• Journal of Korean Society of Industrial and Systems Engineering
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• v.42 no.4
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• pp.91-105
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• 2019

Discrete-time Queueing models are frequently utilized to analyze the performance of computing and communication systems. The length of busy period is one of important performance measures for such systems. In this paper, we consider the busy period of the Geo/Geo/1/K queue with a single vacation. We derive the moments of the length of the busy (idle) period, the number of customers who arrive and enter the system during the busy (idle) period and the number of customers who arrive but are lost due to no vacancies in the system for both early arrival system (EAS) and late arrival system (LAS). In order to do this, recursive equations for the joint probability generating function of the busy period of the Geo/Geo/1/K queue starting with n, 1 ≤ n ≤ K, customers, the number of customers who arrive and enter the system, and arrive but are lost during that busy period are constructed. Using the result of the busy period analysis, we also numerically study differences of various performance measures between EAS and LAS. This numerical study shows that the performance gap between EAS and LAS increases as the system capacity K decrease, and the arrival rate (probability) approaches the service rate (probability). This performance gap also decreases as the vacation rate (probability) decrease, but it does not shrink to zero.

• 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.

• Journal of the Korean Operations Research and Management Science Society
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• v.32 no.2
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• pp.141-147
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• 2007

We consider a GI/M/1 queue with vacations such that the server works with different rate rather than completely stops working during a vacation period. We derive the transform of the joint distribution of the length of a busy period, the number of customers served during the busy period, and the length of the subsequent idle period.

• Journal of the Korean Operations Research and Management Science Society
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• v.31 no.1
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• pp.83-90
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• 2006

By defining the partial busy period of the M/M/c/K queueing system as the time interval during which at least one server is in service, we derive the first two moments of both the partial busy period and the number of customers served during it. All expressions are given in explicit forms.

• Communications for Statistical Applications and Methods
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• v.7 no.2
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• pp.489-497
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• 2000

The M/M/1 queue with impatient customers is studied. Impatient customers wait for service only for limited time K/0 and leave the system if their services do not start during that time. Notice that in the analysis of virtual waiting time, the impatient customer can be considered as the customer who enters the system only when his/her waiting time does not exceed K. In this paper, we apply martingale methods to the virtual waiting time and obtain the expected period from origin to the point where the virtual waiting time crosses over K or reaches 0, and the variance of this period. With this results, we obtain the expected busy period of the queue, the distribution, expectation and variance of the number of times the virtual waiting time exceeding level K during a busy period, and the probability of there being no impatient customers in a busy period.

• Journal of the Korea Society for Simulation
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• v.27 no.3
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• pp.45-52
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• 2018

We consider the busy period of an M/G/1/K queueing system with queue-length-dependent overload control policy. A variant of an oscillating control strategy that was recently analyzed by Choi and Kim (2016) is considered: two threshold values, $L_1({\leq_-}L_2)$ and $L_2({\leq_-}K)$, are assumed, and service rate and arrival rate are adjusted depending on the queue length to alleviate congestion. We investigate the busy period of an M/G/1/K queue with two overload control policies, and present the formulae to obtain the expected length of a busy period for each control policy. Based on the numerical examples, we conclude that the variability and expected value of the service time distribution have the most influence on the length of a busy period.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.38 no.1
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• pp.21-29
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• 2015

A steady-state controllable M/G/1 queueing model operating under the {T:Min(T,N)} policy is considered where the {T:Min(T,N)} policy is defined as the next busy period will be initiated either after T time units elapsed from the end of the previous busy period if at least one customer arrives at the system during that time period, or after T time units elapsed without a customer' arrival, the time instant when Nth customer arrives at the system or T time units elapsed with at least one customer arrives at the system whichever comes first. After deriving the necessary system characteristics including the expected number of customers in the system, the expected length of busy period and so on, the total expected cost function per unit time for the system operation is constructed to determine the optimal operating policy. To do so, the cost elements associated with such system characteristics including the customers' waiting cost in the system and the server's removal and activating cost are defined. Then, procedures to determine the optimal values of the decision variables included in the operating policy are provided based on minimizing the total expected cost function per unit time to operate the queueing system under considerations.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.39 no.1
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• pp.81-90
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• 2016

Different from general operating policies to be applied for controllable queueing models, two of three well-known simple N, T and D operating policies are applied alternatingly to the single server controllable queueing models, so called alternating (NT), (ND) and (TD) policies. For example, the alternating (ND) operating policy is defined as the busy period is initiated by the simple N operating policy first, then the next busy period is initiated by the simple D operating policy and repeats the same sequence after that continuously. Because of newly designed operating policies, important system characteristic such as the expected busy and idle periods, the expected busy cycle, the expected number of customers in the system and so on should be redefined. That is, the expected busy and idle periods are redefined as the sum of the corresponding expected busy periods and idle periods initiated by both one of the two simple operating policies and the remaining simple operating policy, respectively. The expected number of customers in the system is represented by the weighted or pooled average of both expected number of customers in the system when the predetermined two simple operating policies are applied in sequence repeatedly. In particular, the expected number of customers in the system could be used to derive the expected waiting time in the queue or system by applying the famous Little's formulas. Most of such system characteristics derived would play important roles to construct the total cost functions per unit time for determination of the optimal operating policies by defining appropriate cost elements to operate the desired queueing systems.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.37 no.1
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• pp.96-103
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• 2014

A steady-state controllable M/G/1 queueing model operating under the (TN) policy is considered where the (TN) policy is defined as the next busy period will be initiated either after T time units elapsed from the end of the previous busy period if at least one customer arrives at the system during that time period, or the time instant when Nth customer arrives at the system after T time units elapsed without customers' arrivals during that time period. After deriving the necessary system characteristics such as the expected number of customers in the system, the expected length of busy period and so on, the total expected cost function per unit time in the system operation is constructed to determine the optimal operating policy. To do so, the cost elements associated with such system characteristics including the customers' waiting cost in the system and the server's removal and activating cost are defined. Then, the optimal values of the decision variables included in the operating policies are determined by minimizing the total expected cost function per unit time to operate the system under consideration.

• International Journal of Reliability and Applications
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• v.12 no.1
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• pp.15-39
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• 2011

This paper deals with the reliability analysis of a complex system with three possibilities at the time of repair. The considered system consists of two subsystems A and Bin series configuration (1-out-of-2: F). Subsystem A has n units which are connected in series whereas subsystem B consists of n units in parallel configuration. The configuration of subsystem A is of 1-out-of-n: F whereas subsystem B is of k-out-of-n: D and k+1-out-of-n: F nature. System has three states: Good, degraded and failed. Supplementary variable technique has been used for mathematical formulation of the model. Laplace transform is being utilized to solve the mathematical equation. Reliability, Availability, M.T.T.F., Busy Period and Cost effectiveness of the system have been computed. The repairs from state $S_7$ to $S_0$, $S_8$ to $S_0$, $S_9$ to $S_0$ and $S_{11}$ to $S_0$ have two types namely exponential and general. Joint probability distribution of repair rate from $S_7$ to $S_0$, $S_8$ to $S_0$, $S_9$ to $S_0$ and $S_{11}$ to $S_0$ is computed by Gumbel-Hougaard family of copula. Some particular cases of the system have also been derived to see the practical importance of the model.