• Journal of Korean Institute of Industrial Engineers
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• v.27 no.3
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• pp.260-266
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• 2001

The characteristic of the random-request availability is that random task arrival is included as one of system elements. If the mean number of task arrivals grows, the computational complexity for deriving the random-request availability becomes extremely high. Using a simulation method, the effect of 'random task arrival' elements on the random-request availability is investigated. Some approximation methods are also discussed.

• The Journal of Natural Sciences
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• v.17 no.1
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• pp.51-54
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• 2006

A complete convergence theorem for randomly weighted sums of random variables is obtained. No conditions are imposed on the joint distributions of the random variables.

• Journal of the Korean Mathematical Society
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• v.43 no.4
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• pp.815-828
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• 2006

Let {$V_{nk},\;k\;{\geq}\;1,\;{\geq}\;1$} be an array of rowwise independent random elements which are stochastically dominated by a random variable X with $E\|X\|^{\frac{\alpha}{\gamma}+{\theta}}log^{\rho}(\|X\|)\;<\;{\infty}$ for some ${\rho}\;>\;0,\;{\alpha}\;>\;0,\;{\gamma}\;>\;0,\;{\theta}\;>\;0$ such that ${\theta}+{\alpha}/{\gamma}<2$. Let {$a_{nk},k{\geq}1,n{\geq}1$) be an array of suitable constants. A complete convergence result is obtained for the weighted sums of the form $\sum{^\infty_k_=_1}\;a_{nk}V_{nk}$.

• Structural Engineering and Mechanics
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• v.20 no.3
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• pp.293-312
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• 2005

In this paper, free vibration of rotating beam with random properties is studied. The cross-sectional area, elasticity modulus, moment of inertia, shear modulus and density are modeled as random fields and the rotational speed as a random variable. To study uncertainty, stochastic finite element method based on second order perturbation method is applied. To discretize random fields, the three methods of midpoint, interpolation and local average are applied and compared. The effects of rotational speed, setting angle, random property variances, discretization scheme, number of elements, correlation of random fields, correlation function form and correlation length on "Coefficient of Variation" (C.O.V.) of first mode eigenvalue are investigated completely. To determine the significant random properties on the variation of first mode eigenvalue the sensitivity analysis is performed. The results are studied for both Timoshenko and Bernoulli-Euler rotating beam. It is shown that the C.O.V. of first mode eigenvalue of Timoshenko and Bernoulli-Euler rotating beams are approximately identical. Also, compared to uncorrelated random fields, the correlated case has larger C.O.V. value. Another important result is, where correlation length is small, the convergence rate is lower and more number of elements are necessary for convergence of final response.

• Bulletin of the Korean Mathematical Society
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• v.43 no.3
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• pp.551-558
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• 2006

We establish a weak law of large numbers for sequence of random elements with values in p-uniformly smooth Banach space. Our result is more general and stronger than some well-known ones.

• Journal of the Korean Statistical Society
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• v.31 no.1
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• pp.129-140
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• 2002

In this paper, we obtain a strong law of large numbers for convex tight random elements taking values in the space of fuzzy numbers in R.

• Structural Engineering and Mechanics
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• v.86 no.3
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• pp.349-359
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• 2023

This paper presents a technique for determining the optimal number of elements in stochastic finite element analysis based on reliability analysis. Using the change-of-variable perturbation stochastic finite element approach, the probability density function of the dynamic responses of stochastic structures is explicitly determined. This method combines the perturbation stochastic finite element method with the change-of-variable technique into a united model. To further examine the relationships between the random fields, discretization of the random field parameters, such as the variance function and the scale of fluctuation, is also performed. Accordingly, the reliability index is calculated based on the explicit probability density function of responses with Gaussian or non-Gaussian random fields in any number of elements corresponding to the random field discretization. The numerical examples illustrate the effectiveness of the proposed method for a one-dimensional cantilever reinforced concrete column and a two-dimensional steel plate shear wall. The benefit of this method is that the probability density function of responses can be obtained explicitly without the use simulation techniques. Any type of random variable with any statistical distribution can be incorporated into the calculations, regardless of the restrictions imposed by the type of statistical distribution of random variables. Consequently, this method can be utilized as a suitable guideline for the efficient implementation of stochastic finite element analysis of structures, regardless of the statistical distribution of random variables.

• Bulletin of the Korean Mathematical Society
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• v.47 no.3
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• pp.467-482
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• 2010

For a double array of random elements {$V_{mn};m{\geq}1,\;n{\geq}1$} in a real separable Banach space, some mean convergence theorems and weak laws of large numbers are established. For the mean convergence results, conditions are provided under which $k_{mn}^{-\frac{1}{r}}\sum{{u_m}\atop{i=1}}\sum{{u_n}\atop{i=1}}(V_{ij}-E(V_{ij}|F_{ij})){\rightarrow}0$ in $L_r$ (0 < r < 2). The weak law results provide conditions for $k_{mn}^{-\frac{1}{r}}\sum{{T_m}\atop{i=1}}\sum{{\tau}_n\atop{j=1}}(V_{ij}-E(V_{ij}|F_{ij})){\rightarrow}0$ in probability where {$T_m;m\;{\geq}1$} and {${\tau}_n;n\;{\geq}1$} are sequences of positive integer-valued random variables, {$k_{mn};m{{\geq}}1,\;n{\geq}1$} is an array of positive integers. The sharpness of the results is illustrated by examples.

• Communications for Statistical Applications and Methods
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• v.9 no.3
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• pp.871-876
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• 2002

In this paper, we establish some characterizations of tightness for a sequence of random elements taking values in the space of upper-semicontinuous fuzzy sets with compact support in R.

• Bulletin of the Korean Mathematical Society
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• v.43 no.3
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• pp.543-549
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• 2006

Sung et al. [13] obtained a WLLN (weak law of large numbers) for the array $\{X_{{ni},\;u_n{\leq}i{\leq}v_n,\;n{\leq}1\}$ of random variables under a Cesaro type condition, where $\{u_n{\geq}-{\infty},\;n{\geq}1\}$ and $\{v_n{\leq}+{\infty},\;n{\geq}1\}$ large two sequences of integers. In this paper, we extend the result of Sung et al. [13] to a martingale type p Banach space.