained from direct measurements, and statistical tests show that their agreement with the data is at least as good as that seen for lognormal approximations.

Lognormal Approximations of Fault Tree Uncertainty Distributions.

Fault trees are used in reliability modeling to create logical models of fault combinations that can lead to undesirable events. The output of a fault tree analysis (the top event probability) is expressed in terms of the failure probabilities of basic events that are input to the model. Typically, the basic event probabilities are not known exactly, but are modeled as probability distributions: therefore, the top event probability is also represented as an uncertainty distribution. Monte Carlo methods are generally used for evaluating the uncertainty distribution, but such calculations are computationally intensive and do not readily reveal the dominant contributors to the uncertainty. In this article, a closed-form approximation for the fault tree top event uncertainty distribution is developed, which is applicable when the uncertainties in the basic events of the model are lognormally distributed. The results of the approximate method are compared with results from two sampling-based methods: namely, the Monte Carlo method and the Wilks method based on order statistics. It is shown that the closed-form expression can provide a reasonable approximation to results obtained by Monte Carlo sampling, without incurring the computational expense. The Wilks method is found to be a useful means of providing an upper bound for the percentiles of the uncertainty distribution while being computationally inexpensive compared with full Monte Carlo sampling. The lognormal approximation method and Wilks's method appear attractive, practical alternatives for the evaluation of uncertainty in the output of fault trees and similar multilinear models.

Risk-Adjusted Linearizations of Dynamic Equilibrium Models

We propose a simple risk-adjusted linear approximation to solve a large class of dynamic models with time-varying and non-Gaussian risk. Our approach generalizes lognormal affine approximations commonly used in the macro-finance literature and can be seen as a first-order perturbation around the risky steady state. Therefore, we unify coexisting theories of risk-adjusted linearizations. We provide a formal foundation for approximation methods that remained so far heuristic, and offer explicit formulas for approximate equilibrium objects and conditions for their local existence and uniqueness. Affine approximations are not nested in conventional perturbations of arbitrary order. We apply this technique to models featuring Campbell-Cochrane habits, recursive preferences, and time-varying disaster risk. The proposed affine approximation performs similarly to global solution methods in many applications; risk pricing is accurate at all investment horizons, thereby capturing the main properties of investors’ marginal utility of wealth and measures of welfare costs of fluctuations.

A data science based standardized Gini index as a Lorenz dominance preserving measure of the inequality of distributions.

The Gini index is a measure of the inequality of a distribution that can be derived from Lorenz curves. While commonly used in, e.g., economic research, it suffers from ambiguity via lack of Lorenz dominance preservation. Here, investigation of large sets of empirical distributions of incomes of the World’s countries over several years indicated firstly, that the Gini indices are centered on a value of 33.33% corresponding to the Gini index of the uniform distribution and secondly, that the Lorenz curves of these distributions are consistent with Lorenz curves of log-normal distributions. This can be employed to provide a Lorenz dominance preserving equivalent of the Gini index. Therefore, a modified measure based on log-normal approximation and standardization of Lorenz curves is proposed. The so-called UGini index provides a meaningful and intuitive standardization on the uniform distribution as this characterizes societies that provide equal chances. The novel UGini index preserves Lorenz dominance. Analysis of the probability density distributions of the UGini index of the World’s counties income data indicated multimodality in two independent data sets. Applying Bayesian statistics provided a data-based classification of the World’s countries’ income distributions. The UGini index can be re-transferred into the classical index to preserve comparability with previous research.

Risk aggregation in Solvency II through recursive log-normals

It is argued that the accuracy of risk aggregation in Solvency II can be improved by updating skewness recursively. A simple scheme based on the log-normal distribution is developed and shown to be superior to the standard formula and to adjustments of the Cornish–Fisher type. The method handles tail-dependence if a simple Monte Carlo step is included. A hierarchical Clayton copula is constructed and used to confirm the accuracy of the log-normal approximation and to demonstrate the importance of including tail-dependence. Arguably a log-normal scheme makes the logic in Solvency II consistent, but many other distributions might be used as vehicle, a topic that may deserve further study.

Log-Normal Approximation for VDSL Performance Evaluation

A performance analysis of VDSL2 technologies in FttC or FttC/FttDp hybrid scenarios requires knowledge of the access network geometry, propagation, and crosstalk interference models. A simulation-based approach is commonly adopted for performance assessment. In this paper, we introduce a performance evaluation framework based on the log-normal approximation for the probabilistic description of the signal-to-interference plus noise ratio (SINR), which is then used to obtain the statistics of the achievable bit rate per user. A very good agreement has been observed between the exact and the approximated calculations of the bit rate up to 600 m from the Cabinet. In addition, the log-normal approximation has allowed to define a simple procedure, based on the equivalent number of co-located interferers , to evaluate the bit rate per user for generic access network geometry. Furthermore, two approximated formulas for bit rate have been obtained under high SINR and dominant far-end crosstalk (FEXT) conditions. They evidence the logarithmic decrease of the bit rate with the distance from the Cabinet and provide the accurate results for distances up to 400 m. Finally, when FEXT fluctuations are modeled in accordance with Beta distribution, we have observed the log-normal approximation still provides valid probabilistic description of the SINR.

