誰想計算等候時間？Computing Waiting Time! Who Cares?

1. 誰想計算等候時間？ Computing Waiting Time! Who Cares? 政治大學 應用數學系 陸 行 Hsing Luh Department of Mathematical Sciences

3. 多線排隊

4. 單線排隊 多人服務

7. 到達時間是隨機變數

8. 服務時間是隨機變數

9. 等候時間是隨機變數

11. 狀態平衡方程式 Courtesy to Internet Information

12. n-1 n n+1         Courtesy to Internet Information 狀態平衡方程式

13. 長時間的觀察值 Courtesy to Internet Information n-1 n n+1        

19. State dependent M/M/1 Queueing Systems 0 1 2 3 n-1 n n+1  (0)  (1)  (2)  (3)  (n-2)  (n-1)  (n)  (n+1)  (1)  (2)  (3)  (4)  (n-1)  (n)  (n+1)  (n+2) Courtesy to Internet Information

21. Example 1 Suppose interarrival time is 12 min and service time is 10 minutes per job. . M/M/1:

23. Example 2 Suppose interarrival time is 12 min and service time is 10 minutes per job. . M/M/2:

24. 比比看 Two M/M/1 One M/M/2

25. 平均等候時間 平均服務率 M/M/1 平均到達率 P{idle time} E(waiting time) × 2

26. 平均等候時間 平均服務率 M/M/2 平均到達率 P{idle time} } E(waiting time}

27. Example 3 Suppose interarrival time is 12 min and service time is 10 minutes per job. . Two M/M/1: One M/M/2: × 2

28. M/M/2 與 M/M/1 的比較 < < < × 2 × 2 in queue) <

29. 等候時間的機率分配函數

31. 佇列的等候機率

32. 時間延誤問題 系統模擬

33. 系統模擬 (1)

34. 系統模擬 (2)

35. 系統模擬 (3)

36. 交通號誌

37. 匝道管制

38. 通訊問題

39. 通訊服務

40. An optimization model Minimize Penalty made by unsatisfied traffic aggregates Subject to Fixed resource (Limited Short Path bandwidth) Traffic aggregate QDF budget must be satisfied Allocate resource to traffic aggregates Trading off QDF with bandwidth in each Short Path 最佳化模型

44. A Network Optimization Model

53. Numerical Results

54. Numerical Results

55. Numerical Results

56. Numerical Results

57. Numerical Results

61. 運 OR 籌 資 訊 Information 管 理 Management 科 學 Science

62. By Bertrand Mareschal 1. Introduction History. Methodology. Models. 2. Linear programming Applications. Simplex algorithm. Sensitivity analysis. Integer programming. 3. Network analysis Transportation problems. Maximum flow problems. CPM and PERT – Project management. 4. Inventory management Deterministic EOQ models. Probabilistic models. Production management 5. Multicriteria decision aid Methods. Multicriteria decision support systems. Group decision systems. Applications.

63. 模式與方法 Models and Methodologies Models deterministic probabilistic -------------------------------------------------- optimization / OR OR solutions Methodologies simulation --------------------------------------------------    

66. 供應鏈 Supply Chain Management The Goal: A Process of Ongoing Improvement by Eliyahu M. Goldratt

67. 推銷員問題 Traveling Salesman Problem TSP http://www.ing.unlp.edu.ar/cetad/mos/TSPBIB_home.html O(2^n) 2^50 ~10^15 3*10^7* 10^6 ~ 30 years

68. BIOINFORMATICS 生物資訊 http:// motif.stanford.edu/index.html

69. CRYTOGRAPHY 密碼學 Elliptic Curve http:// www.certicom.ca

70. 資源 Resource 最佳值 Optimum 有效調度 Adjustment 配置 Allocation 目標 Objective 動態系統

71. 最佳解 改進 最佳解 可行解

72. No interac- tion allowed 2 1 Individual sampling model (ISM): Simulated patient-level Markov model (SPLMM) (variations as in quadrant below for patient level models with interaction) Simulated Markov model (SMM) Markov model (evaluated deterministically) Timed Individual sampling model (ISM): Simulated patient-level decision tree (SPLDT) Simulated decision tree (SDT) Decision tree rollback Untimed Non-Markovian, discrete-state, individuals Markovian, discrete state, individuals Markovian, discrete state, stochastic Expected value, continuous state, deterministic Individual level Cohort/aggregate level/counts D C B A

73. Discrete event simulation (CT, DES) Discrete individual simulation (DT, DES) Interaction allowed 4 3 Continuous time Individual event history model (CT, IEH) Continuous time Markov chain model (CTMC) System dynamics (ordinary differential equations, ODE) Continu-ous time Discrete-time individual event history model (DT, IEH) Discrete time Markov chain model (DTMC) System dynamics (finite difference equations, FDE) Discrete time Non-Markovian, discrete-state, individuals Markovian, discrete state, individuals Markovian, discrete state, stochastic Expected value, continuous state, deterministic Individual level Cohort/aggregate level/counts D C B A

75. 謝 謝 大 家