1 / 38

Leistungsanalyse Übung zu 5

Leistungsanalyse Übung zu 5. Example 1 (contd.). The set of all possible values of X is { 1 , 2 , . . , n + 1 } and X = n + 1 for unsuccessful searches.

erma
Download Presentation

Leistungsanalyse Übung zu 5

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related