# Models Of Network Reliability Analysis Combinatorics And Monte Carlo Pdf

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Published: 20.05.2021  We present a brief survey of the current state of the art in network reliability. We survey only exact methods and do not consider Monte Carlo methods.

## *jdL*New* Models of Network Reliability: Analysis, Combinatorics, and Monte

Monte Carlo methods , or Monte Carlo experiments , are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle.

They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes:  optimization , numerical integration , and generating draws from a probability distribution.

In physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom , such as fluids, disordered materials, strongly coupled solids, and cellular structures see cellular Potts model , interacting particle systems , McKean—Vlasov processes , kinetic models of gases.

Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of risk in business and, in mathematics, evaluation of multidimensional definite integrals with complicated boundary conditions.

In application to systems engineering problems space, oil exploration , aircraft design, etc. In principle, Monte Carlo methods can be used to solve any problem having a probabilistic interpretation. By the law of large numbers , integrals described by the expected value of some random variable can be approximated by taking the empirical mean a.

That is, in the limit, the samples being generated by the MCMC method will be samples from the desired target distribution. In other problems, the objective is generating draws from a sequence of probability distributions satisfying a nonlinear evolution equation. These flows of probability distributions can always be interpreted as the distributions of the random states of a Markov process whose transition probabilities depend on the distributions of the current random states see McKean—Vlasov processes , nonlinear filtering equation.

These models can also be seen as the evolution of the law of the random states of a nonlinear Markov chain. In contrast with traditional Monte Carlo and MCMC methodologies, these mean field particle techniques rely on sequential interacting samples. The terminology mean field reflects the fact that each of the samples a. When the size of the system tends to infinity, these random empirical measures converge to the deterministic distribution of the random states of the nonlinear Markov chain, so that the statistical interaction between particles vanishes.

For example, consider a quadrant circular sector inscribed in a unit square. In this procedure the domain of inputs is the square that circumscribes the quadrant. We generate random inputs by scattering grains over the square then perform a computation on each input test whether it falls within the quadrant.

Uses of Monte Carlo methods require large amounts of random numbers, and it was their use that spurred the development of pseudorandom number generators [ citation needed ] , which were far quicker to use than the tables of random numbers that had been previously used for statistical sampling. Before the Monte Carlo method was developed, simulations tested a previously understood deterministic problem, and statistical sampling was used to estimate uncertainties in the simulations.

Monte Carlo simulations invert this approach, solving deterministic problems using probabilistic metaheuristics see simulated annealing.

In the s, Enrico Fermi first experimented with the Monte Carlo method while studying neutron diffusion, but he did not publish this work. In the late s, Stanislaw Ulam invented the modern version of the Markov Chain Monte Carlo method while he was working on nuclear weapons projects at the Los Alamos National Laboratory.

Immediately after Ulam's breakthrough, John von Neumann understood its importance. In , nuclear weapons physicists at Los Alamos were investigating neutron diffusion in fissionable material. Ulam proposed using random experiments. He recounts his inspiration as follows:. The first thoughts and attempts I made to practice [the Monte Carlo Method] were suggested by a question which occurred to me in as I was convalescing from an illness and playing solitaires.

The question was what are the chances that a Canfield solitaire laid out with 52 cards will come out successfully?

After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count the number of successful plays. This was already possible to envisage with the beginning of the new era of fast computers, and I immediately thought of problems of neutron diffusion and other questions of mathematical physics, and more generally how to change processes described by certain differential equations into an equivalent form interpretable as a succession of random operations.

Later [in ], I described the idea to John von Neumann , and we began to plan actual calculations. Being secret, the work of von Neumann and Ulam required a code name. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, and also noted that when it went awry it did so obviously, unlike methods that could be subtly incorrect.

Monte Carlo methods were central to the simulations required for the Manhattan Project , though severely limited by the computational tools at the time. In the s they were used at Los Alamos for early work relating to the development of the hydrogen bomb , and became popularized in the fields of physics , physical chemistry , and operations research.

The Rand Corporation and the U. Air Force were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide application in many different fields. The theory of more sophisticated mean field type particle Monte Carlo methods had certainly started by the mids, with the work of Henry P.

McKean Jr. Harris and Herman Kahn, published in , using mean field genetic -type Monte Carlo methods for estimating particle transmission energies. The origins of these mean field computational techniques can be traced to and with the work of Alan Turing on genetic type mutation-selection learning machines  and the articles by Nils Aall Barricelli at the Institute for Advanced Study in Princeton, New Jersey.

Quantum Monte Carlo , and more specifically diffusion Monte Carlo methods can also be interpreted as a mean field particle Monte Carlo approximation of Feynman — Kac path integrals.

Resampled or Reconfiguration Monte Carlo methods for estimating ground state energies of quantum systems in reduced matrix models is due to Jack H. Hetherington in  In molecular chemistry, the use of genetic heuristic-like particle methodologies a. Rosenbluth and Arianna W. The use of Sequential Monte Carlo in advanced signal processing and Bayesian inference is more recent.

It was in , that Gordon et al. The authors named their algorithm 'the bootstrap filter', and demonstrated that compared to other filtering methods, their bootstrap algorithm does not require any assumption about that state-space or the noise of the system. Particle filters were also developed in signal processing in — by P. Del Moral, J. Noyer, G. Rigal, and G. From to , all the publications on Sequential Monte Carlo methodologies, including the pruning and resample Monte Carlo methods introduced in computational physics and molecular chemistry, present natural and heuristic-like algorithms applied to different situations without a single proof of their consistency, nor a discussion on the bias of the estimates and on genealogical and ancestral tree based algorithms.

The mathematical foundations and the first rigorous analysis of these particle algorithms were written by Pierre Del Moral in Del Moral, A. Guionnet and L. There is no consensus on how Monte Carlo should be defined.

