The two-moment three-parameter decomposition approximation of queueing networks with exponential residual renewal processes

We propose a two-moment three-parameter decomposition approximation of general open queueing networks by which both autocorrelation and cross correlation are accounted for. Each arrival process is approximated as an exponential residual (ER) renewal process that is characterized by three parameters: intensity, residue, and decrement. While the ER renewal process is adopted for modeling autocorrelated processes, the innovations method is used for modeling the cross correlation between randomly split streams. As the interarrival times of an ER renewal process follow a two-stage mixed generalized Erlang distribution, viz., MGE(2), each station is analyzed as an MGE(2)/G/1 system for the approximate mean waiting time. Variability functions are also used in network equations for a more accurate modeling of the propagation of cross correlations in queueing networks. Since an ER renewal process is a special case of a Markovian arrival process (MAP), the value of the variability function is determined by a MAP/MAP/1 approximation of the departure process. Numerical results show that our proposed approach greatly improves the performance of the parametric decomposition approximation of open queueing networks.     

Modelling and analysis of Markovian continuous flow systems with a finite buffer

In this study, a Markovian fluid flow system with two stages separated by a finite buffer is considered. Fluid flow models have been analyzed extensively to evaluate the performance of production, computer, and telecommunication systems. Recently, we developed a methodology to analyze general Markovian continuous flow systems with a finite buffer. The flexibility of this methodology allows us to analyze a wide range of systems by specifying the transition rates and the flow rates associated with each state of each stage. In this study, in order to demonstrate the applicability of our methodology, we model and analyze a range of models studied in the literature. The examples we analyze as special cases of our general model include systems with phase-type failure and repair-time distributions, systems with machines that have multiple up and down states, and systems with multiple unreliable machines in series or parallel in each stage. For each case, the Markovian model is developed, the transition and flow rates are determined, and representative numerical results are obtained by using our methodology.     

Moment decay rates of infinite dimensional stochastic evolution equations with memory and Markovian jumps

The strong solutions approximation approach for mild solutions of stochastic functional differential equations with Markovian switching driven by Lévy martingales in Hilbert spaces is considered. Asymptotic behaviors with a general decay rate for the second moments of mild solutions of the above equations are obtained. An example is given to illustrate our theory.     

Evaluation of inventory policies with unidirectional substitutions

We propose evaluation approaches to multi-item base-stock inventory policies where unidirectional substitutions are allowed. The problems in the paper are in the context of spare parts management and we identify two substitution cases: substitution upon demand arrivals and substitution upon order deliveries. This leads us to three unidirectional substitution policies, for each of which we develop Markovian models. As the number of part types increases, computational effort required to solve the Markovian models increases rapidly. To reduce computation burden, an approximation approach based on the decomposition of multi-dimensional state transition is used for systems with two or more spare part types. Numerical studies show unidirectional substitution improves various system performance measures such as the average inventory level, the average backlogged demand, and the fill rate. The proposed decomposition approach reduces the computation required to compute the performance measures and the approximation errors seems to be quite small.     

Kinetic-fluid derivation and mathematical analysis of a nonlocal cross-diffusion–fluid system

In this article we propose a nonlocal cross-diffusion–fluid system describing the dynamics of multiple interacting populations living in a Newtonian fluid. First, we derive our nonlocal cross-diffusion–fluid system from a nonlocal kinetic-fluid model by the micro-macro decomposition method. Second, we prove the existence of weak solutions for the proposed system by applying the nonlinear Galerkin method with a priori estimates and compactness arguments. On the basis of micro-macro decomposition, we propose and develop an asymptotic-preserving numerical scheme. Finally, we discuss the computational results for the proposed system.     

Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach

In this paper, a numerical method is presented to obtain and analyze the behavior of numerical solutions of distributed order fractional differential equations of the general form in the time domain with the Caputo fractional derivative. The suggested method is based on the Müntz–Legendre wavelet approximation. We derive a new operational vector for the Riemann–Liouville fractional integral of the Müntz–Legendre wavelets by using the Laplace transform method. Applying this operational vector and collocation method in our approach, the problem can be reduced to a system of linear and nonlinear algebraic equations. The arising system can be solved by the Newton method. Discussion on the error bound and convergence analysis for the proposed method is presented. Finally, seven test problems are considered to compare our results with other well‐known methods used for solving these problems. The results in the tabulated tables highlighted that the proposed method is an efficient mathematical tool for analyzing distributed order fractional differential equations of the general form.     

