本研究探討一個具有多重休假、工作故障與兩個異質服務者 (服務者 1 和服務者 2) 之 M/M/2 排隊系統，服務者 1 是可靠的，當系統內沒有顧客時，服務者 1 會離開工作崗位進行休假。而服務者 2 是不可靠的，當服務者 2 提供服務時，隨時可能會發生故障並立即進行修復，當服務者 2 發生故障時，並非完全停止提供服務，而是改以較低的服務率繼續提供顧客服務。對於此排隊模型，我們將利用矩陣幾何法推導系統中顧客數的穩態機率分佈，並發展出不同的系統績效量測。我們建構一個單位時間期望成本函數，並利用典型粒子群最佳化演算法尋找服務者 1 及服務者 2 的近似最佳服務率。我們提供數值結果，說明各個系統參數對系統績效量測及近似最佳服務率之影響。最後，我們提供一個實際案例，來說明此排隊系統在實務上的應用。

This thesis analyzes an M/M/2 queue with multiple vacations, working breakdowns and heterogeneous servers (Server 1 and Server 2). Server 1 is reliable and leaves for a vacation when the system becomes empty. Sever 2 is unreliable and may break down unexpectedly while serving customers. When a breakdown occurs, Server 2 still works rather than halting service. For this queueing system, we use the matrix geometric method to compute the stationary distribution of system size and develop several system performance measures. We formulate a cost optimization model, and employ the canonical particle swarm optimization algorithm to obtain the optimal service rates of Server 1 and Server 2. Numerical examples are given to show the effects of system parameters on the approximated optimal service rates. Finally, we present a practical example to illustrate the application of this system.

摘要 i
Abstract ii
誌謝 iii
目錄 iv
圖目錄 v
表目錄 vii
第一章 緒論 1
1.1 研究背景與動機 1
1.2 文獻探討 2
1.3 系統描述 4
1.4 論文架構 6
第二章 穩態分析 7
2.1 穩態條件 9
2.2 穩態機率解 10
第三章 系統績效評估 13
3.1 系統績效量測 13
3.2 數值結果 16
第四章 成本模型與最佳化分析 40
4.1 成本函數 40
4.2 典型粒子群最佳化演算法 41
4.3 數值結果 45
4.4 應用案例 48
第五章 結論與未來研究 51
5.1 結論 51
5.2 未來研究 51
參考文獻 52

