System Maintainability Modeling & Analysis

1. Systems Reliability, Supportability and Availability Analysis System Maintainability Modeling & Analysis

2. Why Do Maintainability Modeling and Analysis? To identify the important issues To quantify and prioritize these issues To build better design and support systems

3. Bottoms Up Models Provide output to monitor design progress vs. requirements Provide input data for life cycle cost Provide trade-off capability • Design features vs. maintainability requirements • Performance vs. maintainability requirements Provide Justification for maintenance improvements perceived as the design progresses

4. Bottoms Up Models Provide the basis for maintainability guarantees/demonstration Provide inputs to warranty requirements Provide maintenance data for the logistic support analysis record Support post delivery design changes Inputs • Task Time (MH) • Task Frequency (MTBM) • Number of Personnel-Elapsed Time (hours) • For each repairable item

5. Bottoms Up Models Input Data Sources • Task Frequency • Reliability predictions de-rated to account for non-relevant failures • Because many failures are repaired on equipment, the off equipment task frequency will be less than the task frequency for on equipment

6. Bottoms Up Models Input Data Sources (Continued) • Task Time • Touch time vs. total time • That time expended by the technician to effect the repair • Touch time is design controllable • Total Time • Includes the time that the technician expends in “Overhead” functions such as part procurement and paper work • Are developed from industrial engineering data and analyst’s estimates

7. Task Analysis Model Task analysis modeling estimates repair time • MIL-HDK-472 method V • Spreadsheet template • Allow parallel and multi-person tasks estimation • Calculates elapsed time and staff hours • Reports each task element and total repair time • Sums staff hours by repairmen type • Estimates impact of hard to reach/see tasks

8. When considering probability distributions in general, the time dependency between probability of repair and the time allocated for repair can be expected to produce one of the following probability distribution functions: • Normal – Applies to relatively straightforward maintenance tasks and repair actions that consistently require a fixed amount of time to complete with little variation • Exponential – Applies to maintenance tasks involving part substitution methods of failure isolation in large systems that result in a constant repair rate. • Lognormal - Applies to Most maintenance tasks and repair actions comprised of several subsidiary tasks of unequal frequency and time duration.

9. The Exponential Model Definition A random variable X is said to have the Exponential Distribution with parameters , where  > 0, if the probability density function of X is: for x  0 , elsewhere

10. Properties of the Exponential Mode Probability Distribution Function for x < 0 for x  0 Note: the Exponential Distribution is said to be without memory, i.e. P(X > x1 + x2 | X > x1) = P(X > x2)

11. Properties of the Exponential Model Mean or Expected Value Standard Deviation

12. f(x) x Normal Distribution A random variable X is said to have a normal (or Gaussian) distribution with parameters  and , where - < <  and  > 0, with probability density function - < x <  where  = 3.14159… and e = 2.7183...

13. Normal Distribution Mean or expected value of X Mean = E(X) =  Median value of X X0.5 =  Standard deviation

14. Normal Distribution Standard Normal Distribution If X ~ N(, ) and if , then Z ~ N(0, 1). A normal distribution with  = 0 and  = 1, is called the standard normal distribution.

15. f(z) f(x) x z  0 Normal Distribution

16. f(z)  z 0 z Normal Distribution Standard Normal Distribution Table of Probabilities http://www.smu.edu/~christ/stracener/cse7370/normaltable.html Enter table with and find the value of 

17. Normal Distribution - Example The time it takes a field engineer to restore a function in a logistics system can be modeled with a normal distribution having mean value 1.25 hours and standard deviation 0.46 hours. What is the probability that the time is between 1.00 and 1.75 hours? If we view 2 hours as a critically time, what is the probability that actual time to restore the function will exceed this value?

18. Normal Distribution - Example Solution

19. Normal Distribution - Example Solution

20. The Lognormal Model Definition - A random variable X is said to have the Lognormal Distribution with parameters  and , where > 0 and  > 0, if the probability density function of X is: , for x >0 , for x 0

21. Properties of the Lognormal Distribution Probability Distribution Function where (z) is the cumulative probability distribution function of N(0,1) Rule: If T ~ LN(,), then Y = lnT ~ N(,)

22. Properties of the Lognormal Model Mean or Expected Value Median Variance

23. Lognormal Model example The elapsed time (hours) to repair an item is a random variable. Based on analysis of data, elapsed time to repair can be modeled by a lognormal distribution with parameters  = 0.25 and  = 0.50. a. What is the probability that an elapsed time to repair will exceed 0.50 hours? b. What is the probability that an elapsed time to repair will be less than 1.2 hours? c. What is the median elapsed time to repair? d. What is the probability that an elapsed time to repair will exceed the mean elapsed time to repair? e. Sketch the cumulative probability distribution function.

