Mean Value Function Research Articles

Hardy–Stein Type Identities and Rate of Growth of Mean Values of Functions in Generalized Hardy Spaces

In this paper, we prove Hardy–Stein type identities for functions in generalized Hardy spaces. We also prove that rate of growth of mean values of functions in generalized Hardy spaces is at most like \(\circ (1/1 - r)\;\;\text{ as }\;\; r \rightarrow 1.\)

Software Quality Control Truncated Time with Linear Failure Rate Distribution based on NHPP

Statistical process control is a method of monitoring product in its development process using statistical techniques with the presumption that the products produced under identical process condition shall not always be alike with respect to some quality characteristic(s). However, if the observed variations are with in the tolerable limits statistical process control (SPC) methods would pass them for acceptance. This philosophy is adopted to decide the reliability and quality of a developed software by defining some quality measures and proposing a probability model for the quality measurements. The well known linear failure rate distribution(LFRD) is considered to propose a software reliability based on non-homogenous Poisson process (NHPP). Its mean value function is taken as a quality characteristic and SPC limits for it are developed. These control limits are exemplified to a live failure data to detect the out of control signals for the quality of the software based on the software failure data.

Dynamically Weighted Combination Model for Describing Inconsistent Failure Data of Software Projects

Background/Objective: The Software Reliability Engineering discipline is witnessing continued research in the field of Software Reliability Growth Models (SRGMs), in order to develop suitable models so as to match with the phenomenal developments in the software sector. The objective of this paper is to propose a combined dynamically weighed model for reliability analysis. Methods: In general, the ability of a model to describe a dataset adequately is assessed by Goodness of Fit (GoF) measures that a model achieves. We used one of the difficult data sets for evaluation of the GoF performance and the non-linear regression was carried out using Curve fitting application of the MATLABTM software tool. The coefficients of determination R-Square (R2), Sum of Squared Error (SSE), Root Mean Square Error (RMSE) are the different GoF metrics used. Findings: The major challenge for the software projects today, is to measure, analyze and control the level of quality of the delivered software. Undesirably long testing cycles adversely influence time to market and hence software development organizations are very keen on having a good control on the testing cycle. In the last few years many reliability models have been identified and recommended. However, no single model can be considered suitable for all situations. Some models assume the growth of mean value function to be exponential, while other models assume it to follow S-type growth. This poses a challenge to the reliability modeling. The dynamically weighted combination model that we propose in this paper helps to address both the phenomena. The superposition and time transformation property of Non-Homogenous Poisson Process (NHPP) model was considered to derive the proposed model. Application: Using SRGM helps to improve the reliability performance. We have derived a powerful model by combining two well-known models. The proposed model gives excellent GoF performance and provides more confidence both for the customer and for the software development organizations.

Quantum transport in a two-level quantum dot driven by coherent and stochastic fields

Abstract We study theoretically the current and shot noise properties flowing through a two-level quantum dot driven by a strong coherent field and a weak stochastic field. The interaction x(t) between the quantum dot and the stochastic field is assumed to be a Gaussian-Markovian random process with zero mean value and correlation function 〈 x ( t ) x ( t ′ ) 〉 = D κ e − κ | t − t ′ | , where D and κ are the strength and bandwidth of the stochastic field, respectively. It is found that the stochastic field could enhance the resonant effect between the quantum dot and the coherent field, and generate new resonant points. At the resonant points, the state population difference between two levels is suppressed and the current is considerably enhanced. The zero-frequency shot noise of the current varies dramatically between sub- and super-Poissonian characteristics by tuning the stochastic field appropriately.

Maximum-Likelihood Estimation of Parameters of NHPP Software Reliability Models Using Expectation Conditional Maximization Algorithm

Since its introduction in 1977, the expectation maximization (EM) algorithm has been one of the most important and widely used estimation method in estimating parameters of distributions in the presence of incomplete information. In this paper, a variant of the EM algorithm, the expectation conditional maximization (ECM) algorithm, is introduced for the first time and it provides a promising alternative in estimating the parameters of nonhomogeneous poisson (NHPP) software reliability growth models (SRGM). This algorithm circumvents the difficult M-step of the EM algorithm by replacing it by a series of conditional maximization steps. The utility of the ECM approach is demonstrated in the estimation of parameters of several well-known models for both time domain and time interval software failure data. Numerical examples with real-data indicate that the ECM algorithm performs well in estimating parameters of NHPP SRGM with complex mean value functions and can produce a faster rate of convergence.

