ANALYTICAL EVALUATION OF PROCESSING TIME OF A QUERY IN AN INFORMATION SYSTEM

The literature is replete with acknowledgments that spatial autocorrelation (SA) inflates the variance of a random variable (RV), and that it also may alter other RV distributional properties. In most studies, impacts of SA have been examined only for the three most commonly used distributions: the normal, Poisson (and its negative binomial counterpart), and binomial distributions; much less is known about its effects on two other RVs that are utilized in GIScience research: the beta and the multinomial. The beta distribution—which is considered to be very flexible because it can mimic a uniform, exponential, sinusoidal, and normal RV—can be utilized to analyze the radiance of a remotely sensed image, for example. The multinomial distribution, a generalization of the binomial distribution, has been widely used for land use classification, and to describe land use change. The literature also suggests that RV impacts of negative SA, a neglected topic in spatial analysis, may differ from those of positive SA, at least for some RVs (e.g., the Poisson RV). The purpose of this article is to extend the investigation of effects of SA to beta and multinomial RVs, with both positive SA and negative SA assessed and contrasted with each other, using simulation experiments. The simulation experiments are designed to support this assessment. One of the major discoveries is that impacts of positive SA and negative SA behave similarly when a RV conforms to a normal distribution; however, maximum negative SA is unable to materialize for asymmetric RV, whereas positive SA always converges upon its maximum.

A single machine scheduling problem in which both the processing times and due‐dates of the jobs awaiting servicing are random variables is analyzed. It is proved that the properties of the shortest processing time rule and the due‐date rule which are known for the deterministic situation also hold in the probabilistic environment when they are suitably, and reasonably, refined for this context.

A synchronizer is a compiler that transforms a program designed to run in a synchronous network into a program that runs in an asynchronous network. The behavior of a simple synchronizer, which also represents a basic mechanism for distributed computing and for the analysis of marked graphs, was studied by S. Even and S. Rajsbaum (1990) under the assumption that message transmission delays and processing times are constant. We study the behavior of the simple synchronizer when processing times and transmission delays are random. The main performance measure is the rate of a network, i.e., the average number of computational steps executed by a processor in the network per unit time. We analyze the effect of the topology and the probability distributions of the random variables on the behavior of the network. For random variables with exponential distribution, we provide tight (i.e., attainable) bounds and study the effect of a bottleneck processor on the rate.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A generalization of the classic Gaussian random variable to the family of multi-Gaussian (MG) random variables characterized by shape parameter M > 0, in addition to the mean and the standard deviation, is introduced. The probability density function (PDF) of the MG family members is an alternating series of Gaussian functions with suitably chosen heights and widths. In particular, for integer values of M, the series has a finite number of terms and leads to flattened profiles, while reducing to the classic Gaussian PDF for M = 1. For non-integer, positive values of M, a convergent infinite series of Gaussian functions is obtained that can be truncated in practical problems. For all M > 1, the MG PDF has flattened profiles, while for 0 < M < 1, the MG PDF has cusped profiles. Moreover, the multivariate extension of the MG random variable is obtained and the log-multi-Gaussian random variable is introduced. In order to illustrate the usefulness of these new random variables for optics, the application of MG random variables to the characterization of novel speckle fields is discussed, both theoretically and via numerical simulations.

In this paper, the k-μ-g random variable is considered. The k-μ-g random process arises from the κ-μ random process when the power of the k-μ random variable follows Gamma distribution. The closed form expressions for probability density function (PDF) and cumulative distribution function (CDF) of k-μ-g random variable are determined. Then, PDFs of the moment of n-th order of k-μ-g random variable, the ratio and product of two k-μ-g random variables are calculated in the closed form. Random variables are of crucial significance, not only for the statistical but also for the deterministic modeling of wireless mobile radio channels. The main contribution of using derived expressions is possibility to make easier performance analysis of wireless communication systems in the presence of k-μ-g short term fading, Gamma shadowing and k-μ-g cochannel interference.

Consider nested subdivisions of a bounded real set into intervals defining the digits $X_1,X_2,\ldots$ of a random variable X with a probability density function f . If f is almost everywhere lower semi-continuous, there is a non-negative integer-valued random variable N such that the distribution of $R=(X_{N+1},X_{N+2},\ldots)$ conditioned on $S=(X_1,\ldots,X_N)$ does not depend on f . If also the lengths of the intervals exhibit a Markovian structure, $R\mid S$ becomes a Markov chain of a certain order $s\ge0$ . If $s=0$ then $X_{N+1},X_{N+2},\ldots$ are independent and identically distributed with a known distribution. When $s&gt;0$ and the Markov chain is uniformly geometric ergodic, there is a random time M such that the chain after time $\max\{N,s\}+M-s$ is stationary and M follows a simple known distribution.

