Open Access. Powered by Scholars. Published by Universities.®

Physical Sciences and Mathematics Commons

Open Access. Powered by Scholars. Published by Universities.®

Simulation

Discipline
Institution
Publication Year
Publication
Publication Type
File Type

Articles 541 - 570 of 784

Full-Text Articles in Physical Sciences and Mathematics

Tetrahedral Image-To-Mesh Conversion Software For Anatomic Modeling Of Arteriovenous Malformations, Fotis Drakopoulos, Ricardo Ortiz, Andinet Enquobahrie, Deanna Sasaki-Adams, Nikos Chrisochoides Jan 2015

Tetrahedral Image-To-Mesh Conversion Software For Anatomic Modeling Of Arteriovenous Malformations, Fotis Drakopoulos, Ricardo Ortiz, Andinet Enquobahrie, Deanna Sasaki-Adams, Nikos Chrisochoides

Computer Science Faculty Publications

We describe a new implementation of an adaptive multi-tissue tetrahedral mesh generator targeting anatomic modeling of Arteriovenous Malformation (AVM) for surgical simulations. Our method, initially constructs an adaptive Body-Centered Cubic (BCC) mesh of high quality elements. Then, it deforms the mesh surfaces to their corresponding physical image boundaries, hence, improving the mesh fidelity and smoothness. Our deformation scheme, which builds upon the ITK toolkit, is based on the concept of energy minimization, and relies on a multi-material point-based registration. It uses non-connectivity patterns to implicitly control the number of the extracted feature points needed for the registration, and thus, adjusts …


Cyber Security Research Frameworks For Coevolutionary Network Defense, George Daniel Rush Jan 2015

Cyber Security Research Frameworks For Coevolutionary Network Defense, George Daniel Rush

Masters Theses

"Cyber security is increasingly a challenge for organizations everywhere. Defense systems that require less expert knowledge and can adapt quickly to threats are strongly needed to combat the rise of cyber attacks. Computational intelligence techniques can be used to rapidly explore potential solutions while searching in a way that is unaffected by human bias.

Several architectures have been created for developing and testing systems used in network security, but most are meant to provide a platform for running cyber security experiments as opposed to automating experiment processes. In the first paper, we propose a framework termed Distributed Cyber Security Automation …


The Cyber Simulation Terrain: Towards An Open Source Cyber Effects Simulation Ontology, Kent O'Sullivan, Benjamin Turnbull Jan 2015

The Cyber Simulation Terrain: Towards An Open Source Cyber Effects Simulation Ontology, Kent O'Sullivan, Benjamin Turnbull

Australian Information Warfare and Security Conference

Cyber resilience is characterised by an ability to understand and adapt to changing network conditions, including cyber attacks. Cyber resilience may be characterised by an effects-based approach to missions or processes. One of the fundamental preconditions underpinning cyber resilience is an accurate representation of current network and machine states and what missions they are supporting. This research outlines the need for an ontological network representation, drawing on existing literature and implementations in the domain. This work then introduces an open-source ontological representation for modelling cyber assets for the purposes of Computer Network Defence. This representation encompasses computers, network connectivity, users, …


Control Of Synchrotron Radiation Effects During Recirculation With Bunch Compression, D. R. Douglas, S. V. Benson, R. Li, Y. Roblin, C. D. Tennant, Geoffrey A. Krafft, Balŝa Terzić, C. -Y. Tsai Jan 2015

Control Of Synchrotron Radiation Effects During Recirculation With Bunch Compression, D. R. Douglas, S. V. Benson, R. Li, Y. Roblin, C. D. Tennant, Geoffrey A. Krafft, Balŝa Terzić, C. -Y. Tsai

Physics Faculty Publications

Studies of beam quality preservation during recirculation * have been extended to generate a design of a compact arc providing bunch compression with positive momentum compaction ** and control of both incoherent and coherent synchrotron radiation (ISR and CSR) effects using the optics balance methods of diMitri et al.***. In addition, the arc/compressor generates very little micro-bunching gain. We detail the beam dynamical basis for the design, discuss the design process, give an example solution, and provide simulations of ISR and CSR effects. Reference will be made to a complete analysis of micro-bunching effects ****.


