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Simulation

Simulation. Inverse Functions. Actually, we’ve already done this with the normal distribution. 0.1. x. 3.0. 3.38. Inverse Normal. Actually, we’ve already done this with the normal distribution. -. m. X. =. x = m + s z = 3.0 + 0.3 x 1.282 = 3.3846. Z. s. f. (. x.

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Simulation

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  1. Simulation

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

  3. 0.1 x 3.0 3.38 Inverse Normal • Actually, we’ve already done this with the normal distribution. - m X = x = m + sz = 3.0 + 0.3 x 1.282 = 3.3846 Z s

  4. f ( x )   e   x F ( a )  Pr{ X  a } a   e   x dx  0  1  e   a Inverse Exponential Exponential Life 2.0 1.8 1.6 1.4 1.2 f(x) Density 1.0 0.8 0.6 0.4 0.2 0.0 0 0.5 1 1.5 2 2.5 3   e   x a a Time to Fail 0

  5. F(x) - l X e x Inverse Exponential F ( x ) = 1 -

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

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

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

  9. F(x) F(a) - l a e x a 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) = -la

  10. F(x) F(a) x - l a e a 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) = -la a = - ln(0.9)/l

  11. Inverse Exponential Suppose a car battery is governed by an exponential distribution with l = 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)/l = - (-2.3026)/0.005 = 21.07 hrs. F(x) F(a) x 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. - l » l A A e i i - m » m S S e i i M/M/1 Queue M/M/1 Queue assumes exponential interarrival times and exponential service times

  14. - l » l A A e i i - m » m S S e i i M/M/1 Queue M/M/1 Queue assumes exponential interarrival times and exponential service times Exponential Review Expectations

  15. 2.032 1.951 1.349 .795 .539 .347 M/M/1 Queue 0.305 0.074 0.035 0.520 1.535 0.159

  16. 2.032 1.951 1.349 .795 .539 .347 0.305 0.074 0.035 0.520 1.535 0.159 M/M/1 Queue

  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. .795 M/M/1 Queue .539 0.652

  23. .795 M/M/1 Queue .539 0.652 0.074

  24. M/M/1 Event Calendar

  25. .795 M/M/1 Queue 0.726

  26. M/M/1 Event Calendar

  27. .795 M/M/1 Queue

  28. M/M/1 Event Calendar

  29. 1.349 M/M/1 Queue .795 0.830 0.035

  30. 1.349 M/M/1 Queue 0.830

  31. M/M/1 Event Calendar

  32. M/M/1 Queue 1.349

  33. M/M/1 Event Calendar

  34. 1.951 M/M/1 Queue 1.349 1.869 0.520

  35. M/M/1 Queue 1.869 1.951

  36. M/M/1 Event Calendar

  37. 2.032 1.951 1.349 .795 .539 .347 0.305 0.074 0.035 0.520 1.535 0.159 M/M/1 Queue

  38. M/M/1 Event Calendar

  39. M/M/1 Performance Measures

  40. M/M/1 Performance Measures

  41. M/M/1 Performance Measures

  42. Applications; Financial

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