Journal of the Korea Society for Simulation (한국시뮬레이션학회논문지)

The Korea Society for Simulation (한국시뮬레이션학회)
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Spreadsheet Model Approach for Buffer-Sharing Line Production Systems with General Processing Times

일반 공정시간을 갖는 버퍼 공유 라인 생산시스템의 스프레드시트 모형 분석

  • Seo, Dong-Won (School of Management and Management Research Institute, Kyung Hee University)
  • Received : 2019.01.24
  • Accepted : 2019.06.21
  • Published : 2019.06.30
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Abstract

Although line production systems with finite buffers have been studied over several decades, except for some special cases there are no explicit expressions for system performances such as waiting times(or response time) and blocking probability. Recently, a max-plus algebraic approach for buffer-sharing systems with constant processing times was introduced and it can lead to analytic expressions for (higher) moment and tail probability of stationary waiting. Theoretically this approach can be applied to general processing times, but it cannot give a proper way for computing performance measures. To this end, in this study we developed simulation models using @RISK software and the expressions derived from max-plus algebra, and computed and compared blocking probability, waiting time (or response time) with respect to two blocking policies: communication(BBS: Blocking Before Service) and production(BAS: Blocking After Service). Moreover, an optimization problem which determines the minimum shared-buffer capacity satisfying a predetermined QoS(quality of service) is also considered.

유한 버퍼를 갖는 라인 생산시스템은 오랜 기간 동안 연구되어왔지만, 몇몇 특별한 경우 외에는 대기시간(체류시간), 차단 확률과 같은 시스템 성능 값에 대한 분석 결과는 많지 않다. 최근에, max-plus 대수를 활용하여 상수 공정시간을 갖고 버퍼 완전 공유 정책을 따르는 시스템에서 대기시간의 고차평균과 꼬리확률에 대한 분석 결과가 소개되었다. 이와 같은 max-plus 대수를 활용한 분석이 이론적으론 일반 공정시간 모형에도 응용 가능하지만, 도출된 표현식에 대한 적절한 계산방법을 제공하지 못한다. 이러한 이유로, 본 연구에서는 max-plus 대수로 도출된 표현식과 @RISK 소프트웨어를 활용하여 스프레드시트 시뮬레이션 모형을 개발하고, 두 가지 차단정책(통신차단과 제조차단) 하에서 시스템 특성값인 대기시간(또는 체류시간)과 차단확률을 비교 분석하였다. 또한 차단확률에 대한 제약을 만족하는 공유 버퍼의 크기를 결정하는 최적화 문제도 분석하였다.

Keywords

SMROBX_2019_v28n2_119_f0001.png 이미지

Fig. 1. m-node line production system with a finite buffer of size K shared by m nodes

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Fig. 2. 2-node tandem line production system with a infinite buffer between nodes

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Fig. 3. 2-node tandem line production system with a finite buffer of size 3 between nodes

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Fig. 4. m-node line production system with a dummy node and a finite buffer of size K shared by m nodes

Table 1. Starting Epoch at each node under BBS and BAS with K = 4 and K = 3

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Table 2. Starting Epoch difference between finite and infinite K under BBS(4)

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Table 3. 95% Confidence Interval for waiting time at each node and response time at node 4

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Table 4. Mean Actual Waiting Time at each node

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Table 5. Blocking Probabilities at node 1

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Table 6. RISKOptimizer Results for BAS

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Table 7. RISKOptimizer Results for BBS

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References

  1. @RISK (2019) http://www.palisade.com/risk (Accessed January 21, 2019).
  2. Ayhan, H. and D.-W. Seo (2001) "Laplace transform and moments of waiting times in Poisson driven (Max,+) linear systems", Queueing Systems, Vol. 37, No. 4, pp. 405-438. https://doi.org/10.1023/A:1010845618420
  3. Ayhan, H. and D.-W. Seo (2002) "Tail probability of transient and stationary waiting times in (Max,+) linear systems", IEEE Transaction on Automatic Control, Vol. 47, No. 1, pp. 151-157. https://doi.org/10.1109/9.981736
  4. Baccelli, F., G. Cohen, G. J. Olsder and J-P. Quadrat (1992) Synchronization and Linearity: An Algebra for Discrete Event Systems, John Wiley and Sons.
  5. Baccelli, F., S. Hasenfuss and V. Schmidt (1998) "Expansions for steady state characteristics in (Max,+) linear systems", Stochastic Models, Vol. 14, pp. 1-24. https://doi.org/10.1080/15326349808807458
  6. Chao, X, M. Miyazawa and M. Pinedo (1999) Queueing Networks: Customers, Signals and Product Form Solutions, John Wiley & Sons, Inc, New York.
  7. Heidergott, B. (2006) Max-plus linear stochastic systems and perturbation analysis, Springer.
  8. Hillier, F. S. (1995) "On the optimal design of tandem queueing systems with finite buffers", Queueing Systems, Vol. 21, pp. 245-266. https://doi.org/10.1007/BF01149164
  9. Lee, H. and D.-W. Seo (2012) "Comparison of DBR with CONWIP in a Production Line with Constant Processing Times", Journal of the Korea Society for Simulation, Vol. 21, No. 4, pp. 11-24. https://doi.org/10.9709/JKSS.2012.21.4.011
  10. Lee, H. and D.-W. Seo (2015) "Explicit expressions for moments of waiting times in Poisson driven deterministic two-node tandem queues with blocking", Operations Research Letters, Vol. 43, pp. 203-208. https://doi.org/10.1016/j.orl.2015.02.002
  11. Narasimhamu, K. L., V. Venugopal Reddy and C. S. P. Rao (2014) "Optimal Buffer Allocation in Tandem Closed Queuing Network with Single Server using PSO", Procedia Materials Science, Vol. 5, pp. 2084-2089. https://doi.org/10.1016/j.mspro.2014.07.543
  12. Neuts, M. (1998) Advance in Matrix Analytic Methods for Stochastic Models, ch. Some Promising Directions in Algorithmic Probability, Notable Publications Inc.
  13. Seo, D.-W., and H. Lee (2011) "Stationary waiting times in m-node tandem queues with production blocking", IEEE Transactions on Automatic Control, Vol. 56, No. 4, pp. 958-961. https://doi.org/10.1109/TAC.2011.2105290
  14. Seo, D.-W., J. Lee and B.-Y. Chang (2015) "An Approximation Method for Blocking Probabilities in M/D/1/K1${\rightarrow}{\cdot}$/D/1/K2 Queues", Asia-Pacific Journal of Operational Research, Vol. 32, No. 3, 1550017(12 pages). https://doi.org/10.1142/S0217595915500177
  15. Takagi, H. (1993) Queueing Analysis, in: Finite Systems, vol. II, North-Holland.
  16. Tweedie, R. (1982) "Operator-geometric stationary distributions for markov chains, with applications to queueing models", Advanced Applied Probability, Vol. 12, pp. 368-391. https://doi.org/10.2307/1426527
  17. Yang, D. and D.-W. Seo (2017) "Closed-form formulae for moment, tail probability, and blocking probability of waiting time in a buffer-sharing deterministic system", Operations Research Letters, Vol. 45, pp. 403-408. https://doi.org/10.1016/j.orl.2017.06.004
  18. Zhou, W. and Z. Lian Z. (2011) "A tandem network with a sharing buffer", Applied Mathematical Modelling, Vol. 35, pp. 4507-4515. https://doi.org/10.1016/j.apm.2011.03.018