1. Simulation

2. Inverse Functions Actually, we’ve already done this with the normal distribution.

3. Inverse Normal Actually, we’ve already done this with the normal distribution. x 3.0  0.1 x =  +  z = 3.0 + 0.3 x 1.282 = 3.3846    X Z

4. 0 Inverse Exponential Exponential Life 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 00.511.522.53 Time to Fail Density a fxe x ()   f(x) FaXa()Pr{}     edx x a 0   e xa   1e a

5. Inverse Exponential xF X e   1 - )( F(x) x

6. Inverse Exponential aF a e  = 1 - )( F(x) x F(a) a

7. Inverse Exponential Suppose we wish to find a such that the probability of a failure is limited to 0.1. F(x) x F(a) a

8. Inverse Exponential Suppose we wish to find a such that the probability of a failure is limited to 0.1. 0.1 = 1 - a e  F(x) x F(a) a

9. Inverse Exponential Suppose we wish to find a such that the probability of a failure is limited to 0.1. 0.1 = 1 - ln(0.9) = - a F(x) x F(a) a a e 

10. Inverse Exponential Suppose we wish to find a such that the probability of a failure is limited to 0.1. 0.1 = 1 - ln(0.9) = - a a e  F(x) x F(a) a a = - ln(0.9)/

11. Inverse Exponential Suppose a car battery is governed by an exponential distribution with = 0.005. We wish to determine a warranty period such that the probability of a failure is limited to 0.1. a = - ln(0.9)/ = - (-2.3026)/0.005 = 21.07 hrs. F(x) x F(a) a

12. Model Customers arrive randomly in accordance with some arrival time distribution. One server services customers in order of arrival. The service time is random following some service time distribution.

13. M/M/1 Queue M/M/1 Queue assumes exponential interarrival times and exponential service times Ae i A i   Se i S i    

14. M/M/1 Queue M/M/1 Queue assumes exponential interarrival times and exponential service times Ae i A i   Se i S i     Exponential Review Expectations

15. M/M/1 Queue 2.032 1.951 1.349.795.539.347 0.305 0.074 0.035 0.520 1.535 0.159

16. M/M/1 Queue 2.032 1.951 1.349.795.539.347 0.305 0.074 0.035 0.520 1.535 0.159

17. M/M/1 Queue.347

18. M/M/1 Queue.347

19. M/M/1 Event Calendar

20. M/M/1 Queue.539.347 0.305

21. M/M/1 Event Calendar

22. M/M/1 Queue.5390.652.795

23. M/M/1 Queue.539 0.074 0.652.795

24. M/M/1 Event Calendar

25. M/M/1 Queue 0.726.795

26. M/M/1 Event Calendar

27. M/M/1 Queue.795

28. M/M/1 Event Calendar

29. M/M/1 Queue.795 0.035 0.8301.349

30. M/M/1 Queue 0.830 1.349

31. M/M/1 Event Calendar

32. M/M/1 Queue 1.349

33. M/M/1 Event Calendar

34. M/M/1 Queue 1.349 0.520 1.8691.951

35. M/M/1 Queue 1.869 1.951

36. M/M/1 Event Calendar

37. M/M/1 Queue 2.032 1.951 1.349.795.539.347 0.305 0.074 0.035 0.520 1.535 0.159

38. M/M/1 Event Calendar

39. M/M/1 Performance Measures

42. Applications; Financial

43. Generation of Random Variables Random Numbers Table of Random Numbers Built in Functions - Rand() Congruential Mixed Mulitiplicative Additive Random Obs. from a Prob. Distribution Inverse Transformation Acceptance / Rejection Special Cases

44. Random Generation Rule: Start anywhere and read sequentially down or across to required significant digits.

45. Random Generation Rule: Start anywhere and read sequentially down or across to required significant digits.

46. Mixed Congruential xaxcm nn   1 ()(mod) X 0 =seed a and c < m

47. Mixed Congruential xaxcm nn   1 ()(mod)

48. Mixed Congruential xaxcm nn   1 ()(mod)

49. Mixed Congruential xaxcm nn   1 ()(mod)

50. Mixed Congruential xaxcm nn   1 ()(mod) Cycle Length < m

51. Mixed Congruential xaxm nn   1 ()(mod)xxcm nn   1 ( ) Multiplicative Generator Additive Generator