Semi-Markov failure rates processes

Usually, a reliability function is defined by a failure rate which is a real function taking the non-negative real values. In this paper the failure rate is assumed to be a stochastic process with non-negative and right continuous trajectories. The reliability function is defined as an expectation of a function of that random process. Particularly, the failure rate defined by the semi-Markov processes is considered here. The theorems dealing with the renewal equations for the conditional reliability functions with a semi-Markov process as a failure rate are presented in this paper. A system of that kind of equations for the discrete state space semi-Markov process is applied for calculating the reliability function for the 3-states semi-Markov random walk. Using the introduced system of renewal equations for the countable state space, the reliability function for the Furry-Yule failure rate process is obtained.     

Radicals of 0-regular algebras

Summary We consider a generalisation of the Kurosh--Amitsur radical theory for rings (and more generally multi-operator groups) which applies to 0-regular varieties in which all operations preserve 0. We obtain results for subvarieties, quasivarieties and element-wise equationally defined classes. A number of examples of radical and semisimple classes in particular varieties are given, including hoops, loops and similar structures. In the first section, we introduce 0-normal varieties (0-regular varieties in which all operations preserve 0), and show that a key isomorphism theorem holds in a 0-normal variety if it is subtractive, a property more general than congruence permutability. We then define our notion of a radical class in the second section. A number of basic results and characterisations of radical and semisimple classes are then obtained, largely based on the more general categorical framework of L. M\'arki, R. Mlitz and R. Wiegandt as in [13]. We consider the subtractive case separately. In the third section, we obtain results concerning subvarieties and quasivarieties based on the results of the previous section, and also generalise to subtractive varieties some results for multi-operator group radicals defined by simple equational rules. Several examples of radical and semisimple classes are given for a range of fairly natural 0-normal varieties of algebras, most of which are subtractive.     

Convergence analysis of a global optimization algorithm using stochastic differential equations

We establish the convergence of a stochastic global optimization algorithm for general non-convex, smooth functions. The algorithm follows the trajectory of an appropriately defined stochastic differential equation (SDE). In order to achieve feasibility of the trajectory we introduce information from the Lagrange multipliers into the SDE. The analysis is performed in two steps. We first give a characterization of a probability measure (Π) that is defined on the set of global minima of the problem. We then study the transition density associated with the augmented diffusion process and show that its weak limit is given by Π.     

Component commonality via hierarchical orthogonal arrays and detailing economic decision matrices

Component commonality (CC) implies products using many common parts, desensitized to the range of product applications (noise), and meeting the functionality objectives of the product line. This paper lists a nine-step methodology for developing CC and applies it to a problem. These steps utilize the major concepts of analytical modeling, economic decision matrices (EDM), quality loss functions (QLS) for variates and weighted utilities, stochastic models, finite element (FE) simulations for concurrent engineering, and statistical design of experiments (DOE) for uncertainty in either application, statistics or managerial decisions. The details of the first six steps were illustrated in a previous paper by application to a problem involving a slider link subjected to an extreme range of “noise” (various inertia/pressure loadings). Six candidate designs of steel, aluminum and titanium were generated using an analytical model and a sensitivity study. The DOE utilized Taguchi's orthogonal arrays. These designs were ranked using cost, weight, and factors of safety with respect to yielding. A refining EDM with a three-part robustness criteria selected two candidates (best was steel, followed by aluminum) considering inner noise in the managerial decisions. In the current paper, the last three steps of the nine-step methodology are applied to these two candidates in order to obtain the “optimal” part for CC. The FE stress results are used with a modified Goodman fatigue criteria, and a stochastic model is developed based upon beta (strength) and three-parameter Weibull (stress) distributions. The model is then used in a detailing EDM to determine the stochastic reliability associated with a QLS defined with respect to fatigue reliability. A “fine-tuned” aluminum candidate is shown to meet a priori reliability requirements and have low-quality losses. However, both original candidates exhibited some high-quality losses, even though such losses were acceptable in the preceding refining EDM. The authors demonstrate that this loss of quality can be prevented if a fatigue criteria is used in both the refining and detailing EDM stages of the design process and, “warranty failures” are based on stochastic rather than deterministic definitions of maximum environmental conditions.     