Stochastic resonance for estimation of a signal’s bandwidth under low SNR

Traditional bandwidth estimation has low accuracy under low signal-to-noise ratio (SNR). To solve this problem, a new method of bandwidth estimation based on stochastic resonance is proposed. This method will process the signal twice by means of stochastic resonance, which improves the SNR while reducing fluctuations of the amplitude spectrum edge, and then calculating the local mean value function of the amplitude spectrum. A bandwidth can be estimated by using the starting and ending points where the amplitude value of the local mean function is greater than a certain threshold. The experimental simulations show that this method can accurately estimate the bandwidth of the phase shift key empty, quadrature amplitude modulation, and orthogonal frequency modulation at low SNRs. For example, the estimation accuracy could reach 90 % when the SNR is ź10 dB. It is easy to conclude that the proposed method surpasses traditional bandwidth estimation method in terms of accuracy.

Monitoring Software Failure Process using Half Logistic Distribution

In this paper, Software reliability is the anticipation of operations which are free of error in the software in a stated environment during the detailed time duration. Statistical Process Control can survey the gauging of software failure and thereby devote significantly to the enhancement of software reliability. Such an assessment assists the software development team to pinpoint and diagnose their actions during software failure process and hence, assure superior software reliability. A control mechanism planted on the cumulative observations of interval domain failure data using mean value function of the Half Logistic Distribution (HLD) based on Non Homogeneous Poisson Process (NHPP) is proposed. The maximum likelihood estimation approach is used to estimate the unknown parameters of the model. A new mechanism is coded to analyze the observations instead of using regular control charts. General Terms Decision Rule, Software testing, Software failure data, Quality Software.

The Radon–Kipriyanov transform of the generalized spherical mean of a function

A formula relating the Radon transform of functions of spherical symmetries to the Radon–Kipriyanov transform Kγ for a naturalmulti-index γ is given. For an arbitrary multi-index γ, formulas for the representation of the Kγ-transform of a radial function as fractional integrals of Erdelyi–Kober integral type and of Riemann–Liouville integral type are proved. The corresponding inversion formulas are obtained. These results are used to study the inversion of the Radon–Kipriyanov transform of the generalized (generated by a generalized shift) spherical mean values of functions that belong to a weighted Lebesgue space and are even with respect to each of the weight variables.

Dynamically Weighted Combination of Fault - based Software Reliability Growth Models

Background/Objective: The Software Reliability Growth Models (SRGMs) are mainly used to plan and execute system testing in the Software Development Life Cycle (SDLC). The objective of this research paper is to propose a dynamically weighted fault based combination model for application during this phase of reliability growth testing of software systems. Methods: A dynamically weighted fault based SRGM, which describes equally well the exponential growth and S-shaped growth of mean value function in a software testing process is proposed. Non-linear regression methodology was deployed for parameter estimation of (SRGMs).The curve fitting tool in MATLABTm is used for this purpose. The coefficient of determination R2 , Sum of Squared Errors (SSE), Root Mean Squared Error (RMSE) is the goodness of fit measures used to assess the quality of fitting of the SRGMs. Findings: It is found that the proposed combination model describes the failure data better than the constituent models used for combining as revealed by the goodness of fit measures. Applications/ Improvements: The new model can be applied to model reliability growth during testing in software projects with varying characteristics. The model additionally provides vital quality metrics, which can be used to manage the current and future software projectsKeywords: Dynamically Weighted Combination Model, Exponential Growth, Goodness of Fit, S-shaped Growth, Software Reliability Growth Model

A MEAN VALUE FUNCTION AND ITS COMPUTATIONAL FORMULA RELATED TO D. H. LEHMER'S PROBLEM

Let p be an odd prime and c be a fixed integer with (c, p) = 1. For each integer a with <TEX>$1{\leq}a{\leq}p-1$</TEX>, it is clear that there exists one and only one b with <TEX>$0{\leq}b{\leq}p-1$</TEX> such that <TEX>$ab{\equiv}c$</TEX> mod p. Let N(c, p) denote the number of all solutions of the congruence equation <TEX>$ab{\equiv}c$</TEX> mod p for <TEX>$1{\leq}a$</TEX>, <TEX>$b{{\leq}}p-1$</TEX> in which a and <TEX>$\bar{b}$</TEX> are of opposite parity, where <TEX>$\bar{b}$</TEX> is defined by the congruence equation <TEX>$b{\bar{b}}{\equiv}1$</TEX> mod p. The main purpose of this paper is using the mean value theorem of Dirichlet L-functions and the properties of Gauss sums to study the computational problem of one kind mean value function related to <TEX>$E(c,p)=N(c,p)-{\frac{1}{2}}{\phi}(p)$</TEX>, and give its an exact computational formula.