High-Performance Simulations Of Coherent Synchrotron Radiation On Multicore Gpu And Cpu Platforms, B. Terzić, A. Godunov, K. Arumugam, D. Ranjan, M. Zubair Jan 2015

High-Performance Simulations Of Coherent Synchrotron Radiation On Multicore Gpu And Cpu Platforms, B. Terzić, A. Godunov, K. Arumugam, D. Ranjan, M. Zubair

Physics Faculty Publications

Coherent synchrotron radiation (CSR) is an effect of self-interaction of an electron bunch as it traverses a curved path. It can cause a significant emittance degradation and microbunching. We present a new high-performance 2D, particle-in-cell code which uses massively parallel multicore GPU/GPU platforms to alleviate computational bottlenecks. The code formulates the CSR problem from first principles by using the retarded scalar and vector potentials to compute the self-interaction fields. The speedup due to the parallel implementation on GPU/CPU platforms exceeds three orders of magnitude, thereby bringing a previously intractable problem within reach. The accuracy of the code is verified against …


Modeling Crabbing Dynamics In An Electron-Ion Collider, A. Castilla, V. S. Morozov, T. Satogata, J. R. Delayen Jan 2015

Modeling Crabbing Dynamics In An Electron-Ion Collider, A. Castilla, V. S. Morozov, T. Satogata, J. R. Delayen

Physics Faculty Publications

A local crabbing scheme requires π/2 (mod π) horizontal betatron phase advances from an interaction point (IP) to the crab cavities on each side of it. However, realistic phase advances generated by sets of quadrupoles, or Final Focusing Blocks (FFB), between the crab cavities located in the expanded beam regions and the IP differ slightly from π/2. To understand the effect of crabbing on the beam dynamics in this case, a simple model of the optics of the Medium Energy Electron-Ion Collider (MEIC) including local crabbing was developed using linear matrices and then studied numerically over multiple turns (1000 passes) …


A Semiparametric Estimation For Regression Functions In The Partially Linear Autoregressive Time Series Model, R. Farnoosh, M. Hajebi, S. J. Mortazavi Dec 2014

A Semiparametric Estimation For Regression Functions In The Partially Linear Autoregressive Time Series Model, R. Farnoosh, M. Hajebi, S. J. Mortazavi

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, a semiparametric method is proposed for estimating regression function in the partially linear autoregressive time series model . Here, we consider a combination of parametric forms and nonlinear functions, in which the errors are independent. Semiparametric and nonparametric curve estimation provides a useful tool for exploring and understanding the structure of a nonlinear time series data set to make for a more efficient study in the partially linear autoregressive model. The unknown parameters are estimated using the conditional nonlinear least squares method, and the nonparametric adjustment is also estimated by defining and minimizing the local L2 -fitting …


High Dimensional Non-Linear Optimization Of Molecular Models, Joseph C. Fogarty Nov 2014

High Dimensional Non-Linear Optimization Of Molecular Models, Joseph C. Fogarty

USF Tampa Graduate Theses and Dissertations

Molecular models allow computer simulations to predict the microscopic properties of macroscopic systems. Molecular modeling can also provide a fully understood test system for the application of theoretical methods. The power of a model lies in the accuracy of the parameter values which govern its mathematical behavior. In this work, a new software, called ParOpt, for general high dimensional non-linear optimization will be presented. The software provides a very general framework for the optimization of a wide variety of parameter sets. The software is especially powerful when applied to the difficult task of molecular model parameter optimization. Three applications of …


Some General Guidelines For Choosing Missing Data Handling Methods In Educational Research, Jehanzeb R. Cheema Nov 2014

Some General Guidelines For Choosing Missing Data Handling Methods In Educational Research, Jehanzeb R. Cheema

Journal of Modern Applied Statistical Methods

The effect of a number of factors, such as the choice of analytical method, the handling method for missing data, sample size, and proportion of missing data, were examined to evaluate the effect of missing data treatment on accuracy of estimation. A methodological approach involving simulated data was adopted. One outcome of the statistical analyses undertaken in this study is the formulation of easy-to-implement guidelines for educational researchers that allows one to choose one of the following factors when all others are given: sample size, proportion of missing data in the sample, method of analysis, and missing data handling method.


