• Title/Summary/Keyword: Busy Period

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BUSY PERIOD DISTRIBUTION OF A BATCH ARRIVAL RETRIAL QUEUE

  • Kim, Jeongsim
    • 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.

Busy Period Analysis of the Geo/Geo/1/K Queue with a Single Vacation (단일 휴가형 Geo/Geo/1/K 대기행렬의 바쁜 기간 분석)

  • Kim, Kilhwan
    • 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.

Busy Period Analysis for the GI/M/1 Queue with Working Vacations (워킹 휴가형 GI/M/1 대기행렬의 바쁜기간 분석)

  • Chae, Kyung-Chul;Lim, Dae-Eun
    • 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.

Derivation of the Expected Busy Period for the Controllable M/G/1 Queueing Model Operating under the Triadic Policy using the Pseudo Probability Density Function (삼변수운용방침이 적용되는 M/G/1 대기모형에서 가상확률밀도함수를 이용한 busy period의 기대값 유도)

  • Rhee, Hahn-Kyou;Oh, Hyun-Seung
    • Journal of Korean Society of Industrial and Systems Engineering
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    • v.30 no.2
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    • pp.51-57
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    • 2007
  • The expected busy period for the controllable M/G/1 queueing model operating under the triadic policy is derived by using the pseudo probability density function which is totally different from the actual probability density function. In order to justify the approach using the pseudo probability density function to derive the expected busy period for the triadic policy, well-known expected busy periods for the dyadic policies are derived from the obtained result as special cases.

Development of the Most Generalized Form of the Triadic Operating Policy and Derivation of its Corresponding Expected Busy Period (가장 일반화된 형태의 삼변수 운용방침 개발과 그에 따른 Busy Period 기대값 유도)

  • Rhee, Hahn-Kyou;Oh, Hyun-Seung
    • Journal of Korean Society of Industrial and Systems Engineering
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    • v.32 no.4
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    • pp.161-168
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    • 2009
  • The most generalized form of the triadic operating policy for an M/G/1 queueing model is developed. It consists of three simple N, T and D operating policies and has a peculiar structure possessing concepts of dyadic policies. Using the concept of the pseudo probability density function of the busy period, its expected busy period for the controllable M/G/1 queueing model is derived. Since the obtained result is the most generalized form the triadic polity, the expected busy periods for all known dyadic policies are recovered as special cases from it.

Upper and Lower Bounds of the Expected Busy Period for the Triadic Med(N, T, D) Policy (삼변수 Med(N, T, D) 운용방침에 따른 Busy Period 기대값의 상한과 하한 유도)

  • Rhee, Hahn-Kyou
    • Journal of Korean Society of Industrial and Systems Engineering
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    • v.36 no.1
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    • pp.58-63
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    • 2013
  • Using the known result of the expected busy period for the triadic Med (N, T, D) operating policies applied to a controllable M/G/1 queueing model, its upper and lower bounds are derived to approximate its corresponding actual values. Both bounds are represented in terms of the expected busy periods for the dyadic Min (N, T), Min (N, D) and Min (T, D) or Max (N, T), Max (N, D) and Max (T, D) with the simple N, T and D operating policies without using any other types of triadic operating policies such as Min (N, T, D) and Max (N, T, D) policies. All three input variables N, T and D are equally contributed to construct such bounds for estimation of the expected busy period.

A Busy Period Analysis for the M/M/c/K Queueing System (M/M/c/K 대기행렬 시스템의 바쁜 기간 분석)

  • Lim Dae-Eun;Chae Kyung-Chul
    • 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.

The Busy Period of the M/M/1 Queue with Bounded Workload

  • Bae, Jong-Ho
    • Proceedings of the Korean Statistical Society Conference
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    • 2002.11a
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    • pp.273-277
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    • 2002
  • In this paper, with martingale argument we derive the explicit formula for the Laplace transform of the busy period of M/M/1 queue with bounded workload which is also called finite dam. Much simpler derivation than appeared in former literature provided.

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Development of a New Methodology to find the Expected Busy Periods for Controllable M/G/1 Queueing Models Operating under the Multi-variable Operating Policies: Concepts and applications to the dyadic policies

  • Rhee, Hahn-Kyou
    • Journal of Korean Institute of Industrial Engineers
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    • v.23 no.4
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    • pp.729-739
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    • 1997
  • In this paper, steady-state controllable M/G/1 queueing systems operating under the dyadic policies are considered. A new method to obtain the expected busy period when the D-policy is involved in system operation, is developed. This new method requires derivation of so called 'the pseudo probability density function' of the busy period for the system under consideration, which is completely different from its actual probability density function. However, the proposed pseudo probability density function does generate the correct expected busy period through simple procedures.

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