Application of the Generalized Mean Value Function for Detection of Defects in Metal Cylindrical Slugs

Application of the Generalized Mean Value Function for Detection of Defects in Metal Cylindrical Slugs

Higher order Pizzetti’s formulas

We introduce integral mean value functions which are averages of integral means over spheres/balls and over their images under the action of a discrete group of complex rotations. In the case of real analytic functions we derive higher order Pizzetti’s formulas. As applications we obtain a maximum principle for polyharmonic functions and a characterization of convergent solutions to higher order heat type equations.

On Round-Robin routing with FCFS and LCFS scheduling

We study the Round-Robin (RR) routing to a system of parallel queues, which serve jobs according to FCFS or preemptive LCFS scheduling disciplines. The cost structure comprises two components: a service fee and a queueing delay related component. With Poisson arrivals, the inter-arrival time to each queue obeys the Erlang distribution. This allows us to study the mean and transient behavior of the queues separately. The service fee is independent of the scheduling and queueing, and we obtain the corresponding mean cost rate and value function in closed forms. With respect to queueing delay, we first derive integral expressions enabling efficient computation of the corresponding size-aware value functions. By decomposition, these yield also the value function for the whole system of m parallel queues fed by RR. Given the value function, one can carry out the first policy iteration step with arbitrary holding cost rates (e.g., delay, slowdown, etc.) yielding efficient size-, cost- and state-aware policies. Moreover, the mean waiting and sojourn times in the corresponding systems get resolved at the same time. The results are demonstrated in the numerical examples, where we compute near optimal job routing policies for sample systems.

A Relative Research of the Software NHPP Reliability Based on Weibull Extension Distribution and Power Law Model

Objectives: In this study, the comparative problem of reliability model by means of the Weibull extension distribution and power law pattern that completed out productivity of the software trustworthiness was proposed Methods/Statistical Analysis: In addition, the model choice founded on the mean rectangular error and measurement of fortitude for the proficient model were offered. Examination of the failure period influenced by the Weibull extension distribution and power law model was working for the recommending reliability model. The Laplace tendency test was offered for insurance about the failure period. Findings: In the deviation for the between of the predicted values with the actual observations, the shaping parameter from Weibull extension model regard as the best model and in terms of the prognostic power of the variance for the between the forecast values, the shaping parameter from Weibull extension model can be the efficacy model. From judgment of reliability, the situation of the reliability for the conditional work time shows that the case of the shaping parameter from Weibull extension model than shaping parameter from Weibull extension model, the shaping parameter from Weibull extension model, and power low model was revealed high reliability. Namely, the property of the reliability was reflected to subtle about the mission time. The resulting decisions were gained. In terms of deviation for between of the predicted values with the actual observations, the higher shaping model of Weibull extension model regard as the best model. The result of mean value functions has the tendency of non-decreasing form. The result of an intensity functions has the tendency non-increasing form. The case of the higher shaping model of Weibull extension model was judged more reliable model in reliability ground. Improvements/Applications: The software testing for the debugging to reduce cost in terms of the reliability from software is essential problem. From a research, the software developers must be considered for the growth model by the prior knowledge of the software to identify failure modes which can be able to help.

Large Displacement Analysis of Rectangular Orthotropic Membranes Under Stochastic Impact Loading

This paper studies the calculation method about the displacement response mean function of rectangular orthotropic membranes with four edges fixed under stochastic impact loading. We set up the nonlinear stochastic governing differential equation, solve it according to the perturbation method and the random vibration theory and obtain the displacement response mean value function of the membrane surface. Furthermore, this paper makes a random simulation test for ZZF membrane material which is commonly applied in the membrane structural engineering and obtains abundant deflection response sample curves about the feature points of the membrane surface. For sample curves statistical analysis at some fixed time, sample means can be obtained, which verify the correctness of the theoretical calculation method. The calculation method provides a theoretical basis for vibration control of building membrane structures to control the occurrence of natural disasters.