Operational Strategies for Increasing Secondary Materials in Metals Production Under Uncertainty

Increased use of secondary raw materials in metal production offers several benefits including reduced cost and lowered energy burden. The lower cost of secondary or scrap materials is accompanied by an increased uncertainty in elemental composition. This increased uncertainty for different scraps, if not managed well, results in an increased risk that the elemental concentrations in the final products fall outside customer specifications. Previous results show that incorporating this uncertainty explicitly into batch planning can modify the potential use of scrap materials while managing risk. Chance-constrained formulations provide one approach to uncertainty-aware batch planning; however, typical formulations assume normal distributions to represent the compositional uncertainty of the materials. Compositional variation in scrap materials has been shown to have a skewed distribution, and therefore, the performance of these models, in terms of their ability to provide effective planning, it may then be heavily influenced by the structure of the compositional data used. To address this issue, this work developed several approximations for skewed distributional forms within chance-constrained formulations. We explored a lognormal approximation based on Fenton’s method; a convex approximation based on Bernstein inequalities; and a linear approximation using fuzzy set theory. Each of these methods was formulated and case studies executed using compositional data from an aluminum remelter. Results indicate that the relationship between the underlying structure/distribution of the compositional data and how these distributions are formulated in batch planning can modify the use of secondary raw materials.

Coexistence of Power-Controlled Cellular Networks With Rotating Radar

In this paper, we study spectral coexistence between rotating radar and power-controlled cellular networks in radar bands. For two systems to spectrally coexist, they must be able to operate effectively without causing harmful electromagnetic interference to each other. Very short radar-cellular system separation distances are required during 83.3% of time due to the narrow main beam width of the rotational radar antenna. We propose a spatio-temporal analytical approach with adaptive base station (BS) power control for adjacent spectrum sharing between the two systems. We develop a new model for the aggregate interference from power-controlled cellular BSs using log-normal approximation. The cellular system is allowed to transmit at high power when the radar antenna's main beam is pointing elsewhere from it. On the other hand, the cellular system reduces its transmit power only for a short period when the radar directional antenna main beam is pointing toward it. We use the degradation of the radar signal-to-interference plus noise ratio and cellular outage probability as our performance metrics. Numerical results show that power control of cellular BS highly reduces separation distance between the BS and radar, while yielding marginal degradation of outage performance. In addition, the mathematical results given by the log-normal approximation closely follow our simulated results. Detail system level assessments and investigations are presented to comprehensively understand secondary access to this band opportunistically.

The generalized inverse Lindley distribution: A new inverse statistical model for the study of upside-down bathtub data

ABSTRACTIn this article, a two-parameter generalized inverse Lindley distribution capable of modeling a upside-down bathtub-shaped hazard rate function is introduced. Some statistical properties of proposed distribution are explicitly derived here. The method of maximum likelihood, least square, and maximum product spacings are used for estimating the unknown model parameters and also compared through the simulation study. The approximate confidence intervals, based on a normal and a log-normal approximation, are also computed. Two algorithms are proposed for generating a random sample from the proposed distribution. A real data set is modeled to illustrate its applicability, and it is shown that our distribution fits much better than some other existing inverse distributions.

Analysis of a Stochastic Model for Bacterial Growth and the Lognormality of the Cell-Size Distribution

This paper theoretically analyzes a phenomenological stochastic model for bacterial growth. This model comprises cell division and the linear growth of cells, where growth rates and cell cycles are drawn from lognormal distributions. We find that the cell size is expressed as a sum of independent lognormal variables. We show numerically that the quality of the lognormal approximation greatly depends on the distributions of the growth rate and cell cycle. Furthermore, we show that actual parameters of the growth rate and cell cycle take values that give a good lognormal approximation; thus, the experimental cell-size distribution is in good agreement with a lognormal distribution.

Microscopic Analysis of the Uplink Interference in FDMA Small Cell Networks

In this paper, we analytically derive an upper bound on the error in approximating the uplink (UL) single-cell interference by a lognormal distribution in frequency division multiple access (FDMA) small cell networks (SCNs). Such an upper bound is measured by the Kolmogorov Smirnov (KS) distance between the actual cumulative density function (CDF) and the approximate CDF. The lognormal approximation is important because it allows tractable network performance analysis. Our results are more general than the existing works in the sense that we do not pose any requirement on (i) the shape and/or size of cell coverage areas, (ii) the uniformity of user equipment (UE) distribution, and (iii) the type of multi-path fading. Based on our results, we propose a new framework to directly and analytically investigate a complex network with practical deployment of multiple BSs placed at irregular locations, using a power lognormal approximation of the aggregate UL interference. The proposed network performance analysis is particularly useful for the 5th generation (5G) systems with general cell deployment and UE distribution.

Downlink Average Rate and SINR Distribution in Cellular Networks

The statistical characteristics of the signal-to-interference plus noise ratio (SINR) are closely related to many performance metrics of cellular networks. In this paper, the downlink average rate and SINR distribution are studied for orthogonal frequency division multiple access (OFDMA)-based cellular networks subject to the distance-dependent path loss and shadow fading (SF). With the analytical approximate mean and moment generating function (MGF) of the SINR in the logarithmic domain, the closed-form approximation for the lower and upper bounds of the average rate is obtained. Then the distribution of the SINR in the logarithmic domain is proposed to be approximated as the normal inverse Gaussian (NIG) distribution whose parameters are computed explicitly through moment matching. Also, the closed-form expression for the cumulative distribution function (CDF) of the SINR based on the NIG approximation is derived. Simulation results not only verify the tightness of the bounds, but also show that the NIG approximation is up to one order of magnitude more accurate than the Pearson type IV approximation and at least one order of magnitude more accurate than the lognormal approximation when the SF correlation coefficient is small or the standard deviations of the SF are large or different.