• Journal of the Korean Statistical Society
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• v.33 no.1
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• pp.35-58
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• 2004

For fixed positive integers and $\textit{k}\;(n\;{\geq}\;{\textit{k}}\;+\;2)$, the exact probability distributions of non-overlapping and overlapping patterns of two failures separated by (i) exactly $textsc{k}$ successes, (ii) at least $\textit{k}$ successes and (iii) at most $\textit{k}$ successes have been obtained for Bernoulli independent and Markov dependent trials by using combinatorial technique. The waiting time distributions for the first occurrence and the $r^{th}$ (r > 1) occurrence of the patterns have also been obtained.

• Journal of applied mathematics & informatics
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• v.42 no.3
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• pp.709-723
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• 2024

We study the distributions of waiting times in variations of the geometric distribution of order k. Variation imposes length on the runs of successes and failures. We study two types of waiting time random variables. First, we consider the waiting time for a run of k consecutive successes the first time no sequence of consecutive k failures occurs prior, denoted by T^(k). Next, we consider the waiting time for a run of k consecutive failures the first time no sequence of k consecutive successes occurred prior, denoted by J^(k). In addition, we study the distribution of the weighted average. The exact formulae of the probability mass function, mean, and variance of distributions are also obtained.

• Journal of the Korean Mathematical Society
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• v.38 no.4
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• pp.807-842
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• 2001

This paper summarizes recent development of analytical and algorithmical results for stationary FIFO queues with multiple Markovian arrival streams, where service time distributions are general and they may differ for different arrival streams. While this kind of queues naturally arises in considering queues with a superposition of independent phase-type arrivals, the conventional approach based on the queue length dynamics (i.e., M/G/1 pradigm) is not applicable to this kind of queues. On the contrary, the workload process has a Markovian property, so that it is analytically tractable. This paper first reviews the results for the stationary distributions of the amount of work-in-system, actual waiting time and sojourn time, all of which were obtained in the last six years by the author. Further this paper shows an alternative approach, recently developed by the author, to analyze the joint queue length distribution based on the waiting time distribution. An emphasis is placed on how to construct a numerically feasible recursion to compute the stationary queue length mass function.

• Journal of Korean Institute of Industrial Engineers
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• v.28 no.2
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• pp.187-192
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• 2002

In this paper we consider an M/G/1 queueing system with vacation. The length of vacation period may be controlled by the waiting time of the first customer. The server goes on vacation as soon as the system is empty, and resumes service either when the waiting time of the leading customer reaches a predetermined value, or when the vacation period is expired, whichever comes first. We consider two types of vacation, say, multiple vacation type and N-policy type. We derive the steady-state distributions of the number of customers at arbitrary time and arbitrary customer's waiting time by means of decomposition property. Also, the mean lengths of busy period, idle period and a cycle time are given.

• Journal of the Korean Operations Research and Management Science Society
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• v.40 no.1
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• pp.1-4
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• 2015

This paper presents a heuristic approach to derive the Laplace-Stieltjes transform (LST) and the probability generating function (PGF) of the waiting time distributions of a continuous- and a discrete-time GI/G/1 queue, respectively. This is a new idea to derive the well-known results, the waiting time distribution of GI/G/1 queue, in a different way.

• Journal of the Korean Operations Research and Management Science Society
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• v.19 no.1
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• pp.85-98
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• 1994

The distribution of random sum of i. i. d. exponential random variables is called GHP (Generalized Phase-Type) distribution. The class of GPH distributions is large enough to include PH (Phase-Type) distributions and has several properties which can be applied conveniently for computational purposes. In this paper, we show that any distribution difined on R$^{+}$ can be app-roximated by the GPH distribution and demonstrate the accuracy of the approximation through various numerical examples. Also, we introduce an efficient way to compute the delay and waiting various numerical examples. Also, we introduce an efficient way to compute the delay and waiting time distributions of the GPH/GPH/1 queueing system which can be used as an approximation model for the GI/G/1 system, and validate its accuracy through numerical examples. The theoretical and experimental results of this paper help us accept the usefulness of the approximations based on GPH distribution.n.

• Communications for Statistical Applications and Methods
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• v.24 no.3
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• pp.193-209
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• 2017

The power Lindley distribution with some of its properties is considered in this article. Maximum likelihood, least squares, maximum product spacings, and Bayes estimators are proposed to estimate all the unknown parameters of the power Lindley distribution. Lindley's approximation and Markov chain Monte Carlo techniques are utilized for Bayesian calculations since posterior distribution cannot be reduced to standard distribution. The performances of the proposed estimators are compared based on simulated samples. The waiting times of research articles to be accepted in statistical journals are fitted to the power Lindley distribution with other competing distributions. Chi-square statistic, Kolmogorov-Smirnov statistic, Akaike information criterion and Bayesian information criterion are used to access goodness-of-fit. It was found that the power Lindley distribution gives a better fit for the data than other distributions.

• Kyungpook Mathematical Journal
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• v.64 no.2
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• pp.287-301
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• 2024

From the mathematical and statistical point of view, a segment of a DNA strand can be viewed as a sequence of four-state (A, C, G, T) trials. Herein, we consider the distributions of runs and patterns related to the run lengths of multi-state sequences, especially for four states (A, B, C, D). Let X₁, X₂, . . . be a sequence of four state independent and identically distributed trials taking values in the set 𝒢 = {A, B, C, D}. In this study, we obtain exact formulas for the probability distribution function for the discrete distribution of runs of B's of order k. We obtain longest run statistics, shortest run statistics, and determine the distributions of waiting times and run lengths.

• Journal of the Korean Operations Research and Management Science Society
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• v.19 no.3
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• pp.235-241
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• 1994

We consider a B/G/1 queueing with vacations, where the server closes the gate when it begins a vacation. In this system, customers arrive according to a Bernoulli process. The service time and the vacation time follow discrete distributions. We obtain the distribution of the number of customers at a random point in time, and in turn, the distribution of the residence time (queueing time + service time) for a customer. It is observed that solutions for our discret time B/G/1 gated vacation model are analogous to those for the continuous time M/G/1 gated vacation model.

• Journal of KIISE:Computer Systems and Theory
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• v.36 no.4
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• pp.239-246
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• 2009

In this study, we analyze the mean waiting time on the nested open queueing network, where the population within each subnetwork is controlled by a semaphore queue. The queueing network can be transformed into a simpler queueing network in terms of customers waiting time. A major characteristic of this model is that the lower layer flow is halted by the state of higher layer. Since this type of queueing network does not have exact solutions for performance measure, the lower bound and upper bound on the mean waiting time are checked by comparing them with the mean waiting time in the transformed nested queueing network. Simulation estimates are obtained assuming Poisson arrivals and other phase-type arrival process, i.e., Erlang and hyper-exponential distributions. The bounds obtained can be applied to get more close approximation using the suitable approach.