Solutions of Hereditary Stochastic Equations: A Comparison

The trajectories of two random processes defined by the solutions of stochastic equations with delay are compared.     

Group shops scheduling with makespan criterion subject to random release dates and processing times

This paper deals with a stochastic group shop scheduling problem. The group shop scheduling problem is a general formulation that includes the other shop scheduling problems such as the flow shop, the job shop and the open shop scheduling problems. Both the release date of each job and the processing time of each job on each machine are random variables with known distributions. The objective is to find a job schedule which minimizes the expected makespan. First, the problem is formulated in a form of stochastic programming and then a lower bound on the expected makespan is proposed which may be used as a measure for evaluating the performance of a solution without simulating. To solve the stochastic problem efficiently, a simulation optimization approach is developed that is a hybrid of an ant colony optimization algorithm and a heuristic algorithm to generate good solutions and a discrete event simulation model to evaluate the expected makespan. The proposed approach is tested on instances where the random variables are normally, exponentially or uniformly distributed and gives promising results.     

An M/G/1 retrial G-queue with non-exhaustive random vacations and an unreliable server

This paper deals with an M/G/1 retrial queue with negative customers and non-exhaustive random vacations subject to the server breakdowns and repairs. Arrivals of both positive customers and negative customers are two independent Poisson processes. A breakdown at the busy server is represented by the arrival of a negative customer which causes the customer being in service to be lost. The server takes a vacation of random length after an exponential time when the server is up. We develop a new method to discuss the stable condition by finding absorb distribution and using the stable condition of a classical M/G/1 queue. By applying the supplementary variable method, we obtain the steady-state solutions for both queueing measures and reliability quantities. Moreover, we investigate the stochastic decomposition law. We also analyse the busy period of the system. Some special cases of interest are discussed and some known results have been derived. Finally, an application to cellular mobile networks is provided and the effects of various parameters on the system performance are analysed numerically.     

Frequency stability criteria for non-linear stochastic systems

In this paper general classes of stochastic systems described by stochastic integrals of the Volterra type are considered. Existence and uniqueness of random solutions as well as stochastic stability criteria of the Popov frequency form are established. These frequency stability criteria relate not only to the linear system dynamics (or the kernel of the Volterra integral) of the system, but also to the characteristics of the additive and multiplicative random processes that govern the system.     

Non-stationary stochastic embedding for transfer function estimation

This paper presents a consistent framework for the quantification of noise and undermodelling errors in transfer function model estimation. We use the, so-called, “stochastic embedding” approach, in which both noise and undermodelling errors are treated as stochastic processes. In contrast to previous applications of stochastic embedding, in this paper we represent the undermodelling as a multiplicative error characterised by random walk processes in the frequency domain. The benefit of the present formulation is that it significantly simplifies the estimation of the parameters of the embedded process yielding a closed-form expression for the model error quantification. Simulation and experimental examples illustrate how the random walks effectively capture typical cases of undermodelling found in practice, including underdamped modes. The examples also show how to use the method as a tool in the determination of model order and pole location in fixed denominator model structures.     

Optimal linear estimators for systems with random measurement delays

This paper is concerned with the optimal linear estimation problem for linear discrete-time stochastic systems with random measurement delays. A new model that describes the random delays is constructed where possible the largest delay is bounded. Based on this new model, the optimal linear estimators including filter, predictor and smoother are developed via an innovation analysis approach. The estimators are recursively computed in terms of the solutions of a Riccati difference equation and a Lyapunov difference equation. The steady-state estimators are also investigated. A sufficient condition for the convergence of the optimal linear estimators is given. A simulation example shows the effectiveness of the proposed algorithms.     

A comparative study of the numerical approximation of the random Airy differential equation

The aim of this paper is twofold. First, we deal with the extension to the random framework of the piecewise Fröbenius method to solve Airy differential equations. This extension is based on mean square stochastic calculus. Second, we want to explore the capability to provide not only reliable approximations for both the average and the standard deviation functions associated to the solution stochastic process, but also to save computational time as it happens in dealing with the analogous problem in the deterministic scenario. This includes a comparison of the numerical results with respect to those obtained by other commonly used operational methods such as polynomial chaos and Monte Carlo simulations. To conduct this comparative study, we have chosen the Airy random differential equation because it has highly oscillatory solutions. This feature allows us to emphasize differences between all the considered approaches.     