This book presents the theory of probability and mathematical statistics at a level suitable for researchers at the frontiers of applied disciplines. Examples and exercises make essential concepts in measure theory and analysis accessible to those with preparation limited to vector calculus. Complete, detailed solutions to all the exercises demonstrate techniques of problem solving and provide immediate feedback. Part I, The Theory of Probability, starts with elementary set theory and proceeds through basic measure and probability, random variables, integration and mathematical expectation. It concludes with an extensive survey of models for distributions of random variables. Part II, The Theory of Statistics, begins with sampling theory and distribution theory for statistics from normal populations, proceeds to asymptotic (large-sample) theory, and on to point and interval estimation and tests of parametric hypotheses. The last three chapters cover tests of nonparametric hypotheses, Bayesian methods, and linear and nonlinear regression. Researchers and graduate students in applied fields such as actuarial science, biostatistics, economics, finance, mathematical psychology, and systems engineering will find this book to be a valuable learning tool and an essential reference. Contents: The Theory of Probability: Probability on Abstract Sets Probability on Sets of Real Numbers Mathematical Expectation Models for Distributions The Theory of Statistics: Sampling Distributions Asymptotic Distribution Theory Estimation Tests of Parametric Hypotheses Tests of Nonparametric Hypotheses A Survey of Bayesian Methods Regression Analysis Readership: Researchers and graduate students in applied fields such as actuarial science, biostatistics, economics, finance, mathematical psychology, and systems engineering. Key Features: Present basic concepts in real and complex analysis and measure theory along with the applications to probability Difficult theoretical points are followed directly by examples and exercises with detailed solutions Provide a survey of Bayesian methods and an extensive treatment of goodness-of-fit tests Suitable learning tool for students and researchers in applied fields for whom a rigorous foundation in probability and statistical inference is essential

The risk assessment in many engineering applications is hampered by a lack of hard data. Under these conditions the selection of probability density function (PDF) seems arbitrary. Quite often the data are not only sparse but also vague expert opinion or conflicting. Several non-probabilistic methods have been proposed in the literature to perform a risk assessment under these conditions. We propose to use probabilistic techniques using uncertain PDFs. The uncertainty on the PDF is characterized by treating the parameters in the PDF as random variables. We expand the classical Bayesian updating scheme to make use of vague or imprecise interval data. Each expert opinion is considered to be a random sample from a parent distribution of expert opinions. Consequently, a conflict between experts is accounted for through the likelihood function. The uncertain PDFs can be used in both simulation-based and MPP-based reliability methods. Because of the uncertainty on the PDF of the random variables, the risk or reliability index itself will be a random variable. Design decisions are made on the basis of the risk assessment and an incorrect risk assessment increases the total cost of the design. Since a cost can be associated with either an over or underestimation of the risk, an optimal reliability index can be determined, which minimizes this cost. The probabilistic framework we present in this paper establishes a direct link between the amount and quality of the available data and the optimal reliability estimate. This link allows the decision maker to weigh expected value of additional data collection efforts against the expected optimal reliability index improvement.

For any multi-fractal dynamical system, a precise estimate of the local dimension is essential to infer variations in its number of degrees of freedom. Following extreme value theory, a local dimension may be estimated from the distributions of pairwise distances within the dataset. For absolutely continuous random variables and in the absence of zeros and singularities, the theoretical value of this local dimension is constant and equals the phase-space dimension. However, due to uneven sampling across the dataset, practical estimations of the local dimension may diverge from this theoretical value, depending on both the phase-space dimension and the position at which the dimension is estimated. To explore such variations of the estimated local dimension of absolutely continuous random variables, approximate analytical expressions are derived and further assessed in numerical experiments. These variations are expressed as a function of 1. the random variables’ probability density function, 2. the threshold used to compute the local dimension, and 3. the phase-space dimension. Largest deviations are anticipated when the probability density function has a low absolute value, and a high absolute value of its Laplacian. Numerical simulations of random variables of dimension 1 to 30 allow to assess the validity of the approximate analytical expressions. These effects may become important for systems of moderately-high dimension and in case of limited-size datasets. We suggest to take into account this source of local variation of dimension estimates in future studies of empirical data. Implications for weather regimes are discussed.

