• Journal of the Chungcheong Mathematical Society
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• v.29 no.1
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• pp.95-102
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• 2016

Multi-server queueing systems with retrials are widely used to model problems in a call center. We present an explicit formula for an approximation of the queue length distribution in a multi-server retrial queue, by using the Lerch transcendent. Accuracy of our approximation is shown in the numerical examples.

• Communications for Statistical Applications and Methods
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• v.28 no.5
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• pp.425-445
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• 2021

A generalization of the log-logistic (LL) distribution called exponentiated log-logistic (ELL) distribution on lines of exponentiated Weibull distribution is considered. In this paper, based on progressive type-II censored samples, we have derived the maximum likelihood estimators and Bayes estimators for three parameters, the survival function and hazard function of the ELL distribution. Then, under the balanced squared error loss (BSEL) and the balanced linex loss (BLEL) functions, their corresponding Bayes estimators are obtained using Lindley's approximation (see Jung and Chung, 2018; Lindley, 1980), Tierney-Kadane approximation (see Tierney and Kadane, 1986) and Markov Chain Monte Carlo methods (see Hastings, 1970; Gelfand and Smith, 1990). Here, to check the convergence of MCMC chains, the Gelman and Rubin diagnostic (see Gelman and Rubin, 1992; Brooks and Gelman, 1997) was used. On the basis of their risks, the performances of their Bayes estimators are compared with maximum likelihood estimators in the simulation studies. In this paper, research supports the conclusion that ELL distribution is an efficient distribution to modeling data in the analysis of survival data. On top of that, Bayes estimators under various loss functions are useful for many estimation problems.

• Communications for Statistical Applications and Methods
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• v.14 no.3
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• pp.583-592
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• 2007

Given the moments of a symmetric random variable, its density and distribution functions can be accurately approximated by making use of the algorithm proposed in this paper. This algorithm is specially designed for approximating symmetric distributions and comprises of four phases. This approach is essentially based on the transformation of variable technique and moment-based density approximants expressed in terms of the product of an appropriate initial approximant and a polynomial adjustment. Probabilistic quantities such as percentage points and percentiles can also be accurately determined from approximation of the corresponding distribution functions. This algorithm is not only conceptually simple but also easy to implement. As illustrated by the first two numerical examples, the density functions so obtained are in good agreement with the exact values. Moreover, the proposed approximation algorithm can provide the more accurate quantities than direct approximation as shown in the last example.

• The Korean Journal of Applied Statistics
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• v.29 no.4
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• pp.571-579
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• 2016

Most of studies related to the distributions of quadratic forms are conducted under the assumption of multivariate normal distribution. In this paper, we suggested an approximation to the distribution of quadratic forms based on multivariate skew-normal distribution as alternatives for multivariate normal distribution. Saddlepoint approximations are considered and the accuracy of the approximations are verified through simulation studies.

• Communications for Statistical Applications and Methods
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• v.6 no.3
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• pp.821-830
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• 1999

In This paper we calculate the probabilities of binomial and Poisson distributions when n or${\mu}$ is large. Based on this calculation we consider the normal approximation to the binomial and binomial approximation to Poisson. We implement this approximation via CGI and dynamic graphs. These implementation are made available through the internet.

• Journal of Industrial Technology
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• v.10
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• pp.9-13
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• 1990

The approximation of the radial distribution functions(RDF) of mixture plays an important role in deriving the mixing rules for the corresponding states principle(CSP). The mean density approximation(MDA), one of the most successful approximations, fails to predict the radial distribution functions when the size ratio in terms of the Lennard-Jones size parameters is greater than 1.5. To get a better prediction of important structural integrals over the radial distribution functions that arise in the asymmetrical attraction contribution of the perturbaton theory, the new type of mean density approximation(NTMDA) is proposed. With this NTMDA, quite reliable results for those integrals for systems with comparatively large ratio of the size parameters are obtained.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.30 no.2 s.245
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• pp.164-170
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• 2006

To prevent the numerical instabilities in topology optimization, continuous approximation of material distribution (CAMD) is proposed to the homogenization design method (HDM) and the simple isotropic material with penalization (SIMP) method. The continuous FE approximation of design variables including high order elements is applied to the formulation of SIMP method. Numerical examples are presented to compare the efficiency of CAMD both in HDM and SIMP.

• The Korean Journal of Applied Statistics
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• v.11 no.2
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• pp.287-302
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• 1998

Saddlepoint approximation to the distribution function of sample mean(Daniels, 1987) is extended to the case of general statistic in this paper. The suggested approximation methods are applied to derive the approximations to the distributions of some statistics, including sample valiance and studentized mean. Some comparisons with other methods show that the suggested approximations are very accurate for moderate or small sample sizes. Even in extreme tail the accuracies are also maintained.

• Communications for Statistical Applications and Methods
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• v.13 no.2
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• pp.429-440
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• 2006

In statistical computing, it is often for researchers to need the distribution of a weighted sum of noncentral chi-square variables. In this case, it is very limited to know its exact distribution. There are many works to contribute to this topic, e.g. Imhof (1961) and Solomon-Stephens (1977). Imhof's method gives good approximation to the true distribution, but it is not easy to apply even though we consider the development of computer technology Solomon-Stephens's three moment chi-square approximation is relatively easy and accurate to apply. However, they skipped many details, and their simulation is limited to a weighed sum of central chi-square random variables. This paper gives details on Solomon-Stephens's method. We also extend their simulation to the weighted sum of non-central chi-square distribution. We evaluated approximated powers for homogeneous test and compared them with the true powers. Solomon-Stephens's method shows very good approximation for the case.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.19 no.4
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• pp.443-457
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• 2015

We consider the M/M/c/c queues in which the customers blocked to enter the service facility retry after a random amount of time and some of idle servers can leave the vacation. The vacation time and retrial time are assumed to be of phase type distribution. Approximation formulae for the distribution of the number of customers in service facility and the mean number of customers in orbit are presented. We provide an approximation for M/M/c/c queue with general retrial time and general vacation time by approximating the general distribution with phase type distribution. Some numerical results are presented.