응용통계연구 (The Korean Journal of Applied Statistics)

  • 제28권1호
  • /
  • Pages.33-47
  • /
  • 2015
  • /
  • 1225-066X(pISSN)
  • /
  • 2383-5818(eISSN)
한국통계학회 (The Korean Statistical Society)
권호 표지
DOI QR코드

DOI QR Code

M/En/1 대기모형에서 얼랑분포의 성질을 이용한 오버슛의 분포에 대한 근사

Approximation on the Distribution of the Overshoot by the Property of Erlang Distribution in the M/En/1 Queue

  • Lee, Sang-Gi (Sampling Division, Statistics Korea) ;
  • Bae, Jongho (Department of Information and Statistics, Chungnam National University)
  • 투고 : 2014.10.13
  • 심사 : 2014.12.26
  • 발행 : 2015.02.28
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

본 논문은 $M/E_n/1$ 대기모형에서 작업부하량과정의 오버슛의 분포함수에 대한 근사식을 제안한다. 오버슛이란 작업부하량과정이 미리 정해진 한계점을 처음으로 초과할 때 초과하는 양을 말하는데 정확한 분포함수는 수학적인 표현으로만 얻어졌을 뿐 분포함수를 실제로 계산하는 것은 거의 불가능하다. 그래서 기존 연구에서는 오버슛에 관한 몇가지 성질을 이용하여 오버슛의 분포함수에 대한 근사식이 구해졌다. 본 논문은 고객의 서비스시간의 분포가 얼랑분포라는 점을 활용하여 기존에 얻어진 근사식보다 더 정확한 근사식을 제안한다. 그리고 제안한 근사식이 얼마나 참값에 가까운지 판단하기 위하여 시뮬레이션을 통하여 얻어진 오버슛의 분포함수와 비교한다.

We consider an $M/E_n/1$ queueing model where customers arrive at a facility with a single server according to a Poisson process with customer service times assumed to be independent and identically distributed with Erlang distribution. We concentrate on the overshoot of the workload process in the queue. The overshoot means the excess over a threshold at the moment where the workload process exceeds the threshold. The approximation of the distribution of the overshoot was proposed by Bae et al. (2011); however, but the accuracy of the approximation was unsatisfactory. We derive an advanced approximation using the property of the Erlang distribution. Finally the newly proposed approximation is compared with the results of the previous study.

키워드

참고문헌

  1. Bae, J., Jeong, A. and Kim, S. (2011). An approximation to the overshoot in M/$E_n$/1 queues, The Korean Journal of Applied Statistics, 24, 347{357. https://doi.org/10.5351/KJAS.2011.24.2.347
  2. Bae, J. and Kim, S. (2007). The approximation for the auxiliary renewal function, The Korean Journal of Applied Statistics, 20, 333-342. https://doi.org/10.5351/KJAS.2007.20.2.333
  3. Bae, J., Kim, S. and Lee, E. Y. (2002). A $P_{\lambda}^M$-policy for an M/G/1 queueing system, Applied Mathematical Modelling, 26, 929-939. https://doi.org/10.1016/S0307-904X(02)00045-8
  4. Brandt, A. and Brandt, M. (2004). On the two-classM/M/1 system under preemptive resume and impatience of the prioritized customers, Queueing Systems, 47, 147-168. https://doi.org/10.1023/B:QUES.0000032805.73991.8e
  5. Choi, B. D., Kim, B. and Zhu, D. (2004). MAP/M/c queue with constant impatience time, Mathematics of Operations Research, 29, 309-325. https://doi.org/10.1287/moor.1030.0081
  6. Faddy, M. J. (1974). Optimal control of finite dams: Discrete(2-stage) output procedure, Journal of Applied Probability, 11, 111-121. https://doi.org/10.2307/3212588
  7. Kim, S. and Bae, J. (2008). A G/M/1 queueing system with $P_{\lambda}^M$-service policy, Operations Research Letters, 36, 201-204. https://doi.org/10.1016/j.orl.2007.09.001
  8. Kim, J., Bae, J. and Lee, E. Y. (2006). An optimal $P_{\lambda}^M$-service policy for an M/G/1 queueing system, Applied Mathematical Modelling, 30, 38-48. https://doi.org/10.1016/j.apm.2005.03.007
  9. Lee, E. Y. and Ahn, S. K. (1998). $P_{\lambda}^M$-service policy for a dam with input formed by compound Poisson process, Journal of Applied Probability, 35, 482-488. https://doi.org/10.1239/jap/1032192863
  10. Lee, H. W., Baek, J. W. and Jeon, J. (2005). Analysis of $M^X$/G/1 queue under D-policy, Stochastic Analysis and Applications, 23, 785-808. https://doi.org/10.1081/SAP-200064479
  11. Lee, H. W., Cheon, S. H., Lee, E. Y. and Chae, K. C. (2004). Workload and waiting time analysis of MAP/G/1 queue under D-policy, Queueing Systems, 48, 421-443. https://doi.org/10.1023/B:QUES.0000046584.19533.4b
  12. Lillo, R. E. and Martin, M. (2001). Stability in queues with impatience customers, Stochastic Models, 17, 375-389 https://doi.org/10.1081/STM-100002279
  13. Movaghar, A. (1998). On queueing with customer impatience until the beginning of service, Queueing Systems, 29, 337-350. https://doi.org/10.1023/A:1019196416987
  14. Tijms, H. C. (1986). Stochastic Modelling and Analysis: A Computational Approach, John Wiley & Sons, New York.