In general many real time problems are represented with random variables. But a random variable is characterized by a Probability Density Function(PDF). As per the Central Limit Theorem any stochastic process handled with many random variables can be converged as a normal distribution. Hence normal distribution plays a key role in modeling real time problems contains many random variables. This paper dealt with the generation of Gaussian PDF using uniform random sequence. The generated PDF is validated in LABVIEW.

An accurate model for performance analysis of land mobile satellite (LMS) channels is the Shadowed-Rice (SR) fading model. Recently, an approximate SR model has been proposed by the author for LMS channels. Compared with the existing models, the proposed model is analytically tractable and makes it possible to approximate very accurately the probability density function (PDF) of the SR random variable (RV) by a finite weighted sum of Rice PDFs. In this paper and based on the proposed model, the statistics of the sum of independent SR RVs is studied. While this problem is well understood for common fading models such as the Rayleigh, Rice, and Nakagami, a minor attention has been devoted to the SR case. In this paper and by using the well-known Beaulieu series approach, the PDF and the Cumulative distribution function (CDF) of the sum of independent SR RVs are derived in the form of infinite convergent series. In this regard, analytical, closed-form expressions are derived for the coefficients of the Beaulieu series. To give an application of the presented results, the performance of LMS communication system employing equal-gain combining (EGC) diversity techniques over SR fading channel is investigated and novel closed-form expression is derived for the outage probability of the system.

We propose an analytical approach to evaluate the performance of space‐diversity free‐space optical (FSO) systems over correlated Gamma–Gamma (ΓΓ) fading channels. To do this, we firstly derive an infinite series representation for the moment generating function (MGF) of the sum of arbitrarily correlated ΓΓ random variables (RVs). Then, a closed‐form approximate expression for the probability density function (PDF) of the sum of ΓΓ RVs is obtained from the MGF with the aid of Padé approximation. The performance of the FSO system is then evaluated based on this PDF. Through some numerical results we illustrate the accuracy of the proposed method and further discuss its limitations.

This chapter reviews some subjects in mathematics and physics that are used in different contexts throughout this book. The selection of subjects and the level of their coverage reflect the author’s perception of what potential users of this text were exposed to in their earlier studies. Therefore, only brief overview is given of some subjects while somewhat more comprehensive discussion is given of others. In neither case can the coverage provided substitute for the actual learning of these subjects that are covered in detail by many textbooks. A random variable is an observable whose repeated determination yields a series of numerical values (“realizations” of the random variable) that vary from trial to trial in a way characteristic of the observable. The outcomes of tossing a coin or throwing a die are familiar examples of discrete random variables. The position of a dust particle in air and the lifetime of a light bulb are continuous random variables. Discrete random variables are characterized by probability distributions; Pn denotes the probability that a realization of the given random variable is n. Continuous random variables are associated with probability density functions P(x): P(x1)dx denotes the probability that the realization of the variable x will be in the interval x1 . . . x1+dx.

For a continuous random variable in real number field, there must be a distribution and also a probability density function of this random variable. If there is a known function with this random variable as independent variable, its image is a smooth or piecewise smooth line, there must be at least one function that takes this random variable as its independent variable, these functions are bounded on the image of the first function. Any one of these functions conduct line integral operation to the line segment or arc length of the certain image of the first known function is the cumulative probability of this continuous random variable interval corresponding to the section of the image for line integral operation. A general designation for these functions are linear probability density function of continuous random variables. Conduct line integral operation to the linear probability density function and conduct integral operation to the probability density function have same results of the cumulative probability of continuous random variable. By the way, Line integration including curve integration. According to the uniqueness of the probability, the existence and the number of linear probability density function can be proved and calculated.

Inverse simulation is an inverse process of direct simulation. It determines unknown input variables of the direct simulation for a given set of simulation output variables. Uncertainties usually exist, making it difficult to solve inverse simulation problems. The objective of this research is to account for uncertainties in inverse simulation in order to produce high confidence in simulation results. The major approach is the use of the maximum probability density function (PDF), which determines not only unknown deterministic input variables but also the realizations of random input variables. Both types of variables are solved on the condition that the joint probability density of all the random variables is maximum. The proposed methodology is applied to a traffic accident reconstruction problem where the simulation output (accident consequences) is known and the simulation input (velocities of the vehicle at the beginning of crash) is sought.