Ridge Regression And Ill-Conditioning, Ghadban Khalaf, Mohamed Iguernane Nov 2014

Ridge Regression And Ill-Conditioning, Ghadban Khalaf, Mohamed Iguernane

Journal of Modern Applied Statistical Methods

Hoerl and Kennard (1970) suggested the ridge regression estimator as an alternative to the Ordinary Least Squares (OLS) estimator in the presence of multicollinearity. This article proposes new methods for estimating the ridge parameter in case of ordinary ridge regression. A simulation study evaluates the performance of the proposed estimators based on the Mean Squared Error (MSE) criterion and indicates that, under certain conditions, the proposed estimators perform well compared to the OLS estimator and another well-known estimator reviewed.


Double Bootstrap Confidence Interval Estimates With Censored And Truncated Data, Jayanthi Arasan, Mohd B. Adam Nov 2014

Double Bootstrap Confidence Interval Estimates With Censored And Truncated Data, Jayanthi Arasan, Mohd B. Adam

Journal of Modern Applied Statistical Methods

Traditional inferential procedures often fail with censored and truncated data, especially when sample sizes are small. In this paper we evaluate the performances of the double and single bootstrap interval estimates by comparing the double percentile (DB-p), double percentile-t (DB-t), single percentile (B-p), and percentile-t (B-t) bootstrap interval estimation methods via a coverage probability study when the data is censored using the log logistic model. We then apply the double bootstrap intervals to real right censored lifetime data on 32 women with breast cancer and failure data on 98 brake pads where all the observations were left truncated.


Optical Counterparts Of Two Fermi Millisecond Pulsars: Psr J1301+0833 And Psr J1628–3205, Miao Li, Jules P. Halpern, John R. Thorstensen Oct 2014

Optical Counterparts Of Two Fermi Millisecond Pulsars: Psr J1301+0833 And Psr J1628–3205, Miao Li, Jules P. Halpern, John R. Thorstensen

Dartmouth Scholarship

Using the 1.3 m and 2.4 m Telescopes of the MDM Observatory, we identified the close companions of two eclipsing millisecond radio pulsars that were discovered by the Green Bank Telescope in searches of Fermi Gamma-ray Space Telescope sources, and measured their light curves. PSR J1301+0833 is a black widow pulsar in a 6.5 hr orbit whose companion star is strongly heated on the side facing the pulsar. It varies from R = 21.8 to R > 24 around the orbit. PSR J1628–3205 is a "redback," a nearly Roche-lobe-filling system in a 5.0 hr orbit whose optical modulation in the range …


Compromise Programming Based Scenario Analysis Of Urban Water Systems Management Options: Case Study Of Kerman City, Saeed Nazari, S. Jamshid Mousavi, Koroush Behzadian, Zoran Kapelan Aug 2014

Compromise Programming Based Scenario Analysis Of Urban Water Systems Management Options: Case Study Of Kerman City, Saeed Nazari, S. Jamshid Mousavi, Koroush Behzadian, Zoran Kapelan

International Conference on Hydroinformatics

Supplying adequate water for urban water systems (UWSs) suffering from lack of water resources has always been a major concern in urban water management. Integrated simulation models are useful tools for sustainable planning and management of UWSs. This paper presents an integrated, conceptual modeling approach for simulation and analysis of an UWS by which different envisaged scenarios of water demand and resources are assessed. Other than water flow, the simulation model quantifies flows of energy, GHG emissions and cost in UWS. The performance of the developed model is demonstrated through its application to the UWS of Kerman City located in …


Heterogeneous Computing Paradigm For Parallel Water Distribution System Analysis, Zheng Yi Wu Aug 2014

Heterogeneous Computing Paradigm For Parallel Water Distribution System Analysis, Zheng Yi Wu

International Conference on Hydroinformatics

Hydraulic and water quality models have been developed and used over decades for water distribution system (WDS) analysis. The models prove to be powerful tools for engineers to gain systematic understanding on the WDS conditions and conduct technically-sound WDS management. However, it takes significantly amount of computation time to perform one simulation run for large system with tens or hundreds of thousands of pipes over an adequately long period of duration in order to gain good results for system hydraulic and water quality dynamics. In the meantime, WDS analysis solvers are not developed to take the advantages of available computing …


Examining Flow Paths In Bunter Sediments In Thuringian Basin, Thomas Fischer, Cindy Kunkel, Dmitri Naumov, Reinhard Gaupp, Olaf Kolditz Aug 2014

