System reliability analysis with saddlepoint approximation

System reliability is usually estimated through component reliability, which is commonly computed by the First Order Reliability Method (FORM). The FORM is computationally efficient, but may not be accurate for nonlinear limit-state functions. An alternative system reliability analysis method is proposed based on saddlepoint approximation. Unlike the FORM that linearizes limit-state functions in a transformed random space, the proposed method linearizes the limited-state functions without any transformation. After the linearization, the joint probability density of limit-state functions is estimated by the multivariate saddlepoint approximation. Without the nonnormal-to-normal transformation, the present method is more accurate than the FORM when the transformation increases the nonlinearity of limit-state functions. As demonstrated in the two examples, the new method is also as efficient as the FORM.     

Reliability analysis of structures using stochastic response surface method and saddlepoint approximation

Stochastic response surface method (SRSM) is a technique used for reliability analysis of complex structural systems having implicit or time consuming limit state functions. The main aspects of the SRSM are the collection of sample points, the approximation of response surface and the estimation of the probability of failure. In this paper, sample points are selected close to the most probable point of failure and the actual limit state surface (LSS). The response surface is fitted using the weighted regression technique, which allows the fitting points to be weighted based on their distance from the LSS. The cumulant generating function (CGF) of the response surface is derived analytically. The saddlepoint approximation (SPA) method is utilized to compute the probability of failure of the structural system. Finally, four numerical examples compare the proposed algorithm with the traditional quadratic polynomial SRSM, Kriging based SRSM and direct MCS.     

Heavy traffic analysis of polling models by mean value analysis

In this paper we present a new approach to derive heavy-traffic asymptotics for polling models. We consider the classical cyclic polling model with exhaustive or gated service at each queue, and with general service-time and switch-over time distributions, and study its behavior when the load tends to one. For this model, we explore the recently proposed mean value analysis (MVA), which takes a new view on the dynamics of the system, and use this view to provide an alternative way to derive closed-form expressions for the expected asymptotic delay; the expressions were derived earlier in [R.D. van der Mei, H. Levy, Expected delay in polling systems in heavy traffic, Adv. Appl. Probab. 30 (1998) 586–602], but in a different way. Moreover, the MVA-based approach enables us to derive closed-form expressions for the heavy-traffic limits of the covariances between the successive visit periods, which are key performance metrics in many application areas. These results, which have not been obtained before, reveal a number of insensitivity properties of the covariances with respect to the system parameters under heavy-traffic assumptions, and moreover, lead to simple approximations for the covariances between the successive visit times for stable systems. Numerical examples demonstrate that the approximations are accurate when the load is close enough to one.     

Uncertainty propagation analysis using sparse grid technique and saddlepoint approximation based on parameterized p-box representation

Uncertainty propagation analysis, which assesses the impact of the uncertainty of input variables on responses, is an important component in risk assessment or reliability analysis of structures. This paper proposes an uncertainty propagation analysis method for structures with parameterized probability-box (p-box) representation, which could efficiently compute both the bounds on statistical moments and also the complete probability bounds of the response function. Firstly, based on the sparse grid numerical integration (SGNI) method, an optimized SGNI (OSGNI) is presented to calculate the bounds on the statistical moments of the response function and the cumulants of the cumulant generating function (CGF), respectively. Then, using the bounds on the first four cumulants, an optimization procedure based on the saddlepoint approximation is proposed to obtain the whole range of probability bounds of the response function. Through using the saddlepoint approximation, the present approach can achieve a good accuracy in estimating the tail probability bounds of a response function. Finally, two numerical examples and an engineering application are investigated to demonstrate the effectiveness of the proposed method.