For example, Ripley  defines most probabilistic modeling as stochastic simulation , with Monte Carlo being reserved for Monte Carlo integration and Monte Carlo statistical tests. Sawilowsky  distinguishes between a simulation , a Monte Carlo method, and a Monte Carlo simulation: a simulation is a fictitious representation of reality, a Monte Carlo method is a technique that can be used to solve a mathematical or statistical problem, and a Monte Carlo simulation uses repeated sampling to obtain the statistical properties of some phenomenon or behavior.

Kalos and Whitlock  point out that such distinctions are not always easy to maintain. For example, the emission of radiation from atoms is a natural stochastic process. It can be simulated directly, or its average behavior can be described by stochastic equations that can themselves be solved using Monte Carlo methods. The main idea behind this method is that the results are computed based on repeated random sampling and statistical analysis.

The Monte Carlo simulation is, in fact, random experimentations, in the case that, the results of these experiments are not well known. Monte Carlo simulations are typically characterized by many unknown parameters, many of which are difficult to obtain experimentally. The only quality usually necessary to make good simulations is for the pseudo-random sequence to appear "random enough" in a certain sense. What this means depends on the application, but typically they should pass a series of statistical tests.

Testing that the numbers are uniformly distributed or follow another desired distribution when a large enough number of elements of the sequence are considered is one of the simplest and most common ones. Sawilowsky lists the characteristics of a high-quality Monte Carlo simulation: . Pseudo-random number sampling algorithms are used to transform uniformly distributed pseudo-random numbers into numbers that are distributed according to a given probability distribution.

Low-discrepancy sequences are often used instead of random sampling from a space as they ensure even coverage and normally have a faster order of convergence than Monte Carlo simulations using random or pseudorandom sequences.

Methods based on their use are called quasi-Monte Carlo methods. In an effort to assess the impact of random number quality on Monte Carlo simulation outcomes, astrophysical researchers tested cryptographically-secure pseudorandom numbers generated via Intel's RDRAND instruction set, as compared to those derived from algorithms, like the Mersenne Twister , in Monte Carlo simulations of radio flares from brown dwarfs.

No statistically significant difference was found between models generated with typical pseudorandom number generators and RDRAND for trials consisting of the generation of 10 7 random numbers. A Monte Carlo method simulation is defined as any method that utilizes sequences of random numbers to perform the simulation.

Monte Carlo simulations are applied to many topics including quantum chromodynamics , cancer radiation therapy, traffic flow, stellar evolution and VLSI design. All these simulations require the use of random numbers and therefore pseudorandom number generators , which makes creating random-like numbers very important.

If a square enclosed a circle and a point were randomly chosen inside the square the point would either lie inside the circle or outside it. If the process were repeated many times, the ratio of the random points that lie inside the circle to the total number of random points in the square would approximate the ratio of the area of the circle to the area of the square. From this we can estimate pi, as shown in the Python code below utilizing a SciPy package to generate pseudorandom numbers with the MT algorithm.

There are ways of using probabilities that are definitely not Monte Carlo simulations — for example, deterministic modeling using single-point estimates. Each uncertain variable within a model is assigned a "best guess" estimate. Scenarios such as best, worst, or most likely case for each input variable are chosen and the results recorded. By contrast, Monte Carlo simulations sample from a probability distribution for each variable to produce hundreds or thousands of possible outcomes.

The results are analyzed to get probabilities of different outcomes occurring. The samples in such regions are called "rare events". Monte Carlo methods are especially useful for simulating phenomena with significant uncertainty in inputs and systems with many coupled degrees of freedom. Areas of application include:. Monte Carlo methods are very important in computational physics , physical chemistry , and related applied fields, and have diverse applications from complicated quantum chromodynamics calculations to designing heat shields and aerodynamic forms as well as in modeling radiation transport for radiation dosimetry calculations.

In astrophysics , they are used in such diverse manners as to model both galaxy evolution  and microwave radiation transmission through a rough planetary surface. Monte Carlo methods are widely used in engineering for sensitivity analysis and quantitative probabilistic analysis in process design.

The need arises from the interactive, co-linear and non-linear behavior of typical process simulations. For example,. The Intergovernmental Panel on Climate Change relies on Monte Carlo methods in probability density function analysis of radiative forcing. The PDFs are generated based on uncertainties provided in Table 8. The combination of the individual RF agents to derive total forcing over the Industrial Era are done by Monte Carlo simulations and based on the method in Boucher and Haywood PDF of the ERF from surface albedo changes and combined contrails and contrail-induced cirrus are included in the total anthropogenic forcing, but not shown as a separate PDF. ## Reliability Analysis With Monte Carlo Simulation and

Monte Carlo methods , or Monte Carlo experiments , are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes:  optimization , numerical integration , and generating draws from a probability distribution. In physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom , such as fluids, disordered materials, strongly coupled solids, and cellular structures see cellular Potts model , interacting particle systems , McKean—Vlasov processes , kinetic models of gases. Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of risk in business and, in mathematics, evaluation of multidimensional definite integrals with complicated boundary conditions. In application to systems engineering problems space, oil exploration , aircraft design, etc.

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Reliability analysis and improvement of multilevel converters. Doctoral thesis, Nanyang Technological University, Singapore. It can reduce the operating cost of motor drive system significantly. However, the reliability of power converter is a salient concern for both manufacturers and end users as high failure rate will incur additional repairing cost. In high power drive applications, multilevel converters which utilize mature power semiconductors are superior to conventional two-level converters in efficiency and power quality. But the reliability problem of multilevel converters is more serious due to a large number of vulnerable power semiconductors used. ## A Survey of Network Reliability and Domination Theory

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