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標題: G/G/R機器修理問題含有阻礙及放棄之比較分析
Comparative Analysis for the G/G/R Machine Repair Problem with Balking and Reneging
作者: 鐘啟元
Chung, Chi-Yuan
關鍵字: G/G/R machine repair problem
diffusion approximation
heavy traffic
renewal theory
出版社: 統計學研究所
引用: [1] Heyman, D. P., A Diffusion Model Application for the GI/G/1 Queue in Heavy Traffic, The Bell System Technical Journal, 54, 1637-1646 (1975). [2] Halachmi, B. and Franta, W. R., A Diffusion Approximation to the Multi-Server Queue, Management Science, 24(5), 552-559 (1978). [3] Haryono and Sivazlian, B. D., Analysis of the Machine Repair Problem: A Diffusion Process Approach, Mathematics and Computers in Simulation, 27, 339-364 (1985). [4] Halfin, S. and Whitt, W., Heavy Traffic Limits for Queues with Many Exponential Servers, Operations Research, 29(3), 567-588 (1981). [5] Jain, M., Diffusion approximation for GIx/G/m machine interference problem with spare machines, Microelectonics and Reliability, 33(9), 1415-1418 (1993). [6] Jain, M., An (m,M) Machine Repair Problem with Spares and State Dependent Rate: A Diffusion Process Approach, Microelectonics and Reliability, 37(6), 929-933 (1997). [7] Jain, M., Diffusion process for /G/m queuing system with balking and reneging, International Journal of Engineering, 16(1), 47-54 (2003). [8] Jain, M., Sharma, G.C. and Singh, M., G/G/r machine repair problem with spares and additional repairmen, International Journal of Engineering, 15(1), 57-62 (2002). [9] Jain, M., Sharma, G. C. and Baghel, K.P.S., Diffusion Process for G/G/R machine system with spares, balking and reneging, IJE Transactions A: Basics, 19(1), 49-54 (2006). [10] Jain, M., Singh, M. and Sharma, G. C., Diffusion Process for Multi-Repairmen Machining System with Spares and Balking, International Journal of Engineering,15(1), 49-56 (2002). [11] Kimura, T., Diffusion Approximation for an M/G/m Queue, Operations Research,31(2), 304-321 (1983). [12] Kimura, T., Diffusion Approximations for Queues with Markovian Bases, Annals of Operations Research 113, 27-40 (2002). [13] Kingman, J. F. C., The Heavy Traffic Approximation in the Theory of Queues,Proceedings of Symposium on Congestion Theory, University of North Carolina Press, Chapel Hill, NC, 137-159 (1965). [14] Lee, H. W., Yoon, S. H. and Lee, S. S., Continuous approximations of machine repair system, Applied Mathematical Modeling, 19, 550-559 (1995). [15] Sweet, A. L. and Hardin, J. C., Solutions for Some Diffusion Processes with Two Barriers, Journal of Applied Probability, 7, 423-431 (1970). [16] Sunaga, T., Kondo, E. and Biswas, S. K., An Approximation Method Using Continuous Models for Queueing Problems, Journal of the Operations Research Society of Japan, 21(1), 29-42 (1978). [17] Sivazlian, B. D. and Wang, K. H., System Characteristics and Economic Analysis of the G/G/R Machine Repair Problem with Warm Standbys Using Diffusion Approximation, Microelectonics and Reliability, 29(5), 829-848 (1989). [18] Sivazlian, B. D. and Wang, K. H., Diffusion Approximation to the G/G/R Machine Repair Problem with Warm Standby Spares, Naval Research Logistics, 37, 753-772 (1990). [19] Whitt, W., Refining Diffusion Approximations for Queues, Operations Research Letters, 1(5), 165-169 (1982). [20] Yao, D. D., Refining the Diffusion Approximation for the M/G/m Queue, Operations Research, 33(6), 1266-1277 (1985).
摘要: 此篇論文透過擴散過程研究含有阻礙及放棄的G/G/R機器修理問 題。由受限於兩個反映屏障的Fokker-Planck方程式得到穩態擴散方程式。在交通繁重(heavy traffic)的條件下,也就是說,壞掉機器的數量在修理的狀態下幾乎額滿,藉由更新定理(renewal theory)的基本觀念可得到擴散方程式裡的擴散參數的近似表示式。因此我們可以得到壞掉機器數的近似機率密度函數。另外,我們計算一些系統執行測度,並且和實際的結果做比較。根據數據上顯示,對於複雜的機器修理問題,擴散近似法是一個有效率且有用的方法。
This thesis studies the G/G/R machine repair problem with balking and reneging via diffusion approximation. The steady-state diffusion equations are obtained from the Fokker-Planck equations subject to two reflecting barriers. The approximate expressions for the diffusion parameters of the diffusion equations are obtained by applying the basic concepts of the renewal theory under the heavy traffic conditions, that is, the queue of failed machines in the repair state are nonempty in most case all the time. Expressions for the approximate probability density functions of the number of the failed machines in the system are then obtained. Numerical results indicate that the diffusion approximation is an efficient and powerful method for solving complicated machine repair problems.
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