A Numerical Study of the Exact Evolution Equations for Surface Waves in Water of Finite Depth

We describe a pseudo-spectral numerical method to solve the systems of one-dimensional evolution equations for free surface waves in a homogeneous layer of an ideal fluid. We use the method to solve a system of one-dimensional integro-differential equations, first proposed by Ovsjannikov and later derived by Dyachenko, Zakharov, and Kuznetsov, to simulate the exact evolution of nonlinear free surface waves governed by the two-dimensional Euler equations. These equations are written in the transformed plane where the free surface is mapped onto a flat surface and do not require the common assumption that the waves have small amplitude used in deriving the weakly nonlinear Korteweg–de Vries and Boussinesq long-wave equations. We compare the solution of the exact reduced equations with these weakly nonlinear long-wave models and with the nonlinear long-wave equations of Su and Gardner that do not assume the waves have small amplitude. The Su and Gardner solutions are in remarkably close agreement with the exact Euler solutions for large amplitude solitary wave interactions while the interactions of low-amplitude solitary waves of all four models agree. The simulations demonstrate that our method is an efficient and accurate approach to integrate all of these equations and conserves the mass, momentum, and energy of the Euler equations over very long simulations.     

Stability in distribution of neutral stochastic functional differential equations with Markovian switching

Stability in distribution of stochastic differential equations with Markovian switching and stochastic differential delay equations with Markovian switching have been studied by several authors and this kind of stability is an important property for stochastic systems. There are several papers which study this stability for stochastic differential equations with Markovian switching and stochastic differential delay equations with Markovian switching technically. In our paper, we are concerned with the general neutral stochastic functional differential equations with Markovian switching and we derive the sufficient conditions for stability in distribution. At the end of our paper, one example is established to illustrate the theory of our work.     

Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier

Based on the matrix-analytic approach to fluid flows initiated by Ramaswami, we develop an efficient time dependent analysis for a general Markov modulated fluid flow model with a finite buffer and an arbitrary initial fluid level at time 0. We also apply this to an insurance risk model with a dividend barrier and a general Markovian arrival process of claims with possible dependencies in successive inter-claim intervals and in claim sizes. We demonstrate the implementability and accuracy of our algorithms through a set of numerical examples that could also serve as test cases for comparing other solution approaches.      

The cross-entropy method with patching for rare-event simulation of large Markov chains

There are various importance sampling schemes to estimate rare event probabilities in Markovian systems such as Markovian reliability models and Jackson networks. In this work, we present a general state-dependent importance sampling method which partitions the state space and applies the cross-entropy method to each partition. We investigate two versions of our algorithm and apply them to several examples of reliability and queueing models. In all these examples we compare our method with other importance sampling schemes. The performance of the importance sampling schemes is measured by the relative error of the estimator and by the efficiency of the algorithm. The results from experiments show considerable improvements both in running time of the algorithm and the variance of the estimator.     

A Robin-Robin Domain Decomposition Method for a Stokes-Darcy Structure Interaction with a Locally Modified Mesh

A new numerical method based on locally modified Cartesian meshes is proposed for solving a coupled system of a fluid flow and a porous media flow. The fluid flow is modeled by the Stokes equations while the porous media flow is modeled by Darcy's law. The method is based on a Robin-Robin domain decomposition method with a Cartesian mesh with local modifications near the interface. Some computational examples are presented and discussed.     

A combination of energy method and spectral analysis for study of equations of gas motion

There have been extensive studies on the large time behavior of solutions to systems on gas motions, such as the Navier-Stokes equations and the Boltzmann equation. Recently, an approach is introduced by combining the energy method and the spectral analysis to the study of the optimal rates of convergence to the asymptotic profiles. In this paper, we will first illustrate this method by using some simple model and then we will present some recent results on the Navier-Stokes equations and the Boltzmann equation. Precisely, we prove the stability of the non-trivial steady state for the Navier-Stokes equations with potential forces and also obtain the optimal rate of convergence of solutions toward the steady state. The same issue was also studied for the Boltzmann equation in the presence of the general time-space dependent forces. It is expected that this approach can also be applied to other dissipative systems in fluid dynamics and kinetic models such as the model system of radiating gas and the Vlasov-Poisson-Boltzmann system.      

一阶线性和拟线性双曲型方程的格点模型

一阶线性和拟线性双曲型方程的格点模型李元香（武汉大学软件工程国家重点实验室）黄樟灿（武汉工学院）ＬＡＴＴＩＣＥＭＯＤＥＬＳＦＯＲＦＩＲＳＴＯＲＤＥＲＬＩＮＥＡＲＡＮＤＱＵＡＳＩ－ＬＩＮＥＡＲＨＹＰＥＢＯＬＩＣＥＱＵＡＴＩＯＮＳ￥ＬｉＹｕａｎ－ｘｉａｎ...     