A method for maximizing the reliability coefficient of a communications network

The most common idea of network reliability in the literature is a numerical parameter calledoverall network reliability, which is the probability that the network will be in a successful state in which all nodes can mutually communicate. Most papers concentrate on the problem of calculating the overall network reliability which is known to be an NP hard problem. In the present paper, the question asked is how to find a method for determining a reliable subnetwork of a given network. Givenn terminals and one central computer, the problem is to construct a network that links each terminal to the central computer, subject to the following conditions: (1) each link must be economically feasible; (2) the minimum number of links should be used; and (3) the reliability coefficient should be maximized. We argue that the network satisfying condition (2) is a spanning arborescence of the network defined by condition (1). We define the idea of thereliability coefficient of a spanning arborescence of a network, which is the probability that a node at average distance from the root of the arborescence can communicate with the root. We show how this coefficient can be calculated exactly when there are no degree constraints on nodes of the spanning arborescence, or approximately when such degree constraints are present. Computational experience for networks consisting of up to 900 terminals is given.This report was prepared as part of the activities of the Management Science Research Group, Carnegie-Mellon University, under Contract No. N00014-82-K-0329 NR 047–048 with the U.S. Office of Naval Research. Reproduction in whole or in part is permitted for any purpose of the U.S. Government.     

几类考虑有限理性平衡问题解的稳定性

对Ky Fan点问题定义了理性函数,证明了大多数的KyFan点问题(在Baire分类意义上)都是结构稳定的,对$\varepsilon$-平衡也都是鲁棒的.作为应用,还给出了Nash平衡问题和变分不等式问题的稳定性结论.     

Enhancing software reliability modeling and prediction through the introduction of time-variable fault reduction factor

Over the past three decades, many software reliability models with different parameters, reflecting various testing characteristics, have been proposed for estimating the reliability growth of software products. We have noticed that one of the most important parameters controlling software reliability growth is the fault reduction factor (FRF) proposed by Musa. FRF is generally defined as the ratio of net fault reduction to failures experienced. During the software testing process, FRF could be influenced by many environmental factors, such as imperfect debugging, debugging time lag, etc. Thus, in this paper, we first analyze some real data to observe the trends of FRF, and consider FRF to be a time-variable function. We further study how to integrate time-variable FRF into software reliability growth modeling. Some experimental results show that the proposed models can improve the accuracy of software reliability estimation. Finally, sensitivity analyses of various optimal release times based on cost and reliability requirements are discussed. The analytic results indicate that adjusting the value of FRF may affect the release time as well as the development cost.     

Convergence of a non-failable mean-field particle system

The existing literature contains many examples of mean-field particle systems converging to the distribution of a Markov process conditioned to not hit a given set. In many situations, these mean-field particle systems are failable, meaning that they are not well defined after a given random time. Our first aim is to introduce an original mean-field particle system, which is always well defined and whose large number particle limit is, in all generality, the distribution of a process conditioned to not hit a given set. Under natural conditions on the underlying process, we also prove that the convergence holds uniformly in time as the number of particles goes to infinity. As an illustration, we show that our assumptions are satisfied in the case of a piece-wise deterministic Markov process.     

On Some Special Pseudoconvex Spaces

We present two ways of constructing pseudoconvex spaces. The first theorem states that an interval space is pseudoconvex if the intervals are range sets of continuous functions and a cone erected to a convex set is convex. Secondly, if H is pseudoconvex and in the Cartesian product H × Rthe intervals are defined by going first along the H-segment, then along the R-segment (the other coordinates fixed) then H × Ris pseudoconvex.     