Musa-Okumoto와 Power-law형 NHPP 소프트웨어 신뢰모형에 관한 통계적 공정관리 접근방법 비교연구

There are many software reliability models that are based on the times of occurrences of errors in the debugging of software. It is shown that it is possible to do likelihood inference for software reliability models based on finite failure model and non-homogeneous Poisson Processes (NHPP). For someone making a decision about when to market software, the conditional failure rate is an important variables. The infinite failure model are used in a wide variety of practical situations. Their use in characterization problems, detection of outlier, linear estimation, study of system reliability, life-testing, survival analysis, data compression and many other fields can be seen from the many study. Statistical process control (SPC) can monitor the forecasting of software failure and thereby contribute significantly to the improvement of software reliability. Control charts are widely used for software process control in the software industry. In this paper, proposed a control mechanism based on NHPP using mean value function of Musa-Okumo and Power law type property.

Random Vibration Model in Linear and Non Linear Structure, Application in Engineering Structure

&#x0D; &#x0D; &#x0D; Response of linear or complex nonlinear structures takes form in a characteristic functions and in the deterministic or stochastic external loads. Non linear model with non linear structure stiffness is a type of Duffing equation. Stochastic external loads system is referred to a random signal white noise with a constant power spectral density (So), while non linear system identification of deterministic system's is based on time history, phase plane and Poincare map. Methods of Galerkin and Runge-Kutta are used to solve the partial non linear governing diferential equations. Mean value , Standard deviation and Probability Density Function (PDF) is stated as statistical responses due to stochastic response of random variables. The analysis of random vibration in the solution of non linear stochastic differential equation is solved &#x0D; &#x0D; &#x0D;

Analysis of Fisher Information and the Cramér–Rao Bound for Nonlinear Parameter Estimation After Random Compression

In this paper, we analyze the impact of compression with complex random matrices on Fisher information and the Cramer–Rao Bound (CRB) for estimating unknown parameters in the mean value function of a complex multivariate normal distribution. We consider the class of random compression matrices whose distribution is right-unitarily invariant. The compression matrix whose elements are i.i.d. standard complex normal random variables is one such matrix. We show that for all such compression matrices, the Fisher information matrix has a complex matrix beta distribution. We also derive the distribution of CRB. These distributions can be used to quantify the loss in CRB as a function of the Fisher information of the noncompressed data. In our numerical examples, we consider a direction of arrival estimation problem and discuss the use of these distributions as guidelines for choosing compression ratios based on the resulting loss in CRB.

Steepest Descent Method with Random Step Lengths

The paper studies the steepest descent method applied to the minimization of a twice continuously differentiable function. Under certain conditions, the random choice of the step length parameter, independent of the actual iteration, generates a process that is almost surely R-convergent for quadratic functions. The convergence properties of this random procedure are characterized based on the mean value function related to the distribution of the step length parameter. The distribution of the random step length, which guarantees the maximum asymptotic convergence rate independent of the detailed properties of the Hessian matrix of the minimized function, is found, and its uniqueness is proved. The asymptotic convergence rate of this optimally created random procedure is equal to the convergence rate of the Chebyshev polynomials method. Under practical conditions, the efficiency of the suggested random steepest descent method is degraded by numeric noise, particularly for ill-conditioned problems; furthermore, the asymptotic convergence rate is not achieved due to the finiteness of the realized calculations. The suggested random procedure is also applied to the minimization of a general non-quadratic function. An algorithm needed to estimate relevant bounds for the Hessian matrix spectrum is created. In certain cases, the random procedure may surpass the conjugate gradient method. Interesting results are achieved when minimizing functions having a large number of local minima. Preliminary results of numerical experiments show that some modifications of the presented basic method may significantly improve its properties.

다항 위험함수에 근거한 NHPP 소프트웨어 신뢰모형에 관한 통계적 공정관리 접근방법 비교연구

There are many software reliability models that are based on the times of occurrences of errors in the debugging of software. It is shown that it is possible to do parameter inference for software reliability models based on finite failure model and non-homogeneous Poisson Processes (NHPP). For someone making a decision to market software, the conditional failure rate is an important variables. In this case, finite failure model are used in a wide variety of practical situations. Their use in characterization problems, detection of outlier, linear estimation, study of system reliability, life-testing, survival analysis, data compression and many other fields can be seen from the many study. Statistical process control (SPC) can monitor the forecasting of software failure and thereby contribute significantly to the improvement of software reliability. Control charts are widely used for software process control in the software industry. In this paper, proposed a control mechanism based on NHPP using mean value function of polynomial hazard function.