• 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.

• Communications of the Korean Mathematical Society
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• v.32 no.2
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• pp.425-433
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• 2017

This paper is concerned with the analysis of the busy period distribution in a batch arrival $M^X/G/1$ retrial queue. The expression for the Laplace-Stieltjes transform of the length of the busy period is well known, but from this expression we cannot compute the moments of the length of the busy period by direct differentiation. This paper provides a direct method of calculation for the first and second moments of the length of the busy period.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.19 no.5
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• pp.833-841
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• 1994

In this paper we develop an approximation formalism for the queue length distribution of general queueing models. Our formalism is based on two steps of analytic approximation employing both the lower and upper bound techniques. It is favorable to a fast numerical calcuation for the queue length distribution of a superposition of a superposition of arbitary type traffic sources. In the application. M+ N D /M/1 is considered. The calculated result for queue length distribution measured by arriving or leaving customers show a good agreement with the direct simulation of the system. Especially, we demonstrate that our formula for M/M/1 is equivalent to the exact solution, while that D/M/1 is simplified in an analytic form.

• Journal of the Korean Mathematical Society
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• v.40 no.1
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• pp.87-107
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• 2003

In this paper, we consider a discrete time queueing system fed by a superposition of an ON and OFF source with heavy tail ON periods and geometric OFF periods and a D-BMAP (Discrete Batch Markovian Arrival Process). We study the tail behavior of the queue length distribution and both infinite and finite buffer systems are considered. In the infinite buffer case, we show that the asymptotic tail behavior of the queue length of the system is equivalent to that of the same queueing system with the D-BMAP being replaced by a batch renewal process. In the finite buffer case (of buffer size K), we derive upper and lower bounds of the asymptotic behavior of the loss probability as $K\;\longrightarrow\;\infty$.

• Journal of the Korean Operations Research and Management Science Society
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• v.39 no.3
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• pp.1-5
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• 2014

We study the paper Zhernovyi and Zhernovyi [Zhernovyi, K.Y. and Y.V. Zhernovyi, "An $M^{\Theta}/G/1/m$ system with two-threshold hysteresis strategy of service intensity switching," Journal of Communications and Electronics, Vol.12, No.2(2012), pp.127-140]. In the paper, authors used the Korolyuk potential method to obtain the stationary queue length distribution. Instead, our note makes an attempt to apply the most frequently used methods : the embedded Markov chain and the supplementary variable method. We derive the queue length distribution at a customer's departure epoch and then at an arbitrary epoch.

• Journal of the Korean Operations Research and Management Science Society
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• v.33 no.1
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• pp.107-115
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• 2008

We consider the discrete-time GI/G/1/K queue under the early arrival system. Using a modified supplementary variable technique(SVT), we obtain the distribution of the steady-state queue length. Unlike the conventional SVT, the modified SVT yields transform-free results in such a form that a simple two-moment approximation scheme can be easily established.

• Communications of the Korean Mathematical Society
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• v.14 no.3
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• pp.611-625
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• 1999

We consider ΜΑΡ/G/ 1 finite capacity queue with mul-tiple thresholds on buffer. The arrival of customers follows a Markov-ian arrival process(MAP). The service time of a customer depends on the queue length at service initiation of the customer. By using the embeded Markov chain method and the supplementary variable method, we obtain the queue length distribution ar departure epochs and at arbitrary epochs. This gives the loss probability and the mean waiting time by Little's law. We also give a simple numerical examples to apply the overload control in packetized networks.

• ETRI Journal
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• v.15 no.3
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• pp.35-45
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• 1994

In this paper we develop an approximation formalism on the queue length distribution for general queueing models. Our formalism is based on two steps of approximation; the first step is to find a lower bound on the exact formula, and subsequently the Chernoff upper bound technique is applied to this lower bound. We demonstrate that for the M/M/1 model our formula is equivalent to the exact solution. For the D/M/1 queue, we find an extremely tight lower bound below the exact formula. On the other hand, our approach shows a tight upper bound on the exact distribution for both the ND/D/1 and M/D/1 queues. We also consider the $M+{\Sigma}N_jD/D/1$ queue and compare our formula with other formalisms for the $M+{\Sigma}N_jD/D/1$ and M+D/D/1 queues.

• Journal of Korean Institute of Industrial Engineers
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• v.28 no.4
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• pp.390-396
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• 2002

We present a meaningful interpretation of the equations for the steady-state queue length distribution of the GI/GI/c/K queue so that the equations are better understood and become more applicable. As a byproduct, we present an exact expression of the mean queue waiting time for the M/GI/c queue.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.469-481
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• 2004

In this paper, we investigate the mean queue length and the mean packet delay of a dynamic bandwidth allocation(DBA) scheme in an Ethernet passive optical network(EPON). We focus on the interleaved polling system with a gated service discipline. We assume that input packets arrive at an optical network unit(ONU) according to Poisson process. We use a continuous time queueing model in order to find the queue length distribution of the gated interleaved polling system with the first stage input queue and the second stage transmission queue. We give some numerical results to investigate the mean queue lengths and mean packet delays for the symmetric polling system with statistically identical stations.