1. 1 Evaluation of Safety Critical Software David L. Parnas, C ACM, June 1990

2. Software Reliability u Nonetheless, our practical experience is that software appears to exhibit stochastic properties. It is quite useful to associate reliability figures such as MTBF (Mean Time Between Failures) with an operating system or other software product. Some software experts attribute the apparently random behavior to our ignorance. They believe that all software failures would be predictable if we fully understood the software, but our failure to understand our own creations justifies the treatment of software failures as random. 2 841f07hamlet19oct30

3. MTTF u Mean is an average. u What is the relationship to Expected Value? 3 841f07hamlet19oct30

5. 5 Table 1 Table I shows that, if our design target was to have the probability of failure be less than 1 in 1000, performing between 4500 and 5000 tests (randomly chosen from the appropriate test case distribution) without failure would mean that the probability of an unacceptable product passing the test was less than 1 in a hundred. 5 841f07hamlet19oct30

6. 6 Table II 6 841f07hamlet19oct30

7. 7 Practical ultra-reliability for abstract data types Borislav Nikolik and Dick Hamlet Softw. Test. Verif. Reliab. 2007; 17:183–203 7 841f07hamlet19oct30

8. 8 Term Redundancy Method (TRM) 8 841f07hamlet19oct30

9. 9 Boolean Stack ADT 9 841f07hamlet19oct30 r1: pop(push(s, b))→s r2: top(push(s, b))→b Figure 1. Stack of boolean values TRS.

10. 10 Rewriting 10 841f07hamlet19oct30 pop(push(pop(push(x, y)), b)) = pop(push(x, y))

11. 11 Post-release testing 11 841f07hamlet19oct30 The relatively poor reliability estimate from the pre-release testing phase can now be used to obtain ultra-reliable term evaluations. In the self-checking phase the additional equivalent terms are drawn from the test-phase distribution for which the 10−4 bound was obtained. Therefore, if three randomly chosen terms agree, the probability that all of them are failures is less than (10−4)3 = 10−12.

12. 12 Reliability 12 841f07hamlet19oct30 Suppose a constant failure rate θ of π, and n random terms drawn from π, executed on δ without failure. The probability that δ fails on a randomly chosen term from π is θ, and 1 − θ that it will succeed. Given that the n terms are independent, the probability that δ succeeds on all the terms is (1 − θ)n. The confidence bound α on θ is defined as the probability that the failure rate of δ is below θ. The confidence bound is related to the testset size n and the failure rate θ by α ≤ 1 − (1 − θ)n

13. 13 Failure rate 13 841f07hamlet19oct30 The confidence bound of Equation (1) is used to quantify the probability of failure of a majority values of a self-check. Equation (1) could be used to estimate the confidence bound on the failure rate of δ on a majority of N random terms generated by RBTR. Suppose a successful test (no failures occurred during the test) of δ on n terms is conducted at test time. Half or more of N terms (majority) falsely agreeing at run-time gives a failure rate of at least N/2n. Therefore, substituting N/2n for θ in Equation (1) yields α ≤ 1 − 1 − N 2nn

15. 15 Equation 2 15 841f07hamlet19oct30 Equation (1) yields α ≤ 1 − 1 − N 2nn (2) The meaning of the confidence bound is the probability that the failure rate is below N/2n for a repetition of the test. For example, 1 − α = 6.0 × 10−8 with N = 33 and n = 104.

16. 16 How do we evaluate Hamlet? 16 841f07hamlet19oct30