Nonintrusive reduced‐order modeling of parametrized time‐dependent partial differential equations

We propose a nonintrusive reduced‐order modeling method based on the notion of space‐time‐parameter proper orthogonal decomposition (POD) for approximating the solution of nonlinear parametrized time‐dependent partial differential equations. A two‐level POD method is introduced for constructing spatial and temporal basis functions with special properties such that the reduced‐order model satisfies the boundary and initial conditions by construction. A radial basis function approximation method is used to estimate the undetermined coefficients in the reduced‐order model without resorting to Galerkin projection. This nonintrusive approach enables the application of our approach to general problems with complicated nonlinearity terms. Numerical studies are presented for the parametrized Burgers' equation and a parametrized convection‐reaction‐diffusion problem. We demonstrate that our approach leads to reduced‐order models that accurately capture the behavior of the field variables as a function of the spatial coordinates, the parameter vector and time. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

基于Fuzzy推理的时变系统建模

提出一种基于Fuzzy推理的时变系统建模方法,其基本思想是:对时间维度进行分割,在每个较短的时间间隔内用时不变模型代替时变模型,将这些时不变模型组合在一起,最终获得一个整体非线性时变的微分方程模型.分别研究了输入输出型时变系统和状态空间型时变系统的模型建立方法,除了从理论上保证了所获得的模型对系统的逼近性,还从仿真实验验证了用该方法建立的模型对非线性时变系统有很好的逼近效果.     

An improved approximate analytic solution for Riccati equations over extended intervals

In this paper, we present a new effective approach based on our previous method for solving Riccati equations [Vahidi and Didgar, An improved method for determining the solution of Riccati equations, Neural Comput & Applic DOI 10.1007/s00521-012-10645, 2012]. The proposed technique combines transformation of variables with the Laplace Adomian decomposition method, which is a variant of the classic Adomian decomposition method. The method examined practicality for several specific examples of the Riccati equation. Numerical results show that the proposed method is more efficient and accurate than other applied methods over extended intervals, and stands in good agreement to the exact solutions. The obtained results provide a rapidly convergent series, from which we achieve a high degree of accuracy using only a few terms of the recursion scheme. Thus we have developed a natural sequence of rational approximations as a technique of solution continuation for the Riccati equation.     

Decomposition results for general polling systems and their applications

In this paper we derive decomposition results for the number of customers in polling systems under arbitrary (dynamic) polling order and service policies. Furthermore, we obtain sharper decomposition results for both the number of customers in the system and the waiting times under static polling policies. Our analysis, which is based on distributional laws, relaxes the Poisson assumption that characterizes the polling systems literature. In particular, we obtain exact decomposition results for systems with either Mixed Generalized Erlang (MGE) arrival processes, or asymptotically exact decomposition results for systems with general renewal arrival processes under heavy traffic conditions. The derived decomposition results can be used to obtain the performance analysis of specific systems. As an example, we evaluate the performance of gated Markovian polling systems operating under heavy traffic conditions. We also provide numerical evidence that our heavy traffic analysis is very accurate even for moderate traffic. This revised version was published online in June 2006 with corrections to the Cover Date.     

Dimension reduction in fluid dynamics equations

A method for transforming the Euler and Navier-Stokes equations and a complete system of fluid dynamics equations in three dimensions to a closed system on any moving surface is proposed. As a result, for an arbitrary geometric configuration, the dimension of the equations is reduced by one, which makes them convenient for numerical simulation. The general principles of the method are described, and verifying examples are presented.     

Substructured two-grid and multi-grid domain decomposition methods

Two-level Schwarz domain decomposition methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level preconditioner (or its corresponding smoothing iterative method), which is based on domain decomposition techniques, and a coarse correction step, which relies on a coarse space. The coarse space must properly represent the error components that the chosen one-level method is not capable to deal with. In the literature, most of the works introduced efficient coarse spaces obtained as the span of functions defined on the entire space domain of the considered PDE. Therefore, the corresponding two-level preconditioners and iterative methods are defined in volume. In this paper, we use the excellent smoothing properties of Schwarz domain decomposition methods to define, for general elliptic problems, a new class of substructured two-level methods, for which both Schwarz smoothers and coarse correction steps are defined on the interfaces (except for the application of the smoother that requires volumetric subdomain solves). This approach has several advantages. On the one hand, the required computational effort is cheaper than the one required by classical volumetric two-level methods. On the other hand, our approach does not require, like classical multi-grid methods, the explicit construction of coarse spaces, and it permits a multilevel extension, which is desirable when the high dimension of the problem or the scarce quality of the coarse space prevents the efficient numerical solution. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.