Representation of Nonlinear Random Transformations by Non-Gaussian Stochastic Neural Networks

The learning capability of neural networks is equivalent to modeling physical events that occur in the real environment. Several early works have demonstrated that neural networks belonging to some classes are universal approximators of input-output deterministic functions. Recent works extend the ability of neural networks in approximating random functions using a class of networks named stochastic neural networks (SNN). In the language of system theory, the approximation of both deterministic and stochastic functions falls within the identification of nonlinear no-memory systems. However, all the results presented so far are restricted to the case of Gaussian stochastic processes (SPs) only, or to linear transformations that guarantee this property. This paper aims at investigating the ability of stochastic neural networks to approximate nonlinear input-output random transformations, thus widening the range of applicability of these networks to nonlinear systems with memory. In particular, this study shows that networks belonging to a class named non-Gaussian stochastic approximate identity neural networks (SAINNs) are capable of approximating the solutions of large classes of nonlinear random ordinary differential transformations. The effectiveness of this approach is demonstrated and discussed by some application examples.     

基于随机模拟与PSO算法相结合的随机机会约束规划算法

随机机会约束规划作为一类重要的随机规划,广泛存在于许多领域中.为了寻找更有效的求解随机机会约束规划的算法,通过采用随机模拟来逼近随机函数,并在微粒群算法PSO(Particle Swarm Optimization)中利用随机模拟实现估计适应值和检验解的可行性,从而给出了求解随机机会约束规划的新算法,最后,测试其性能并与遗传算法进行了比较,实例结果表明该算法的正确性和有效性.     

Optimal and suboptimal solutions to stochastically uncertain problems of quintile optimization

In this paper, problems of stochastic optimization under incomplete information on distribution of random perturbations with the quintile and probability criteria are considered. The minimax approach is used when optimal solutions are chosen. Conditions for equivalency of direct and inverse problems of stochastic optimization under incomplete statistical information are studied. The solution method for statistically uncertain problems of optimization with the quintile criterion basing on the use of generalized confidence sets for statistically uncertain random quantities is proposed. The use of confidence sets for finding suboptimal solutions to the problem of stochastic optimization under incomplete information is considered. Examples of the application of obtained relations are represented.     

Optimal run lengths in deteriorating production processes in random fuzzy environments

This paper presents a random fuzzy economic manufacturing quantity (EMQ) model in a deteriorating process. It is assumed that the setup cost and the average holding cost are characterized as fuzzy variables and the elapsed time until shift is a random fuzzy variable. As a function of these parameters, the average total cost is also a random fuzzy variable, and the unimodality of its expected value is studied. To obtain the optimal run length and the minimum average cost, simultaneous perturbation stochastic approximation (SPSA) algorithm based on random fuzzy simulation is provided. Random fuzzy EMQ models with fuzzy deterioration, fuzzy linear deterioration and fuzzy exponential deterioration are presented, respectively. These models can be solved by the proposed algorithm. Numerical examples are presented in the end.     

Random fuzzy mean-absolute deviation models for portfolio optimization problem with hybrid uncertainty

Absolute deviation is a commonly used risk measure, which has attracted more attentions in portfolio optimization. The existing mean-absolute deviation models are devoted to either stochastic portfolio optimization or fuzzy one. However, practical investment decision problems often involve the mixture of randomness and fuzziness such as stochastic returns with fuzzy information. Thus it is necessary to model portfolio selection problem in such a hybrid uncertain environment. In this paper, we employ random fuzzy variable to describe the stochastic return on individual security with ambiguous information. We first define the absolute deviation of random fuzzy variable and then employ it as risk measure to formulate mean-absolute deviation portfolio optimization models. To find the optimal portfolio, we design random fuzzy simulation and simulation-based genetic algorithm to solve the proposed models. Finally, a numerical example for synthetic data is presented to illustrate the validity of the method.     

Existence theory for a stochastic differential equation

A non-linear function stochastic differential equation was studied where t𝛆R₊={t;t ? 0},ω𝛆 Ω, Ω being the underlying sot of a complete probability measure space ( Ω,A,P) The random process x(t;ω) is the unknown stochastic function defined on R+ × Ω h(t, x;ω ) ) is the stochastic term defined for t𝛆 R+ and x ( t;ω)εG(a Branch Space); and n(t, x ω) is a random variable defined for tε R+ω 𝛆 Ω, and x 𝛆 F{Grcub; (a Frcchet space). The purpose of this paper is to develop sufficient conditions for the existence of random solutions, second order stochastic processes, for the above equation and to place bounds upon these random solutions. Several examples are also presented which illustrate the usefulness of the theoretical findings.     

