1. Systems Engineering Program Department of Engineering Management, Information and Systems EMIS 7305/5305 Systems Reliability, Supportability and Availability Analysis • Reliability Models & Applications (continued) Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering

2. The Normal or Gaussian Model: • Definition • A random variable T is said to have the Normal • (Gaussian) Distribution with parameters  and , • where  > 0, if the density function of T is: • , for - < t < • Definition • If T ~ N(,) and if , then Z ~ N(0,1) • the Standard Normal Distribution and Cumulative • Probability is tabulated for various values of z.

3. Properties of the Normal Model: • Probability Distribution Function • Where (Z) is the Cumulative Probability Distribution • Function of the Standard Normal Distribution. • Reliability Function • provided that P(T < 0)  0

4. Properties of the Normal Model: • MTBF (Mean Time Between Failure) • Standard Deviation of Time to Failure = • Failure Rate

5. Properties of the Normal Model - Failure Densities:

6. The Normal Model - Example Examplem=1,000s=100

7. Normal Distribution: Standard Normal Distribution ~ X ~ N (, )  z  0

8. Properties of the Normal Model - Standard Normal Distribution: Table of Probabilities p

9. Standard Normal Distribution Cumulative Probability Distribution Function F(x)

10. The Lognormal Model: • Definition • A random variable T is said to have the Lognormal • Distribution with parameters  and , where -  < < • and  > 0, if the density function of T is: • , for t >0 • , for t 0 • Remark • The Lognormal Model is often used as the • failure distribution for mechanical items and for • the distribution of repair times.

11. Properties of the Lognormal Model: • Failure Distribution • where (z) is the cumulative distribution function • Reliability Function • If T ~ LN(,), then Y = lnT ~ N(,)

12. Properties of the Lognormal Model: • MTBF (Mean Time Between Failures) • Variance of Time to Failure • Failure Rate

13. The Lognormal Model: • Failure rate • functions • for various • values of •  and 

14. The Lognormal Model:

15. Example - Ductile Strength A theoretical justification based on a certain material failure mechanism underlies the assumption that ductile strength X of a material has a lognormal distribution. Suppose the parameters are  = 5 and  = 0.1 (a) Compute E(X) and Var(X) (b) Compute P(X > 120) (c) Compute P(110  X  130) (d) What is the value of median ductile strength? (e) If ten different samples of an alloy steel of this type were subjected to a strength test, how many would you expect to have strength at least 120? (f) If the smallest 5% of strength values were unacceptable, what would the minimum acceptable strength be?

16. Example - Solution (a)

17. Example - Solution (b)

18. (c) Example - Solution (d)

19. Example - Solution (e)

20. Example - Solution (f)

21. Example - Solution

22. The Binomial Model: • Definition • - If X is the number of successes in n trials, where: • (1) The trials are identical and independent, • (2) Each trial results in one of two possible outcomes • success or failure, • (3) The probability of success on a single trial is p, • and is constant from trial to trial, then X has the • binomial Distribution with Probability Mass Function • given by:

23. The Binomial Model Probability Mass Function , for x = 0, 1, 2, ... , n , otherwise where =

24. Binomial Distribution Rule:

25. Binomial Distribution • Mean or Expected Value •  = np • Standard Deviation •  = (npq)1/2 , • where q=1-p

26. The Binomial Model - Example Application 1: • Four Engine Aircraft • Engine Unreliability Q(t) = p = 0.1 • Mission success: At least two engines survive • Find RS(t)

27. The Binomial Model - Example Application 1- Solution • X = number of engines failing in time t • RS(t) = P(x  2) = b(0) + b(1) + b(2) • = 0.6561 + 0.2916 + 0.00486 = 0.9963

28. Number of Failures Model: • Definition • - If T ~ E() and if X is the number of failures occurring • in an interval of time, t, then X ~ P(t/ ), the Poisson • Distribution with Probability Mass Function given by: • for x = 0, 1, ... • Where is the failure rate • The expected number of failures in time t is

29. The Poisson Model: X ~ P(2)

30. The Poisson Model: p(x) Number of Failures ~ x

31. The Poisson Model - example continued 1.00 0 1 2 3 4 5 6 7 8 Number of Failures ~ x

32. The Poisson Model - Example Application: An item has a failure rate of  = 0.002 failures per hour if the item is being put into service for a period of 1000 hours. What is the probability that 4 spares in stock will be sufficient?

33. The Poisson Model - Example Application - Solution Expected number of failures (spares required) = t = 2 P(enough spares) = P(X  4) = p(0) + p(1) + p(2) + p(3) + p(4) = 0.945 or about a 5% chance of not having enough spares!