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

다국어 초록 (Multilingual Abstract)

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.

참고문헌 (Reference)

1 김길환, "엄격한 T‐축출 우선순위 대기행렬을 이용한 기회 주파수 접근 방식의 성능 분석" 한국산업경영시스템학회 35 (35): 162-170, 2012

2 Cohen, J.W., "The single server queue" North-Holland Amsterdam 1982

3 Harris, T. J., "The remaining busy period of a finite queue" 19 (19): 219-223, 1971

4 Gravey, A., "Simultaneity in discretetime single server queues with Bernoulli inputs" 14 (14): 123-131, 1992

5 Takagi, H, "Queueing analysis, Volume 3 : Discrete-time systems" North-Holland 1993

6 Takagi, H, "Queueing analysis, Volume 1 : Vacation and priority systems, part 1" North-Holland 1991

7 Harchol-Balter, M., "Performance modeling and design of computer systems : Queueing theory in action" Cambridge University Press 2013

8 Cooper, R. B., "On the relationship between the distribution of maximal queue length in the M/G/1queue and the mean busy period in the M/G/1/N queue" 13 (13): 195-199, 1976

9 Shanthikumar, J. G., "On the busy-period distributions of M/G/1/K queues by state-dependent arrivals and FCFS/LCFS-p service disciplines" 22 (22): 912-919, 1985

10 Pacheco, A., "Moments of the duration of busy periods of Mx/G/1/N systems" 22 (22): 347-354, 2008

11 Ferreira, F., "Moments of losses during busy-periods of regular and nonpreemptive oscillating MX/G/1/n systems" 252 (252): 191211-, 2017

12 Lee, T. T., "M/G/1/N queue with vacation time and exhaustive service discipline" 32 (32): 774-784, 1984

13 Wilf, H. S., "Generating functionology" AK Peters/CRC Press 2005

14 Chaudhry, M. L., "First-passage-time and busy-period distributions of discrete-time Markovian queues : Geom (n)/Geom (n)/1/n" 18 (18): 526-, 1994

15 Takagi, H., "Explicit probability density function for the length of a busy period in an M/M/1/K queue" Springer 213-226,

16 Bruneel, H., "Discrete-time models for communication systems including ATM" Kluwer Academic Publishers 1993

17 Al Hanbali, A., "Busy period analysis of the state dependent M/M/1/K queue" 38 (38): 1-6, 2010

18 Al Hanbali, A., "Busy period analysis of the level dependent PH/PH/1/K queue" 67 (67): 221-249, 2011

19 Yu, M., "A simple method to obtain the stochastic decomposition structure of the busy period in Geo/Geo/1/N vacation queue" 13 (13): 361380-, 2015

20 Miller, L. W., "A note on the busy period of an M/G/1finite queue" 23 (23): 1179-1182, 1975

21 김길환, "(N, n)-Preemptive Repeat-Different Priority Queues" 한국산업경영시스템학회 40 (40): 66-75, 2017