1. Stracener_EMIS 7305/5305_Spr08_02.28.08 1 Systems Reliability Growth Planning and Data Analysis Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7305/5305 Systems Reliability, Supportability and Availability Analysis Systems Engineering Program Department of Engineering Management, Information and Systems

2. Stracener_EMIS 7305/5305_Spr08_02.28.08 2 Systems Reliability Growth Planning

3. Stracener_EMIS 7305/5305_Spr08_02.28.08 3 The Duane Model The instantaneous MTBF as a function of cumulative test time is obtained mathematically from MTBF C (t) and is given by where MTBF i (t) is the instantaneous MTBF at time t and is interpreted as the equipment MTBF if reliability development testing was terminated after a cumulative amount of testing time, t.

4. Stracener_EMIS 7305/5305_Spr08_02.28.08 4 Reliability Growth Factors Initial MTBF, K, depends on –type of equipment –complexity of the design and equipment operation –Maturity Growth rate, α –TAAF implementation and FRACAS Management –type of equipment –complexity of the design and equipment operation –Maturity

5. Stracener_EMIS 7305/5305_Spr08_02.28.08 5 10100 1000 10000 100 1000 10 Cumulative Test Hours MTBF Cumulative Instantaneous MTBF Growth Curves

6. Stracener_EMIS 7305/5305_Spr08_02.28.08 6 The Duane Model The Duane Model may also be formulated in terms of equipment failure rate as a function of cumulative test time as follows: and where C (t) is the cumulative failure rate after test time t k* is the initial failure rate and k*=1/k, B is the failure rate growth (decrease rate) i (t) is the instantaneous failure rate at time t

7. Stracener_EMIS 7305/5305_Spr08_02.28.08 7 Determination of Reliability Growth Test Time Specified MTBF at system maturity, θ 0 Solve Duane Model for t

8. Stracener_EMIS 7305/5305_Spr08_02.28.08 8 Determination of Reliability Growth Test Time - Example Determine required test time to achieve specified MTBF, θ 0 =1000, if α=0.5 and K=10 Solution

9. Stracener_EMIS 7305/5305_Spr08_02.28.08 9 Determination of Reliability Growth Test Time – Example (continued) Interpretation of θ 0 =1000 hours at t=2499.88 hours If no further reliability growth testing is conducted, the systems failure rate at t=2499.88 hours is Failures per hour, and is constant for time beyond t=2499.88 hours

10. Stracener_EMIS 7305/5305_Spr08_02.28.08 10 Determination of Reliability Growth Test Time – Example (continued) Check since Cumulative MTBF at t = 2499.88 hours Since =1000 =500

11. Stracener_EMIS 7305/5305_Spr08_02.28.08 Determination of Reliability Growth Test Time – Example Test Time a)Determine the test time required to develop (grow) the reliability of a product to  0 if the required reliability is 0.9 based on a 100-hour mission and the initial MTBF is 20% of  0 and  =0.3. How many failures would you expect to occur during the test? Investigate the effect on the test time needed to achieve the required MTBF of deviations in initial MTBF and growth rate. 11

12. Stracener_EMIS 7305/5305_Spr08_02.28.08 Determination of Reliability Growth Test Time – Example (continued) The test time required depends on the starting point. The usual convention is to start the growth curve at t=100. We will determine the test time based on this. Also, we will show the effect on the requirement test time if the starting point of 0.2 0 is at t=1 hour. 1 100 t R1 t R100 =0.3 Required MTBF Required Test Time 20% of Required MTBF 12

13. Stracener_EMIS 7305/5305_Spr08_02.28.08 Determination of Reliability Growth Test Time – Example (continued) Since Using the Duane Model since =0.3 13

14. Stracener_EMIS 7305/5305_Spr08_02.28.08 Determination of Reliability Growth Test Time – Example (continued) But and so that or so that the model is 14

15. Stracener_EMIS 7305/5305_Spr08_02.28.08 Determination of Reliability Growth Test Time – Example (continued) To find the test time required to grow the MTBF to  0 set so that and 15

16. Stracener_EMIS 7305/5305_Spr08_02.28.08 Determination of Reliability Growth Test Time – Example (continued) Since at t=21373.4 hours, 16

17. Stracener_EMIS 7305/5305_Spr08_02.28.08 Determination of Reliability Growth Test Time – Example (continued) If the initial MTBF I (t) is interpreted to be at t=1 hour, then and so that or so that the model is 17

18. Stracener_EMIS 7305/5305_Spr08_02.28.08 Determination of Reliability Growth Test Time – Example (continued) To find the test time required to grow the MTBF to  0 set so that and 18

19. Stracener_EMIS 7305/5305_Spr08_02.28.08 Determination of Reliability Growth Test Time – Example (continued) Since at t=213.747 hours, 19

20. Stracener_EMIS 7305/5305_Spr08_02.28.08 20 Systems Reliability Growth Data Analysis

21. Stracener_EMIS 7305/5305_Spr08_02.28.08 21 Reliability Growth Test Data Analysis Duane Model Parameter Estimation Maximum Likelihood Estimation Least Squares Estimation Confidence Bounds Plotting the estimated MTBF Growth Curves and Confidence Bounds Procedures from MIL-HDBK-189, Feb. 13. 1981

22. Stracener_EMIS 7305/5305_Spr08_02.28.08 Least Squares for the Duane Model Since MTBF c (t)=t , And for simplicity in the calculations, let: so that Transforming the data (t i, MTBF c (t i )) to (x i, y i ) for i=1, 2, …, n use the method of least squares to estimate the equation y= 0 +  1 x+ 22

23. Stracener_EMIS 7305/5305_Spr08_02.28.08 23 Least Squares for the Duane Model The Least squares estimates of  0 and  1 are:

24. Stracener_EMIS 7305/5305_Spr08_02.28.08 24 Reliability Growth Data Analysis A test is conducted to growth the reliability of a system. At the end of 100 hours of testing the results are as follows: 12.52.0 212.8 353.5 604.8 1006.0 t c MTBF C

25. Stracener_EMIS 7305/5305_Spr08_02.28.08 25 Estimate MTBF I (t) and MTBF C (t) as a function test time t and plot. What is the estimated MTBF of the system if testing is stopped at 200 hours? Reliability Growth Data Analysis - continued

26. Stracener_EMIS 7305/5305_Spr08_02.28.08 Solution

27. Stracener_EMIS 7305/5305_Spr08_02.28.08 Solution tctc XiXi X i ^2 MTBF c YiYi X i *Y i 12.52.536.38 20.69 1.75 213.049.27 2.81.03 3.13 353.5612.64 3.51.25 4.45 604.0916.76 4.81.57 6.42 1004.6121.21 61.79 8.25 Sum(X i )Sum(X i ^2) Sum(Y i ) Sum(X i *Y i ) 17.8366.26 6.34 24.01

28. Stracener_EMIS 7305/5305_Spr08_02.28.08 Solution

29. Stracener_EMIS 7305/5305_Spr08_02.28.08 Solution 200

30. Stracener_EMIS 7305/5305_Spr08_02.28.08 Solution At t=200 hours, the system failure rate becomes a constant =Failures per hour

31. Stracener_EMIS 7305/5305_Spr08_02.28.08 31 Duane Model Parameter Estimation Estimate the “Best Fit” reliability growth equation by using the observed failure times t 1, t 2, …, t n to estimate the Duane Model parameters Use the failure rate version of the Duane Model since most of the theory is based on it