Leistungsanalyse Übung zu 5

1. LeistungsanalyseÜbung zu 5

2. Example 1 (contd.) • The set of all possible values of X is {1, 2, . . , n + 1} and X = n + 1 for unsuccessful searches. • Consider a random variable Y denoting the number of comparisons on a successful search. The set of all possible values of Y is {1, 2, . . , n}. • Assume pmf of Y to be uniform over the range

3. Example 1 (contd.) • Thus, on the average, approximately half the table needs to be searched.

4. Example 2-Zipf’s law • Zipf’s law has been used to model the distribution of Web page requests. • pY (i), the probability of a request for the i th most popular page is inversely proportional to i

5. Example 2-Zipf’s law (contd.) • Assumption • Web page requests are independent • The cache can hold only m Web pages regardless of the size of each Web page. • Adopting “least frequently used” removal policy, hit ratio h (m) -the probability that a request can find its page in cache-is given by (using Eq. (4.2) on p. 195 of text) • Hit ratio increases logarithmically as a function of cache size.

6. Moment Generating Property • Except for the sign of s, the Laplace transform is the moment generating function used in mathematical statistics: • Therth moment of X about the origin, if it exists, is given by the coefficient of (-sr)/r! in the Taylor series expansion of f*(s). • If X denotes the time to failure of a system, then from a knowledge of the transform f*(s) we can obtain the system MTTF E[X], while it is more difficult to obtain the pdf f(t) and the reliability R(t).

7. Moment Generating Property • Example: • Failure-time distribution is exponential with parameter λ:

8. MTTF Computation • R(t) = P(X > t), X: Lifetime of a component • Expected life time or MTTF is • In general, kthmoment is, • Simplified formula above can be derived using integration by parts and the fact that X is a non-negative random variable

9. MTTF Computation-Series System • Series of components, component i lifetime is EXP(λi) • Thus lifetime of the system is EXP with parameter • and series system MTTF =

10. Series SystemMTTF (contd.) • rv Xi : ith comp’s life time (arbitrary distribution) • Case of weakest link. To prove above

11. Parallel System-MTTF Computation • Parallel system: lifetime of ith component is rv Xi • X = max{X1, X2, ..,Xn} • If all Xi’s are EXP(λ), then, • As n increases, MTTF increases • and so does the Variance.

12. Variation of expected life with degree of parallel redundancy with each component having failure rate λ=10-6

13. Standby Redundancy • A system with 1 component and (n-1) cold spares. • System lifetime, • If all Xi’s same EXP() X has Erlang distribution. • TMR and ‘k of n’.

14. Triple Mode Redundancy (TMR) • Assuming that the reliability of a single component is given by, • we get: • Comparing with expected life of a single component.

15. TMR (Continued) • Thus TMR actually reduces (by 16%) the MTTF over the simplex system. • Although TMR has lower MTTF than does Simplex, it has higher reliability than Simplex for “short” missions, defined by mission time t<(ln2)/λ.

16. EXP(3) EXP() EXP(3) EXP(2) TMR and TMR/simplexas hypoexponentials TMR/Simplex TMR

17. Homework 1: • Derive & compare reliability expressions for two component Cold, Warm and Hot standby cases. • Also find MTTF in each case.

18. EXP() EXP() Cold standby X Y Lifetime in Spare state EXP() Lifetime in Active state EXP() • Total lifetime 2-Stage Erlang • Assumptions: • Detection & Switching perfect • Spare does not fail

19. EXP(+) EXP() Warm standby: • With Warm spare, we have: • Time-to-failure in active state: EXP() • Time-to-failure in spare state: EXP() • 2-stage hypoexponential distribution

20. Warm standby

21. EXP(2) EXP() Hot standby: • With hot spare, we have: • Time-to-failure in active state: EXP() • Time-to-failure in spare state: EXP() • 2-stage hypoexponential

22. Hot standby

23. Comparison graph:

24. The WFS Example File Server Computer Network Workstation 1 Workstation 2

25. RBD for the WFS Example Workstation 1 File Server Workstation 2

26. Rw(t): workstation reliability Rf (t): file-server reliability System reliability Rsys(t) is given by: Note: applies to any time-to-failure distributions RBD for the WFS Example (contd.)

27. RBD for the WFS Example (contd.) • Assuming exponentially distributed times to failure: • failure rate of workstation • failure rate of file-server • The system mean time to failure (MTTF) is given by:

28. Homework 2: • For a 2-component parallel redundant system with EXP( ) and EXP( ) behavior, write down expressions for: • Rp(t) • MTTFp

29. Solution 2:

30. Homework 3 :Series-Parallel system (Example) Example: 2 Control Channels and 3 Voice Channels voice control voice control voice

31. Homework 3 (Contd.): • Specialize formula to the case where reliability of control and voice are given as : • Derive expressions for system reliability and system meantime to failure.

32. Control channels-Voice channels

33. Homework 4: • Specialize the bridge reliability formula to the case where Ri(t) = • Find Rbridge(t) and MTTF for the bridge.

34. Bridge: conditioning C1 C2 C3 fails S T C1 C2 C4 C5 C3 S T C3 is working C4 C5 C1 C2 S T Factor (condition) on C3 C4 C5 Non-series-parallel block diagram

35. C1 C2 S T C4 C5 Bridge: Rbridge(t) When C3 is working

36. C1 C2 S T C4 C5 Bridge: Rbridge(t) When C3 fails

37. Bridge: Rbridge(t)

38. Bridge: MTTF