Examining Flow Paths In Bunter Sediments In Thuringian Basin, Thomas Fischer, Cindy Kunkel, Dmitri Naumov, Reinhard Gaupp, Olaf Kolditz

International Conference on Hydroinformatics

We would like to present a study on the Thuringian Basin, a large area in central Germany. Within this basin exist areas with very different interesting geo- and hydrological properties. We set up models and simulations for four subdomains within the basin, which describe different depositional environments - Eolian, Fluvial, Sandflat and Lacustrine. Researchers would like to investigate which paths is the water taking in the subsurface and how fast it is moving. The aim of this study is to gain knowledge about flow paths within these four model areas with regard to their architectural design. First simulation results suggest …


Enabling Access To Non-Point Source Risk Mapping Tools Using Open Source Software And Open Geospatial Consortium (Ogc) Standards : The Development Of The Scimap Webapp, Peter Wells, Sim Reaney Aug 2014

Enabling Access To Non-Point Source Risk Mapping Tools Using Open Source Software And Open Geospatial Consortium (Ogc) Standards : The Development Of The Scimap Webapp, Peter Wells, Sim Reaney

International Conference on Hydroinformatics

No-point (diffuse) pollution is a key environmental pressure effecting water quality and ecology in lakes and rivers. Many governmental and NGO bodies are working to tacking the problem but their efforts are constrained by the complexities of the problem. To enable the spatial targeting of the mitigation measures at the landscape scale, the SCIMAP risk mapping approach was developed by Durham and Lancaster Universities in the UK. The approach was well received but the desktop implementation created problems for users. NGO users generally did not have the required datasets or GIS skills. Governmental users within a corporate managed IT environment …


High Performance Gpu Speed-Up Strategies For The Computation Of 2d Inundation Models, Asier Lacasta, Mario Morales-Hernández, Javier Murillo, Pilar García-Navarro, Reinaldo Garcia Aug 2014

High Performance Gpu Speed-Up Strategies For The Computation Of 2d Inundation Models, Asier Lacasta, Mario Morales-Hernández, Javier Murillo, Pilar García-Navarro, Reinaldo Garcia

International Conference on Hydroinformatics

Two-dimensional (2D) models are increasingly used for inundation assesements in situations involving large domains of millions of computational elements and long-time scales of several months. Practical applications often involve a compromise between spatial accuracy and computational efficiency and to achieve the necessary spatial resolution, rather fine meshes become necessary requiring more data storage and very long computer times comparable to the real simulated process (e.g. 1 month of real-time simulation may require 1 month of computations approximately). Obviously, using conventional 2D non-parallelized models (CPU based) make simulations impractical in real project applications, but improving the performance of such complex models …


Gate Monte Carlo Simulations In A Cloud Computing Environment, Blake Austin Rowedder Aug 2014

Gate Monte Carlo Simulations In A Cloud Computing Environment, Blake Austin Rowedder

UNLV Theses, Dissertations, Professional Papers, and Capstones

The GEANT4-based GATE is a unique and powerful Monte Carlo (MC) platform, which provides a single code library allowing the simulation of specific medical physics applications, e.g. PET, SPECT, CT, radiotherapy, and hadron therapy. However, this rigorous yet flexible platform is used only sparingly in the clinic due to its lengthy calculation time. By accessing the powerful computational resources of a cloud computing environment, GATE's runtime can be significantly reduced to clinically feasible levels without the sizable investment of a local high performance cluster. This study investigated a reliable and efficient execution of GATE MC simulations using a commercial cloud …


Quantitative Evidence For The Use Of Simulation And Randomization In The Introductory Statistics Course, Nathan L. Tintle, Ally Rogers, Beth Chance, George Cobb, Allan Rossman, Soma Roy, Todd Swanson, Jill Vanderstoep Jul 2014

Quantitative Evidence For The Use Of Simulation And Randomization In The Introductory Statistics Course, Nathan L. Tintle, Ally Rogers, Beth Chance, George Cobb, Allan Rossman, Soma Roy, Todd Swanson, Jill Vanderstoep

Faculty Work Comprehensive List

The use of simulation and randomization in the introductory statistics course is gaining popularity, but what evidence is there that these approaches are improving students’ conceptual understanding and attitudes as we hope? In this talk I will discuss evidence from early full-length versions of such a curriculum, covering issues such as (a) items and scales showing improved conceptual performance compared to traditional curriculum, (b) transferability of findings to different institutions, (c) retention of conceptual understanding post-course and (d) student attitudes. Along the way I will discuss a few areas in which students in both simulation/randomization courses and the traditional course …