Mathematical Analysis of a Soccer Game. Part I: Individual and Collective Behaviors

Detailed time series analysis of a soccer match is given based on the detailed data of the 2D motions of all 22 players and of the ball for the match. The whole analysis includes two parts. In Part I, the individual and collective behaviors of the players of the two teams as well as the motion of the ball are presented as various time series. Geometrical centers, radii, expansion speeds, possession functions of the two teams are defined and calculated as functions of time. Major ranges of all players as well as of different groups of players (defenders, midfielders, forwards) of the two teams during the entire first half, the attacking phase of team A and the attacking phase of team B are calculated, respectively, showing the structures of the two teams during different phases. Distance coverage of each player and the mean distances covered by different groups of players (defenders, midfielders, forwards) during different phases are calculated. The time portions of possession of the ball by each team and the time portions of different phases are also calculated. In Part II, energy and spectral analysis and various correlations will be derived. The relation between various parameters and potential indicators will be discussed. The major purpose of the present study is to offer some general mathematical tools for the detailed analysis and to reveal some general features of soccer match when the detailed 2D data are available. The results would offer the raw materials for various potential indicators which may eventually be used in the coaching process to enhance the performance and in the prediction of the results of soccer matches.     

A deteriorating cold standby repairable system with priority in use

In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is studied. In this system, it is assumed that the working time distributions and the repair time distributions of the two components are both exponential and component 1 is given priority in use. After repair, component 2 is “as good as new” while component 1 follows a geometric process repair. Under these assumptions, using the geometric process and a supplementary variable technique, some important reliability indices such as the system availability, reliability, mean time to first failure (MTTFF), rate of occurrence of failure (ROCOF) and the idle probability of the repairman are derived. A numerical example for the system reliability R(t) is given. And it is considered that a repair-replacement policy based on the working age T of component 1 under which the system is replaced when the working age of component 1 reaches T. Our problem is to determine an optimal policy T^∗ such that the long-run average cost per unit time of the system is minimized. The explicit expression for the long-run average cost per unit time of the system is evaluated, and the corresponding optimal replacement policy T^∗ can be found analytically or numerically. Another numerical example for replacement model is also given.     

多源点多汇点网络系统可靠度的解析表示

本文研究了基于最小路径描述的多源点多汇点网络系统可靠性问题。定义了最小路径矩阵的几种运算,利用所定义的运算,将多源点多汇点网络系统转化为等价的单源点单汇点网络系统,并给出了由子系统可靠度精确表示网络系统可靠度的解析表达式。这种解析表达是非常重要的,它是系统可靠性的理论研究与实际应用的一个极为有效的工具。     

Estimating Stochastic Dynamical Systems Driven by a Continuous-Time Jump Markov Process

We discuss the use of a continuous-time jump Markov process as the driving process in stochastic differential systems. Results are given on the estimation of the infinitesimal generator of the jump Markov process, when considering sample paths on random time intervals. These results are then applied within the framework of stochastic dynamical systems modeling and estimation. Numerical examples are given to illustrate both consistency and asymptotic normality of the estimator of the infinitesimal generator of the driving process. We apply these results to fatigue crack growth modeling as an example of a complex dynamical system, with applications to reliability analysis.      

Duane-LR模型下复杂系统的动态可靠性增长评定

基于复杂系统可靠性增长试验的特点，运用Duane可靠性增长模型结合数理统计中的线性回归方法对新批次产品的可靠性参数进行预测。结合产品的少量现场试验数据，利用Bayes方法对系统的可靠性增长试验结果进行评定。文中首先给出了可靠性增长分析的模型，然后运用历次阶段试验中的可靠性增长数据建立动态参数的递推估计模型，在此基础上，运用随机变量函数的分布，给出各阶段可靠性增长试验中可靠性参数的Bayes估计。文中对Weibull、指数和二项分布三种试验结果进行分析，给出计算公式。     

Predictive control of nonlinear dynamic processes

Predictive control can be applied if the reference value of the process is known in advance and the deterministic disturbances can be predicted. A cost function defined in the future horizon is minimized. The control signal is calculated for a control horizon, but only the first one is applied and the procedure is repeated (receding horizon strategy). Processes with mild analytical nonlinear characteristics are considered. The possible process models are either nonparametric (linear, Hammerstein, and Volterra weighting function series) or parametric ones (generalized Hammerstein, parametric Volterra, and bilinear models). The algorithms of the optimal and suboptimal predictive control based on the nonparametric and the parametric models mentioned are derived. Several simulations present how effective these methods are. The adaptive case is dealt with as well.     