24. Lognormal Model example - solution a. What is the probability that an elapsed time to repair will exceed 0.50 hours? X ~ LN(, ) where  = 0.25 and  = 0.50 note that: Y = lnX ~ N(, ) P(X > 0.50) = P(lnX > -0.693)

25. Lognormal Model example b. What is the probability that an elapsed time to repair will be less than 1.2 hours? P(X < 1.20) = P(lnX < ln1.20)

26. Lognormal Model example c. What is the median elapsed time to repair? P(X < x0.5) = 0.5 therefore

27. Lognormal Model example d. What is the probability that an elapsed time to repair will exceed the mean elapsed time to repair?

28. Lognormal Model example P(X > MTTR) = P(X > 1.455) = P(lnX > 0.375)

29. Lognormal Model example e. Sketch the cumulative probability distribution function. tmax

30. 95th Percentile / MTTR Ratio If repair time, T, has a lognormal distribution with parameters μ and σ, then 95th percentile of time to repair Mean Time To Repair Ratio of 95th percentile to mean time to repair

31. 95th Percentile / MTTR Ratio

32. 95th Percentile / MTTR Ratio Since Then and

33. Analysis of Combination of Repair Times

34. Corrective Maintenance Cycle Failure Occurs Detection Failure Confirmed Preparation for Maintenance Active Maintenance Commences Location and Isolation Faulty Item Identified Disassembly (Access) Disassembly Complete or Repair of Equipment Removal of Fault Item Installation of Spare/Repair Part Re-assembly Re-assembly Complete Alignment and Adjustment Condition Verification Repair Completed

35. Linear Combinations of Random Variables If X1, X2, ..., Xn are independent random variables with means 1, 2, ..., n and variances 12, 22, ..., n2, respectively, and if a1, a2, … an are real numbers then the random variable has mean and variance

36. then and Linear Combinations of Random Variables If X1, X2, ..., Xn are independent random variables having Normal Distributions with means 1, 2, ..., n and variances 12, 22, ..., n2, respectively, and if where where a1, a2, … an are real numbers

37. Generating Random Samplesusing Monte Carlo Simulation

38. Population vs. Sample Population the total of all possible values (measurement, counts, etc.) of a particular characteristic for a specific group of objects. Sample a part of a population selected according to some rule or plan. Why sample? • - Population does not exist • - Sampling and testing is destructive

39. Sampling Characteristics that distinguish one type of sample from another: the manner in which the sample was obtained the purpose for which the sample was obtained

40. Types of Samples Simple Random Sample The sample X1, X2, ... ,Xn is a random sample if X1, X2, ... , Xn are independent identically distributed random variables. Remark: Each value in the population has an equal and independent chance of being included in the sample. Stratified Random Sample The population is first subdivided into sub-populations for strata, and a simple random sample is drawn from each strata

41. Types of Samples (continued) Censored Samples Type I Censoring - Sample is terminated at a fixed time, t0. The sample consists of K times to failure plus the information that n-k items survived the fixed time of truncation. Type II Censoring - Sampling is terminated upon the Kth failure. The sample consists of K times to failure, plus information that n-k items survived the random time of truncation, tk. Progressive Censoring - Sampling is reduced in stage.

42. Uniform Probability Integral Transformation For any random variable Y with probability density function f(y), the variable is uniformly distributed over (0, 1), or F(y) has the probability density function

43. Uniform Probability Integral Transformation Remark: the cumulative probability distribution function for any continuous random variable is uniformly distributed over the interval (0, 1).

44. f(y) y F(y) 1.0 0.8 0.6 0.4 0.2 0 ri y yi Generating Random Numbers

45. Generating Random Numbers Generating values of a random variable using the probability integral transformation to generate a random value y from a given probability density function f(y): 1. Generate a random value rU from a uniform distribution over (0, 1). 2. Set rU = F(y) 3. Solve the resulting expression for y.

46. Generating Random Numbers with Excel From the Tools menu, look for Data Analysis.

47. Generating Random Numbers with Excel If it is not there, you must install it.

48. Generating Random Numbers with Excel Once you select Data Analysis, the following window will appear. Scroll down to “Random Number Generation” and select it, then press “OK”

49. Generating Random Numbers with Excel Choose which distribution you would like. Use uniform for an exponential or weibull distribution or normal for a normal or lognormal distribution

50. Generating Random Numbers with Excel Uniform Distribution, U(0, 1). Select “Uniform” under the “Distribution” menu. Type in “1” for number of variables and 10 for number of random numbers. Then press OK. 10 random numbers of uniform distribution will now appear on a new chart