• 대한수학회논문집
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• 제12권1호
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• pp.177-192
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• 1997

The basic theory of the quasi-birth-and-death process is extended to a process which is inhomogeous in levels. Several key results in the standard homogeneous theory hold in a more general context than that usually stated, in particular not requiring positive recurrence. Theser results are subsumed under our development. The treatment is entirely probabilistic.

• Journal of applied mathematics & informatics
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• 제29권3_4호
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• pp.803-811
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• 2011

We present an algorithmic solution for the stationary distribution of the M/M/c retrial queue in which the retrial times of each customer in orbit are of phase type distribution of order 2. The system is modeled by the level dependent quasi-birth-and-death (LDQBD) process.

• Journal of applied mathematics & informatics
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• 제7권1호
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• pp.83-100
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• 2000

Many queueing systems such as M/M/s/K retrial queue with impatient customers, MAP/PH/1 retrial queue, retrial queue with two types of customers and MAP/M/$\infty$ queue can be modeled by a level dependent quasi-birth-death(LDQBD) process with linear transition rates of the form ${\lambda}_k$={\alpga}{+}{\beta}k$ at each level $\kappa$. The purpose of this paper is to propose an algorithm to find transient distributions for LDQBD processes with linear transition rates based on the adaptive uniformization technique introduced by van Moorsel and Sanders [11]. We apply the algorithm to some retrial queues and present numerical results.

• Journal of applied mathematics & informatics
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• 제39권5_6호
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• pp.601-616
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• 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.

• 대한산업공학회지
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• 제14권1호
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• pp.17-30
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• 1988

The aim of this project is to analyze the queueing model with the two types of customers which either can wait unrestrictedly or wait restrictively in the system depending on the condition of service facility unless they may be served immediately. This model consists of the three-dimensional state space and then quasi birth-death process is formulated. The steady-state probabilities and measures of performance of this system are derived by using Matrix Geometric method.

• 한국경영과학회지
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• 제30권2호
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• pp.131-142
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• 2005

This paper proposes queueing models for a Web server system which is composed of an infinite-buffer main server and finite-buffer auxiliary server(s). The system is modeled by the level-dependent quasi-birth- death (QBD) process. Utilizing the special structure of the QBD, we convert the infinite level-dependent QBD into a finite level-independent QBD and compute the state probabilities. We then explore the operational characteristics of the proposed web-server models and draw some useful conclusions.

• 정보처리학회논문지C
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• 제9C권3호
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• pp.399-406
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• 2002

본 논문에서는 고정 전송율 서비스 및 가변 전송율 서비스 호들이 동적으로 네트워크에 들어와서 데이터를 전송하고, 데이터를 모두 전송하면 네트워크에서 사라지는 동적인 다중 서비스 네트워크에서, 고정 전송율 및 가변 전송율 서비스의 요구 성능을 만족하기 위한 네트워크 스위치(링크)의 성능 분석 및 설계 방법을 제안한다. 먼저 동적인 다중 서비스 네트워크의 링크를 Harkov 체인을 이용해 모델하고, QBD (Quasi-Birth-Death) 프로세스와 행렬 방정식을 이용해 가변 전송율 서비스의 호들에 할당될 전송율의 성능을 구한다. 즉, 가변 전송율 서비스의 평균 호수를 추정하고 이를 바탕으로, 호당 평균 전송율 및 평균 전송 지연 시간을 구한다. 고정 전송율 서비스의 성능은 고정 전송율 서비스 호들의 차단 확률(blocking probability)로 구해진다. 이렇게 해서 구해진 고정 전송율 및 가변 전송율 서비스의 성능을 이용해서, 고정 전송율 및 가변 전송율 서비스 호들의 요구 성능(전송율, 전송 지연 시간, 호의 차단 확률 등)을 만족하기 위한 네트워크 링크의 용량을 설계하는 방법을 제안한다. 이러한 방법을 데이터 및 음성(비디오)을 동시에 지원하는 네트워크 스위치의 성능 분석 및 용량 설계에 이용할 수 있다.

• 한국경영과학회지
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• 제41권2호
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• pp.35-51
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• 2016

This paper proposes a new approximate approach to calculate the order fill rate and the probability of filling an entire customer order immediately from the shelf in a business environment under purchase dependence characterized by customer purchase patterns observed in such areas as marketing, manufacturing systems, and distribution systems. The new approximate approach divides customer orders into item orders and calculates fill rates of all order types to approximate the order fill rate. We develop a greed iterative search algorithm (GISA) based on the Gauss-Seidel method to avoid dimensionality and prevent the solution divergence for larger instances. Through the computational analysis that compares the GISA with the simulation, we demonstrate that the GISA is a dependable algorithm for deriving the stationary joint distribution of on-hand inventories in the type-K pure system. We also present some managerial insights.

• Journal of applied mathematics & informatics
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• 제33권1_2호
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• pp.77-87
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• 2015

Parallel lines are often used to increase production rate in many manufacturing systems where the main line splits into several lines in parallel, and after some operations, they merge into a main line again. Queueing networks with finite buffers have been widely used for modeling and analyzing manufacturing systems. This paper provides an approximation technique for multi-server two-stage networks with merge configuration and blocking which will be a building block for analysis of general manufacturing systems with parallel lines and merge configuration. The main idea of the method is to decompose the original system into subsystems that have two service stations with multiple servers, two buffers and external arrivals to the second stage are allowed. The subsystems are modeled by level dependent quasi-birth-and-death (LDQBD) process.

• 산업경영시스템학회지
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• 제40권4호
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• pp.137-146
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• 2017

For the competitive business environment under purchase dependence, this paper proposes a new approximate calculation of order fill rate which is a probability of satisfying a customer order immediately using the existing inventory. Purchase dependence is different to demand dependence. Purchase dependence treats the purchase behavior of customers, while demand dependence considers demand correlation between items, between regions, or over time. Purchase dependence can be observed in such areas as marketing, manufacturing systems, and distribution systems. Traditional computational methods have a difficulty of the curse of dimensionality for the large cases, when deriving the stationary joint distribution which is utilized to calculate the order fill rate. In order to escape the curse of dimensionality and protect the solution from diverging for the large cases, we develop a greedy iterative search algorithm based on the Gauss-Seidel method. We show that the greedy iterative search algorithm is a dependable algorithm to derive the stationary joint distribution of on-hand inventories in the retailer system by conducting a comparison analysis of a greedy iterative search algorithm with the simulation. In addition, we present some managerial insights such as : (1) The upper bound of order fill rate can be calculated by the one-item pure system, while the lower bound can be provided by the pure system that consists of all items; (2) As the degree of purchase dependence declines while other conditions remain same, it is observed that the difference between the lower and upper bounds reduces, the order fill rate increases, and the order fill rate gets closer to the upper bound.