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Leistungsanalyse Übung zu 5

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LeistungsanalyseÜbung zu 5

- 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

- Thus, on the average, approximately half the table needs to be searched.

- 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

- 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.

- 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).

- Example:
- Failure-time distribution is exponential with parameter λ:

- 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

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

- rv Xi : ith comp’s life time (arbitrary distribution)
- Case of weakest link. To prove above

- 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.

- 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’.

- Assuming that the reliability of a single component is given by,
- we get:
- Comparing with expected life of a single component.

- 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.

EXP(3)

EXP()

EXP(3)

EXP(2)

TMR/Simplex

TMR

- Derive & compare reliability expressions for two component Cold, Warm and Hot standby cases.
- Also find MTTF in each case.

EXP()

EXP()

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

EXP(+)

EXP()

- With Warm spare, we have:
- Time-to-failure in active state: EXP()
- Time-to-failure in spare state: EXP()

- 2-stage hypoexponential distribution

EXP(2)

EXP()

- With hot spare, we have:
- Time-to-failure in active state: EXP()
- Time-to-failure in spare state: EXP()

- 2-stage hypoexponential

File Server

Computer Network

Workstation 1

Workstation 2

Workstation 1

File Server

Workstation 2

Rw(t): workstation reliability

Rf (t): file-server reliability

System reliability Rsys(t) is given by:

Note: applies to any time-to-failure distributions

- 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:

- For a 2-component parallel redundant system
with EXP( ) and EXP( ) behavior, write down expressions for:

- Rp(t)
- MTTFp

Example: 2 Control Channels and 3 Voice Channels

voice

control

voice

control

voice

- Specialize formula to the case where reliability of control and voice are given as :
- Derive expressions for system reliability and system meantime to failure.

- Specialize the bridge reliability formula to the case where
Ri(t) =

- Find Rbridge(t) and MTTF for the bridge.

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

C1

C2

S

T

C4

C5

When C3 is working

C1

C2

S

T

C4

C5

When C3 fails