Random fuzzy delayed renewal processes

In renewal processes, fuzziness and randomness often coexist intrinsically. Based on the random fuzzy theory, a delayed renewal process with random fuzzy interarrival times is proposed in this paper. Relations between the renewal number and interarrival times in such a process are investigated. Useful theorems such as the elementary renewal theorem, the Blackwell renewal theorem and the Smith key renewal theorem in a conventional delayed renewal process are extended to their counterparts for random fuzzy delayed renewal processes.     

Qualitative properties of stochastic iterative processes under random structural perturbations

In this work, under different modes of stochastic convergence, several convergence and stability results for stochastic iterative processes are developed. Difference inequalities and a comparison method in the context of Lyapunov-like functions are utilized. The presented method does not demand the knowledge of the probability distributions of solution processes. By decomposing random perturbations in nonlinear iterative processes into internal and external random perturbations, effects of these stochastic disturbances on the convergence and the stability of the the iterative processes are investigated. In fact, it is shown that the convergence and stability analysis is robust under random structural perturbations. The presented conditions are easy to verify, algebraically simple, and computationally attractive. The results provide new tests for distributed iterative processes in decentralized external regulation, adaptation, parameter estimation and the numerical analysis schemes.     

Finite fractal dimension of random attractor for stochastic non-autonomous strongly damped wave equation

In this paper, we first prove the existence of a random attractor for stochastic non-autonomous strongly damped wave equations with additive white noise. Then we apply a criteria to obtain an upper bound of fractal dimension of the random attractor of considered system.     

Building Blocks for Computer Vision with Stochastic Partial Differential Equations

We discuss the basic concepts of computer vision with stochastic partial differential equations (SPDEs). In typical approaches based on partial differential equations (PDEs), the end result in the best case is usually one value per pixel, the “expected” value. Error estimates or even full probability density functions PDFs are usually not available. This paper provides a framework allowing one to derive such PDFs, rendering computer vision approaches into measurements fulfilling scientific standards due to full error propagation. We identify the image data with random fields in order to model images and image sequences which carry uncertainty in their gray values, e.g. due to noise in the acquisition process. The noisy behaviors of gray values is modeled as stochastic processes which are approximated with the method of generalized polynomial chaos (Wiener-Askey-Chaos). The Wiener-Askey polynomial chaos is combined with a standard spatial approximation based upon piecewise multi-linear finite elements. We present the basic building blocks needed for computer vision and image processing in this stochastic setting, i.e. we discuss the computation of stochastic moments, projections, gradient magnitudes, edge indicators, structure tensors, etc. Finally we show applications of our framework to derive stochastic analogs of well known PDEs for de-noising and optical flow extraction. These models are discretized with the stochastic Galerkin method. Our selection of SPDE models allows us to draw connections to the classical deterministic models as well as to stochastic image processing not based on PDEs. Several examples guide the reader through the presentation and show the usefulness of the framework.     

Mean square stability of the solutions of autonomous dynamic diffusion systems with finite aftereffect with regard for random factors

The necessary and sufficient conditions are obtained for the asymptotic mean square stability of strong solutions of autonomous diffusion stochastic functional-differential equations with finite aftereffect and random factors (random functions with different distribution) taken into account.     

Stochastic models for performance analysis of multistate flexible manufacturing cells

Performance analysis of flexible manufacturing cells (FMCs) can help companies find the pros and cons of production processes. However, the emphasis has been on issues like cell formation, layout design and scheduling optimization. Little seems to have been done to assess the reliability of an FMC. In this paper, we develop the stochastic models for the performance analysis mainly on the reliability of two different FMCs configured from a set of teaching intelligent flexible manufacturing system (TIFMS). The closed form solutions of probabilities of system states are obtained. Then, utilization rate of equipment in the cell and productivities of the two FMCs as the performance indexes are calculated and optimized. Compared to simulation methods, the closed form solutions make calculations of the performance indexes faster and more accurate. When random variables in the stochastic models are assumed to follow non-exponential distributions, the effects of them on the performance indexes are discussed. The objective of this paper is to fill up the gap that the closed form solutions are difficult to obtain as the number of machine tool increases. Another objective is to optimize the performance indexes to help engineers better evaluate the performance of FMC. Numerical analysis cases are used to illustrate the proposed stochastic models.     

Non-constant discounting and differential games with random time horizon

Previous results on non-constant discounting in continuous time are extended to the field of deterministic differential games with a stochastic terminal time. A dynamic programming equation is derived for problems with general time inconsistent preferences and random duration. Different cooperative and non-cooperative solution concepts for differential games with random duration are analyzed. The results are illustrated by solving the cake-eating problem describing the classical model of management of a nonrenewable resource.