考虑n中连续取k失效准则的树状系统可靠性求解方法

树状网络系统在管道运输,网络通信中较为常见,对其进行可靠性评估对系统设计及优化具有重要意义。针对树状冗余系统,在n中连续取k失效准则下,通过有限马尔可夫嵌入法并对其进行变形,研究了树状系统可靠性求解方法。本文对树状系统建模加以定义,提出了基于层数参数,层-节点向量,父-子节点矩阵三元参数的树状系统表示方法,研究了变形有限马尔可夫嵌入法的树状系统n中连续取k失效准则下的可靠性求解方法,给出了三个数值算例应用并分析了算法的运算复杂度。最后,本文对比讨论了基于概率母函数法的树状系统在n中连续取k准则下系统可靠性求解方法的研究,得出结论本文算法针对树状冗余系统n中连续取k失效准则下系统可靠性求解应用范围更广,求解效率较高。     

Brillinger type mixing conditions for a simple branching diffusion process

A simple branching diffusion process is described. Formulae for intensity functions and factorial cumulant density functions at several times are given. Mixing conditions in terms of integrals of these cumulants are defined and proven for this stochastic evolutionary point process. The mixing conditions then allow a spatial central limit theorem and a strong law of large numbers to be easily obtained.     

网络概率模型的建立与特征量概率公式的推证

基于网络系统状态的设置,建立起该系统网络可靠性概率模型.按再生过程给出可靠性特征量tw的初步估计式,通过定理进一步推证出tw的概率估计与极限分布公式.并在两种条件下论证故障概率θ(1θ和θ2)的估计公式,又以两种极限监理情况为例分别对θ进行双侧估计,给出该估计的上下限.同时在两种条件下分别提出可靠性特征量twk(tw′k,t″wk)概率估计的极限定理.最后评价了网络系统概率估算的主要结果.     

Solving a novel multiobjective placement problem of recloser and distributed generation sources in simultaneous mode by improved harmony search algorithm

This article deals with optimal placement of Distributed Generation (DG) sources and recloser in simultaneous mode and develops an improved harmony search (iHS) algorithm to solve it. For this, two important control parameters have been adjusted to reach better solution from simple HS algorithm to obtain better solution from simple HS algorithm. The proposed multiobjective function consists of two parts; first is improving reliability indices and second is minimizing power loss. The reliability indices have been selected based on satisfactory requirements of costumer and electric company as well as response to transient and permanent faults. Then, four reliability indices has been used in objective function; that is, system average interruption duration index (SAIDI), cost of energy not supplied, momentary average interruption frequency index, and system average interruption frequency index (SAIFI). Simulation has been performed on a practical distribution network in North West of Iran. Three scenarios have been introduced; that is, scenario (i) First, placement of DGs, and then recloser, scenario (ii) First, placement of recloser, and then DG, and scenario (iii) simultaneous placement of DG and recloser. Also, three cases are defined based on the number of used DG and recloser. Results of the proposed algorithm have been compared with related values of particle swarm optimization and simple HS algorithms. The core contribution of the presented study is introducing several novel indices to analyze and discuss the obtained results from simulation. © 2014 Wiley Periodicals, Inc. Complexity 21: 328–339, 2015     

On the Default Probability in a Regime-Switching Regulated Market

This paper considers asset dynamics in a regulated (controlled) market, where the macroeconomic environment is taken into account. A regime-switching reflected stochastic process with two-sided barriers is proposed for modeling asset price dynamics. We study a default problem with the default time being defined as the first passage time of the price dynamics. By solving a pair of interacting ordinary differential equations (ODEs), we obtain an explicit formula for the Laplace transform (LT) of the default time. Some numerical results are given for illustration.