     

Approximate mean value analysis for stochastic marked graphs

An iterative technique for the computation of approximate performance indices of a class of stochastic Petri net models is presented. The proposed technique is derived from the mean value analysis algorithm for product-form solution stochastic Petri nets. In this paper, we apply the approximation technique to stochastic marked graphs. In principle, the proposed technique can be used for other stochastic Petri net subclasses. In this paper, some of these possible applications are presented. Several examples are presented in order to validate the approximate results     

Experiments with improved approximate mean value analysis algorithms

Approximate mean value analysis (MVA) is a popular technique for analyzing queueing networks because of the efficiency and accuracy that it affords. In this paper, we present a new software package, called the improved approximate mean value analysis library (IAMVAL), which can be easily integrated into existing commercial and research queueing network analysis packages. The IAMVAL packages include two new approximate MVA algorithms, the queue line (QL) algorithm and the fraction line (FL) algorithm, for analyzing multiple class separable queueing networks. The QL algorithm is always more accurate than, and yet has approximately the same computational efficiency as, the Bard–Schweitzer proportional estimation (PE) algorithm, which is currently the most widely used approximate MVA algorithm. The FL algorithm has the same computational cost and, in noncongested separable queueing networks where queue lengths are quite small, yields more accurate solutions than both the QL and PE algorithms.     

Remarks on periodic boundary value problems of first order discrete systems

A new monotone iterative method for the periodic boundary value problems of first order discrete systems is developed. The numerical results are given.     

Limited descent-based mean value method for inverse reliability analysis

Engineering with Computers - The robustness and efficiency of inverse reliability methods are important issues in reliability-based design optimization (RBDO) using performance measure approach...     

Adaptive probability analysis using an enhanced hybrid mean value method

This paper proposes an adaptive probability analysis method that can effectively generate the probability distribution of the output performance function by identifying the propagation of input uncertainty to output uncertainty. The method is based on an enhanced hybrid mean value (HMV+) analysis in the performance measure approach (PMA) for numerical stability and efficiency in search of the most probable point (MPP). The HMV+ method improves numerical stability and efficiency especially for highly nonlinear output performance functions by providing steady convergent behavior in the MPP search. The proposed adaptive probability analysis method approximates the MPP locus, and then adaptively refines this locus using an a posteriori error estimator. Using the fact that probability levels can be easily set a priori in PMA, the MPP locus is approximated using the interpolated moving least-squares method. For refinement of the approximated MPP locus, additional probability levels are adaptively determined through an a posteriori error estimator. The adaptive probability analysis method will determine the minimum number of necessary probability levels, while ensuring accuracy of the approximated MPP locus. Several examples are used to show the effectiveness of the proposed adaptive probability analysis method using the enhanced HMV+ method.     

Rational approximation of fractional order systems by vector fitting method

International Journal of Control, Automation and Systems - A novel method for approximating fractional order systems is presented. Vector fitting is involved in this method. As the basis of...     

Integral boundary value problems for first order integro-differential equations with impulsive integral conditions

This paper is concerned with the integral boundary value problems for first order integro-differential equations with impulsive integral conditions. Sufficient conditions are obtained for the existence of extremal solutions.     

Convergence analysis of first order reliability method using chaos theory

First order reliability method (FORM) for the computation of reliability index has been used widely because of its advantages of the efficiency and effectiveness as well simplicity for many years. There exists the phenomenon of convergent failure in the FORM in calculating the reliability index iteratively for some limit state functions, for which the essential factor is investigated using chaotic dynamics theory in the present paper. The bifurcation plots of reliability index are presented for several typical limit state functions, and the computational results from those mapping functions due to FORM iterations show the complicated dynamics phenomena such as the periodic oscillation, bifurcation and chaos. Moreover, the Lyapunov exponents of non-linear map from FORM are calculated. From the numerical investigation of presented examples, it is concluded that the convergence of FORM does not depend on the curvature of design points of the limit state function, and the quantitative index for identifying the convergence of FORM iterative computation is the Lyapunov exponents of non-linear map corresponding to that limit state function.     