Monte Carlo Simulation: When Should A Contestant Stop Spinning?, Gregory Horn Jul 2014

Monte Carlo Simulation: When Should A Contestant Stop Spinning?, Gregory Horn

All Student Theses and Dissertations

Every episode of the popular game show The Price Is Right contains two rounds called The Showcase Showdown or The Big Wheel. During these rounds, three contestants spin a large wheel that consists of monetary values from five cents through one dollar in 5 cent increments. The object of this game is to get closest to one dollar without going over in one or a combination of two spins. The two winners of these rounds get to compete for the most valuable prizes at the end of each show. Monte Carlo simulation will be used to find the range of …


Simulating Influenza Transmission With Network Data, Henry V. Bongiovi Jun 2014

Simulating Influenza Transmission With Network Data, Henry V. Bongiovi

Statistics

Simulating Influenza Transmission with Real Network Data

Henry Bongiovi BS Statistics, California Polytechnic State University, San Luis Obispo

[email protected]

Keywords: Network Data, Simulation, Education, Influenza, Epidemic

Disease has been humanities arch rival since the dawn of our existence. As such, we have been trying our best to understand its spread and proliferation. One of the most common diseases, Influenza, is also one of the most complex. To understand the complexities of its spread would greatly improve our ability to combat it and other diseases like it. Using R in conjunction with the package statnet, I have created a simulation of …


Smarticles: A Method For Identifying And Correcting Instability And Error Caused By Explicit Integration Techniques In Physically Based Simulations, Susan Aileen Marano Jun 2014

Smarticles: A Method For Identifying And Correcting Instability And Error Caused By Explicit Integration Techniques In Physically Based Simulations, Susan Aileen Marano

Master's Theses

Using an explicit integration method in physically based animations has many advantages including conceptual and computational simplicity, however, it re- quires small time steps to ensure low numerical instability. Simulations with large numbers of individually interacting components such as cloth, hair, and fluid models, are limited by the sections of particles most susceptible to error. This results in the need for smaller time steps than required for the majority of the system. These sections can be diverse and dynamic, quickly changing in size and location based on forces in the system. Identifying and handling these trou- blesome sections could allow …


Design Of Cpu Simulation Software For Armv7 Instruction Set Architecture, Dillon Tellier Jun 2014

Design Of Cpu Simulation Software For Armv7 Instruction Set Architecture, Dillon Tellier

Computer Engineering

Simulations have long been a part of the engineering process in both the professional and academic domain. From a pedagogic standpoint, simulations allow students to explore the dynamics of engineering scenarios by controlling variables, taking measurements, and observing behavior which would be difficult or impossible without simulation. One such tool is a CPU simulator used in Cal Poly’s Computer Architecture classes; this software simulates an instruction accurate operation of a computer processor and reports statistics regarding the execution of the supplied compiled machine code. For the last several years Cal Poly’s computer architecture classes have used a previous version of …


Bias And Precision Of The Squared Canonical Correlation Coefficient Under Nonnormal Data Condition, Lesley F. Leach, Robin K. Henson May 2014

Bias And Precision Of The Squared Canonical Correlation Coefficient Under Nonnormal Data Condition, Lesley F. Leach, Robin K. Henson

Journal of Modern Applied Statistical Methods

Monte Carlo methods were employed to investigate the effect of nonnormality on the bias associated with the squared canonical correlation coefficient (Rc2). The majority of Rc2 estimates were found to be extremely biased, but the magnitude of bias was impacted little by the degree of nonnormality.


Comparison Of Different Methods For Estimating Log-Normal Means, Qi Tang May 2014

Comparison Of Different Methods For Estimating Log-Normal Means, Qi Tang

Electronic Theses and Dissertations

The log-normal distribution is a popular model in many areas, especially in biostatistics and survival analysis where the data tend to be right skewed. In our research, a total of ten different estimators of log-normal means are compared theoretically. Simulations are done using different values of parameters and sample size. As a result of comparison, ``A degree of freedom adjusted" maximum likelihood estimator and Bayesian estimator under quadratic loss are the best when using the mean square error (MSE) as a criterion. The ten estimators are applied to a real dataset, an environmental study from Naval Construction Battalion Center (NCBC), …


