• Korean Journal of Mathematics
• /
• v.29 no.2
• /
• pp.371-386
• /
• 2021

This paper investigates two discrete time queueing inventory models with positive service time and lead time. Customers arrive according to a Bernoulli process and service time and lead time follow geometric distributions. The first model under discussion based on replenishment of order upto S policy where as the second model is based on order placement by a fixed quantity Q, where Q = S - s, whenever the inventory level falls to s. We analyse this queueing systems using the matrix geometric method and derive an explicit expression for the stability condition. We obtain the steady-state behaviour of these systems and several system performance measures. The influence of various parameters on the systems performance measures and comparison on the cost analysis are also discussed through numerical example.

• Journal of the Korea Society for Simulation
• /
• v.13 no.3
• /
• pp.1-10
• /
• 2004

The cell loss probability required in the ATM network is in the range of 10$^{-9}$ ∼10$^{-12}$ . If Monte Carlo simulation is used to analyze the performance of the ATM node, an enormous amount of computer time is required. To obtain large speed-up factors, importance sampling may be used. Since the Markov-modulated processes have been used to model various high-speed network traffic sources, we consider discrete time single server queueing systems with Markov-modulated arrival processes which can be used to model an ATM node. We apply importance sampling based on the Large Deviation Theory for the performance evaluation of, MMBP/D/1/K, ∑MMBP/D/1/K, and two stage tandem queueing networks with Markov-modulated arrival processes and deterministic service times. The simulation results show that the buffer overflow probabilities obtained by the importance sampling are very close to those obtained by the Monte Carlo simulation and the computer time can be reduced drastically.

• The Journal of Korean Institute of Communications and Information Sciences
• /
• v.28 no.9B
• /
• pp.783-792
• /
• 2003

We consider a finite-buffer discrete-time queueing system with fixed-size bulk-service discipline: Geo/ $G^{B}$1/K＋B. The main purpose of this paper is to present a performance analysis of this system that has a wide range of applications in Asynchronous Transfer Mode (ATM) and other related telecommunication systems. For this purpose, we first derive the departure-epoch probabilities based on the embedded Markov chain method. Next, based on simple rate in and rate out argument, we present stable relationships for the steady-state probabilities of the queue length at different epochs: departure, random, and arrival. Finally, based on these relationships, we present various useful performance measures of interest such as the moments of number of packets in the system at three different epochs and the loss probability. The numerical results are presented for a deterministic service-time distribution - a case that has gained importance in recent years.s.

• Journal of Korea Society of Industrial Information Systems
• /
• v.23 no.2
• /
• pp.29-39
• /
• 2018

In this paper, we discuss a discrete-time queueing system with dyadic server control policy that combines workload control and multiple vacations. Customers arrive at the system with Bernoulli arrival process. If there is no customer to serve in the system, an idle single server spends a vacation of discrete random variable V and returns. The server repeats the vacation until the total service time of waiting customers exceeds the predetermined workload threshold D. In this paper, we derived the steady-state workload distribution of a discrete-time queueing system which is operating under a more realistic and flexible server control policy. Mean workload is also derived as a performance measure. The results are basis for the analysis of system performance measures such as queue lengths, waiting time, and sojourn time.

• ETRI Journal
• /
• v.25 no.4
• /
• pp.238-246
• /
• 2003

This paper introduces the modeling and analysis of a discrete-time, two-phase queueing system for both exhaustive batch service and gated batch service. Packets arrive at the system according to a Bernoulli process and receive batch service in the first phase and individual services in the second phase. We derive the probability generating function (PGF) of the system size and show that it is decomposed into two PGFs, one of which is the PGF of the system size in the standard discrete-time Geo/G/1 queue without vacations. We also present the PGF of the sojourn time. Based on these PGFs, we present useful performance measures, such as the mean number of packets in the system and the mean sojourn time of a packet.

• 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
• /
• 2002.05a
• /
• pp.859-862
• /
• 2002

In this paper, we consider a discrete-time two-phase queueing system. We derive the PCF of the system size and show that it is decomposed into two probabilty generating functions (PCFs), one of which is the PGF of the system size in the standard Ceo/G/1 queue. Based on this PGF, we present the performance measure of interest such as the mean number of customers In the system.

• 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.23 no.1
• /
• pp.53-63
• /
• 2018

In this paper, we analyze the waiting time of a queueing system with D-BMAP (discrete-time batch Markovian arrival process) and D-policy. Customer group or packets arrives at the system according to discrete-time Markovian arrival process, and an idle single server becomes busy when the total service time of waiting customer group exceeds the predetermined workload threshold D. Once the server starts busy period, the server provides service until there is no customer in the system. The steady-state waiting time distribution is derived in the form of a generating function. Mean waiting time is derived as a performance measure. Simulation is also performed for the purpose of verification and validation. Two simple numerical examples are shown.

• 제어로봇시스템학회:학술대회논문집
• /
• 2004.08a
• /
• pp.1758-1763
• /
• 2004

This paper focuses on congestion control for ATM network with uncertain time-variant delays. The time-variant delays can be distinguished into two distinct components. The first one is represented by time-variant queueing delays in the intermediate switches that are occurred in the return paths of RM cells. The next one is a forward path delay. It is solved by the VBR model which quantifies the data propagation from the sources to the switch. Robust $H_8$ control is studied for solving congestion problem with norm-bounded time-varying uncertain parameters. The suitable robust $H_8$ controller is obtained from the solution of a convex optimization problem through LMI technique.