A note on the use of first order quadrilateral elements in axisymmetric analysis

The first order quadrilateral element is known to suffer from some “defects”, such as parasitic or false shear phenomena in bending and locking for nearly incompressible materials. Some improvements can be achieved using additional shape functions or a particular integration method. However, some limitations remain, especially for axisymmetric problems. Commercial Finite Element (FE) codes use various element formulations, and therefore can lead to rather different results, also for very simple problems. In this study, a linear elastic pressurised pipe was assumed as a test case, to compare Ansys and Abaqus results from a FE-user point of view. Differences in the element shape functions and in the integration methods produced a rather significant discrepancy in the results at nodes and at Gauss points. It is interesting to underline that, although four elements in the thickness suffice to obtain displacements with an error lower than 0.02%, about thirty elements are required to reduce the error on the radial stress to 2%. Numerical results were confirmed and generalised by means of analytical verification. Moreover further simulations showed the extendibility of the main conclusions to similar cases.     

Highly accurate method for solving initial boundary value problem for first order hyperbolic differential equations

Restrictive Taylor approximation is a new approach similar to Restrictive Pade' approximation mentioned in [2] is proposed. An explicit method for solving a first order wave equation is treated. The suggested method is of high accuracy whatever the exact solution is too large. It treats also the discontinuous behavior of the initial data for the function or its derivative. The stability condition and comparison between the considered method and Lax-Wendroff method are discussed.     

Development of a second order approximation for the Navier-Stokes equations

The purpose of this paper is the development of a 2nd order finite difference approximation to the steady state Navier-Stokes equations governing flow of an incompressible fluid in a closed cavity. The approximation leads to a system of equations that has proved to be very stable. In fact, numerical convergence was obtained for Reynolds numbers up to 20,000. However, it is shown that extremely small mesh sizes are needed for excellent accuracy with a Reynolds number of this magnitude. The method uses a nine point finite difference approximation to the convection term of the vorticity equation. At the same time it is capable of avoiding values at corner nodes where discontinuities in the boundary conditions occur. Figures include level curves of the stream and vorticity functions for an assortment of grid sizes and Reynolds numbers.     

A linear approximation method for the Shapley value

The Shapley value is a key solution concept for coalitional games in general and voting games in particular. Its main advantage is that it provides a unique and fair solution, but its main drawback is the complexity of computing it (e.g., for voting games this complexity is #p-complete). However, given the importance of the Shapley value and voting games, a number of approximation methods have been developed to overcome this complexity. Among these, Owen's multi-linear extension method is the most time efficient, being linear in the number of players. Now, in addition to speed, the other key criterion for an approximation algorithm is its approximation error. On this dimension, the multi-linear extension method is less impressive. Against this background, this paper presents a new approximation algorithm, based on randomization, for computing the Shapley value of voting games. This method has time complexity linear in the number of players, but has an approximation error that is, on average, lower than Owen's. In addition to this comparative study, we empirically evaluate the error for our method and show how the different parameters of the voting game affect it. Specifically, we show the following effects. First, as the number of players in a voting game increases, the average percentage error decreases. Second, as the quota increases, the average percentage error decreases. Third, the error is different for players with different weights; players with weight closer to the mean weight have a lower error than those with weight further away. We then extend our approximation to the more general k-majority voting games and show that, for n players, the method has time complexity O(k²n) and the upper bound on its approximation error is .     

Performance evaluation of re-entrant manufacturing system with production loss using mean value analysis

This paper proposes an approximation method based on mean value analysis (MVA) technique for estimating the performance measures of re-entrant manufacturing system with production loss. The model is an extension of the one proposed by Park et al. (Comput. Oper. Res. 29 (2002) 1009). A unique feature in the extended model is that random production losses due to machine failures and yields are considered. Considering such losses is critical in performance evaluation, because it may often cause significant errors in the results compared to the real values if the analysis does not explicitly consider them. However, such random losses substantially increase the complexity of the analysis, due to the fact that even through simulation it requires not only extra modeling efforts, but also a number of replications. As a result, it requires bigger efforts and data, and significantly longer computational times. For an analytical approach, such random losses also prohibit exact analysis of the system. Therefore, a methodology for analyzing the system approximately is proposed using the iterative procedures based upon the MVA and some heuristic adjustments. The performance measures of interest are the steady-state average of the cycle time of each job class, the queue length of each buffer, and the throughput of the system. Numerical tests are presented to show the performance of the proposed approach against the simulation results. Also, the comparisons with the earlier test results summarize the insights from the overall research thus far.