1. Al-Seedy, R. O., El-Sherbiny, A. A., El-Shehawy, S. A. and Ammar, S. I. (2009). Transient solution of the M/M/c queue with balking and reneging. Computers & Mathematics with Applications, 57, 1280-1285.
2. Ammar, S. I. (2014). Transient analysis of a two-heterogeneous servers queue with impatient behavior. Journal of the Egyptian mathematical society, 22, 90-95.
3. Carlisle, A. and Dozier, G. (2001). An off-the-shelf PSO. In Proceeding of the workshop on particle swarm optimization, Indianapolis: Purdue School of Engineering and Technology.
4. Chen, J.-Y., Yen, T.-C. and Wang, K.-H. (2016). Cost optimization of a single-server queue with working breakdowns under the N policy. Journal of Testing and Evaluation, 44, 2059-2067.
5. Clerc, M. and Kennedy, J. (2002). The particle swarm - explosion, stability, and convergence in a multidimensional complex space. IEEE Transactions on Evolutionary Computation, 6, 58-73.
6. Del Valle, Y., Venayagamoorthy, G. K., Mohagheghi, S., Hernandez, J. C. and Harley, R. G. (2008). Particle swarm optimization: basic concepts, variants and applications in power systems. IEEE Transactions on evolutionary computation, 12, 171-195.
7. Doshi, B. T. (1986). Queueing systems with vacations––a survey. Queueing Systems, 1, 29-66.
8. Eberhart, R. C. and Shi, Y. (2001). Particle swarm optimization: developments, applications and resources. In Proceeding IEEE international conference on Evolutionary Computation, 1, 81–86.
9. Goswami, V. (2014). Analysis of discrete-time multi-server queue with balking. International Journal of Management Science and Engineering Management, 9, 21-32.
10. Gun, L. (1989). Experimental results on matrix-analytical solution techniques–extensions and comparisons. Stochastic Models, 5, 669-682.
11. Jain, M. and Bhagat, A. (2016). Mx/G/1 retrial vacation queue for multi-optional services, phase repair and reneging. Quality Technology & Quantitative Management, 13, 263-288.
12. Jiang, T. and Xin, B. (2018). Computational analysis of the queue with working breakdowns and delaying repair under a Bernoulli-schedule-controlled policy. Communications in Statistics-Theory and Methods, In Press.
13. Kalidass, K. and Kasturi, R. (2012). A queue with working breakdowns. Computers & Industrial Engineering, 63, 779-783.
14. Ke, J.-C., Wu, C.-H. and Pearn, W. L. (2013). Analysis of an infinite multi-server queue with an optional service. Computers & Industrial Engineering, 65, 216-225.
15. Ke, J.-C., Wu, C.-H. and Zhang, Z.-G. (2010). Recent developments in vacation queueing models: a short survey. International Journal of Operations Research, 7, 3-8.
16. Kennedy, J. and Eberhart, R. (1995). Particle swarm optimization. In Proceedings of the IEEE international conference on neural networks (pp. 1942–1948). Washington DC: IEEE Computer Society.
17. Kim, B. K. and Lee, D. H. (2014). The M/G/1 queue with disasters and working breakdowns. Applied Mathematical Modelling, 38, 1788-1798.
18. Krishnamoorthy, A. and Sreenivasan, C. (2012). An M/M/2 queueing system with heterogeneous servers including one with working vacation. International Journal of Stochastic Analysis, 2012, doi:10.1155/2012/145867.
19. Kumar, B. K. and Madheswari, S. P. (2005). An M/M/2 queueing system with heterogeneous servers and multiple vacations. Mathematical and Computer Modelling, 41, 1415-1429.
20. Latouche, G. and Ramaswami, V. (1993). A logarithmic reduction algorithm for quasi-birth-death processes. Journal of Applied Probability, 30, 650-674.
21. Levy, Y. and Yechiali, U. (1975). Utilization of idle time in an M/G/1 queueing system. Management Science, 22, 202-211.
22. Li, T. and Zhang, L. (2017). An M/G/1 retrial G-queue with general retrial times and working breakdowns. Mathematical and Computational Applications, 22, 15.
23. Liou, C.-D. (2015). Markovian queue optimisation analysis with an unreliable server subject to working breakdowns and impatient customers. International Journal of Systems Science, 46, 2165-2182.
24. Liu, Z. and Song, Y. (2014). The MX/M/1 queue with working breakdown. RAIRO-Operations Research, 48, 399-413
25. Neuts, M. (1981). Matrix-geometric Solutions in Stochastic Models. Baltimore: John Hopkins University Press.
26. Shi, Y. and Eberhart, R. C. (1998). A modified particle swarm optimizer. In Proceedings of the IEEE International Conference on Evolutionary Computation, 69–73.
27. Sudhesh, R., Savitha, P. and Dharmaraja, S. (2017). Transient analysis of a two-heterogeneous servers queue with system disaster, server repair and customers’ impatience. Top, 25, 179-205.
28. Takagi, H. (1991). Queueing Analysis, vol. 1. Elsevier Science Publishers, Amsterdam.
29. Teghem Jr, J. (1986). Control of the service process in a queueing system. European Journal of Operational Research, 23, 141-158.
30. Tian, N. and Zhang, Z.-G. (2006). Vacation queueing models: Theory and Applications (Vol. 93). Springer Science & Business Media.
31. Upadhyaya, S. (2016). Queueing systems with vacation: an overview. International Journal of Mathematics in Operational Research, 9, 167-213.
32. Wang, J. and Zhang, F. (2011). Equilibrium analysis of the observable queues with balking and delayed repairs. Applied Mathematics and Computation, 218, 2716-2729.
33. Wang, J., Zhang, X. and Huang, P. (2017). Strategic behavior and social optimization in a constant retrial queue with the N-policy. European Journal of Operational Research, 256, 841-849.
34. Yang, D.-Y. and Chen, Y.-H. (2018). Computation and optimization of a working breakdown queue with second optional service. Journal of Industrial and Production Engineering, 35, 181-188.
35. Yang, D.-Y., Chiang, Y.-C. and Tsou, C.-S. (2013). Cost analysis of a finite capacity queue with server breakdowns and threshold-based recovery policy. Journal of Manufacturing Systems, 32, 174-179.
36. Yang, D.-Y. and Wu, C.-H. (2015). Cost-minimization analysis of a working vacation queue with N-policy and server breakdowns. Computers & Industrial Engineering, 82, 151-158.
37. Yang, D.-Y. and Wu, Y.-Y. (2017). Analysis of a finite-capacity system with working breakdowns and retention of impatient customers. Journal of Manufacturing Systems, 44, 207-216.
38. Ye, Q. and Liu, L. (2018). Analysis of MAP/M/1 queue with working breakdowns. Communications in Statistics-Theory and Methods, 47, 3073-3084.
39. Yen, T.-C., Chen, W.-L. and Chen, J.-Y. (2016). Reliability and sensitivity analysis of the controllable repair system with warm standbys and working breakdown. Computers & Industrial Engineering, 97, 84-92.
40. Yue, D., Yu, J. and Yue, W. (2009). A Markovian queue with two heterogeneous servers and multiple vacations. Journal of Industrial & Management Optimization, 5, 453-465.
41. Zhang, Y. and Wang, J. (2017). Equilibrium pricing in an M/G/1 retrial queue with reserved idle time and setup time. Applied Mathematical Modelling, 49, 514-530.
42. Zhang, X., Wang, J. and Ma, Q. (2017). Optimal design for a retrial queueing system with state-dependent service rate. Journal of Systems Science and Complexity, 30, 883-900.