• Proceedings of the Korean Operations and Management Science Society Conference
• /
• 2006.05a
• /
• pp.1101-1103
• /
• 2006

For a broad class of discrete-time FIFO queueing systems with D-MAP (discrete-time Markovian arrival process) arrivals, we present a distributional Little's law that relates the distribution of the stationary number of customers in system (queue) with that of the stationary number of slots a customer spends in system (queue). Taking the multi-server D-MAP/D/c queue for example, we illustrate how to utilize this relation to get the desired distribution of the number of customers.

• Journal of Korea Society of Industrial Information Systems
• /
• v.11 no.1
• /
• pp.1-6
• /
• 2006

Queueing models are good models for fragments of communication systems and networks, so their investigation is interesting for theory and applications. Theses queues may play an important role for the validation of different decomposition algorithms designed for investigating more general queueing networks. So, in this paper we illustrate that the batch size distribution affects the loss probability, which is the main performance measure of a finite buffer queues.

• Journal of Korea Society of Industrial Information Systems
• /
• v.22 no.1
• /
• pp.23-32
• /
• 2017

In this paper, We suggest a unified approach for the analysis of discrete-time MAP/G/1 queueing system. Many researches on the D-MAP/G/1 queue have been used different approach to analyze system queue length and waiting time for the same system. Therefore, a unified framework for analyzing a system is necessary from a viewpoint of system design and management. We first derived steady-state workload distribution, and then waiting time and sojourn time are derived by the result of workload analysis. Finally, system queue length distribution is derived with generating function from the sojourn time distribution.

• Journal of Korea Society of Industrial Information Systems
• /
• v.29 no.1
• /
• pp.63-76
• /
• 2024

In this study, we analyzed queue length and sojourn time of discrete-time BMAP/G/1 queues under the workload control. Group customers (packets) with correlations arrive at the system following a discrete-time Markovian arrival process. The server starts busy period when the total service time of the arrived customers exceeds a predetermined workload threshold D and serves customers until the system is empty. From the analysis of workload and waiting time, distributions of queue length at the departure epoch and arbitrary time epoch and system sojourn time are derived. We also derived the mean value as a performance measure. Through numerical examples, we confirmed that we can obtain results represented by complex forms of equations, and we verified the validity of the theoretical values by comparing them with simulation results. From the results, we can obtain key performance measures of complex systems that operate similarly in various industrial fields and to analyze various optimization problems.

• Journal of applied mathematics & informatics
• /
• v.39 no.5_6
• /
• pp.601-616
• /
• 2021

Retrial queueing models have been studied extensively in the literature. These have many practical applications, especially in service sectors. However, retrial queueing models have their own limitations. Typically, analyzing such models involve level-dependent quasi-birth-and-death processes, and hence some form of a truncation or an approximate method or simulation approach is needed to study in steady-state. Secondly, in general, the customers are not served on a first-come-first-served basis. The latter is the case when a new arrival may find a free server while prior arrivals are waiting in the retrial orbit due to the servers being busy during their arrivals. In this paper, we take a different approach to the study of multi-server retrial queues in which the signals are generated in such a way to provide a reasonably fair treatment to all the customers seeking service. Further, this approach makes the study to be level-independent quasi-birth-and-death process. This approach is different from any considered in the literature. Using matrix-analytic methods we analyze MAP/M/c-type retrial queueing models along with Poisson signals in steady-state. Illustrative numerical examples including a comparison with previously published retrial queues are presented and they show marked improvements in providing a quality of service to the customers.

• Journal of the Korean Mathematical Society
• /
• v.40 no.1
• /
• pp.87-107
• /
• 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 $K\;\longrightarrow\;\infty$.

• Proceedings of the Korean Operations and Management Science Society Conference
• /
• 2005.05a
• /
• pp.866-872
• /
• 2005

본 연구는 비축출형 우선순위 2-계층 대기행렬을 다룬다. 각 계층별 고객들은 마코비안 도착과정(Markovian arrival process, MAP)에 의하여 시스템에 도착하고, 각 계층마다 고유의 임계점을 갖는다. 시스템 내에 고객들이 존재하지 않으면 서어버는 유휴해지고 어느 계층이든지 상관없이 계층에 부여된 임계점에 먼저 도달하면 서어버는 서비스를 시작한다. 우선순위가 높은 고객들을 먼저 서비스하는 비축출형 우선순위 서비스규칙을 따른다. 본 연구에서는 각 계층별 고객들의 대기시간분포에 대한 라플라스(Laplace-Stieltjes) 변환과 평균 대기시간을 유도한다.

• Proceedings of the Korean Operations and Management Science Society Conference
• /
• 2006.05a
• /
• pp.1093-1100
• /
• 2006

본 연구는 D-정책을 갖는 BMAP/G/1 대기행렬 시스템의 일량(workload) 및 대기시간(waiting time)을 분석한다. 유휴한 서버는 도착하는 고객들의 서비스 시간의 총합이 주어진 임계점 D를 넘어야만 서비스를 시작한다. 고객의 도착과정은 집단마코비안도착과정(BMAP, Batch Markovian Arrival Process)을 따른다. 본 논문에서는 이러한 시스템의 일량 및 대기시간에 대한 LST를 구하고, 이로부터 평균일량 및 평균대기시간을 유도한다. 또한 BMAP/G/1의 특별한 경우인 $M^X/G/1$인 경우와 대기시간의 비교를 행한다.