Numerical Simulation Of Hydrogen Absorption/Desorption Processes In Cylindrical Metal-Hydrogen Reactors For Hydrogen Storage, Fei Han Apr 2014

Numerical Simulation Of Hydrogen Absorption/Desorption Processes In Cylindrical Metal-Hydrogen Reactors For Hydrogen Storage, Fei Han

Doctoral Dissertations

Hydrogen is one of the best fuels because of its high calorific value and environmental friendliness. However, because of its low density, it has storage problems such as high pressure, large volume requirements, heavy weight and safely risks; this quality prevents its wide usage and commercialization.

Researchers have found that some metal/inter-metallic compounds/alloys, such as Mg, La, LaNi5, ZrV2, Mg2Ni and Ti 2Ni, can react with hydrogen and attain relatively large amounts of hydrogen at a relatively low pressure and near normal temperature. Under certain conditions, hydride can desorb hydrogen quickly. Thus, the metal- hydrogen reaction could be a practical …


M.A.E.G.U.S: Measuring Alternate Energy Generation Via Unity Simulation, Kavin Muhilan Nataraja Apr 2014

M.A.E.G.U.S: Measuring Alternate Energy Generation Via Unity Simulation, Kavin Muhilan Nataraja

Open Access Theses

This paper presents the MAEGUS serious game and a study to determine its efficacy as a pedagogical tool. The MAEGUS serious game teaches sustainable energy concepts through gameplay simulating wind turbines and solar arrays. Players take the role of an energy manager for a city and use realistic data and information visualizations to learn the physical factors of wind and solar energy generation. The MAEGUS serious game study compares game assisted learning to a more traditional teaching method such as reading material in a crossover study, the results of which can inform future serious game development for educational purposes.


Engineering Collaborations In Medical Modeling And Simulation, Frederic D. Mckenzie, Salim Chemlal, Tom Hubbard, Robert E. Kelly, Roderick C. Borgie, David A. Besachio, Michel Audette Jan 2014

Engineering Collaborations In Medical Modeling And Simulation, Frederic D. Mckenzie, Salim Chemlal, Tom Hubbard, Robert E. Kelly, Roderick C. Borgie, David A. Besachio, Michel Audette

Computational Modeling & Simulation Engineering Faculty Publications

Fifty years ago computer science was just beginning to see common acceptance as a growing discipline and very few universities had a computer science department although other departments were utilizing computers and software to enhance their methodologies. We believe modeling and simulation (M&S) is on a similar path. Many other disciplines utilize M&S to enhance their methodologies but we also believe that M&S fundamentals can be essential in making better decisions by utilizing the appropriate model for the problem at hand, expanding the solution space through simulation, and understanding it through visualization and proper analyses. After our students learn these …


Assessment Of The Perchertal Avalanche In Tyrol, Austria, Tayfun Kurt Jan 2014

Assessment Of The Perchertal Avalanche In Tyrol, Austria, Tayfun Kurt

Turkish Journal of Earth Sciences

The present study has been conducted to analyze the Perchertal avalanche area near Bärenkopf Mountain, which has several avalanche-prone areas on its slopes, within the area of Pertisau, Tyrol, in Austria. The main focus is on identifying the characteristics of the avalanche process itself to determine the potential risk to endangered objects, which include an important road and a hotel. Another focus is to evaluate the current local hazard map. Based on the dynamic avalanche models (Samos-AT, Ramms), several important action parameters such as impact pressure, avalanche velocity, and run-out length of the Perchertal avalanche track are presented and discussed. …


Exponentially Weighted Moving Average Charts For Monitoring The Process Generalized Variance, Anna Khamitova Jan 2014

Exponentially Weighted Moving Average Charts For Monitoring The Process Generalized Variance, Anna Khamitova

College of Graduate Studies: Theses & Dissertations

The exponentially weighted moving average chart based on the sample generalized variance is studied under the independent multivariate normal model for the vector of quality measurements. The performance of the chart is based on an analysis of the chart's initial and steady-state run length distributions. The three methods that are commonly used to determinate run length distribution, simulation, the integral equation method, and the Markov chain approximation are discussed. The integral equation and Markov chain approaches are analytical methods that require a nu- merical method for determining the probability density and cumulative distribution functions describing the distribution of the sample …