Basic Concepts Reliability, MTTF, Availability, etc.

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Basic ConceptsReliability, MTTF, Availability,
etc.
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Definitions

• Reliability of a system is defined to be the
  probability that the given system will perform
  its required function under specified conditions
  for a specified period of time.
• MTBF (Mean Time Between Failures) Average time a
  system will run between failures. The MTBF is
  usually expressed in hours. This metric is more
  useful to the user than the reliability measure.

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Approaches to increase the reliability of a system
Increasing reliability of a system

1. Worst case design
2. Using high quality components
3. Strict quality control procedures

1. Redundancy
2. Typically employed
3. Less expensive

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Reliability expressions

• Exponential Failure Law
• Reliability of a system is often modeled as
• R(t) exp(-?t)
• where ? is the failure rate expressed as
  percentage failures per 1000 hours or as failures
  per hour.
• When the product ?t is small,
• R(t) 1 - ?t

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Relation between MTBF and the Failure rate

• MTBF is the average time a system will run
  between failures and is given by
• MTBF ?0 R(t) dt ?0 exp(-?t) dt 1 / ?
• In other words, the MTBF of a system is the
  reciprocal of the failure rate.
• If ? is the number of failures per hour, the
  MTBF is expressed in hours.

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A simple example

• A system has 4000 components with a failure rate
  of 0.02 per 1000 hours. Calculate ? and MTBF.
• ? (0.02 / 100) (1 / 1000) 4000 8 10-4
  failures/hour
• MTBF 1 / (8 10-4 ) 1250 hours

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Relation between Reliability and MTBF

• R(t) (1 ?t) (1 t / MTBF)
• Therefore,
• MTBF t / (1 R(t))

1.0
0.8
Reliability R(t)
0.6
0.4
0.36
0.2
0
2 MTBF
1 MTBF
Time t
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An example

• A first generation computer contains 10000
  components each with ? 0.5/(1000 hours). What
  is the period of 99 reliability?
• MTBF t / (1 R(t)) t / (1 0.99)
• t MTBF 0.01 0.01 / ?av
• Where ?av is the average failure rate
• N No. of components 10000
• ? failure rate of a component
• 0.5 / (1000 hours) 0.005/1000 5 10-6 per
  hour
• Therefore, ?av N ? 10000 5 10-6 5
  10-2 per hour
• Therefore, t 0.01 / (5 10-2 ) 12 minutes

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Reliability for different configurations
1. Series Configuration
1
2
3
4
N
R
R
R
R
R
Overall reliability Ro R R R. R RN
2. Parallel Configuration
1
R
2
Ro 1 (probability that all of the components
fail) Ro 1 (1 - R)N
R
N
R
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Reliability for different configurations
3. Hybrid Configuration
1
R
1
2
N
2
R
R
R
R
M
R
Overall reliability Ro ?
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Reliability for different configurations
4. Triple Modular Redundancy (TMR)
1
R
2
Voting
R
M
R
Overall reliability Ro 3C2 R2 (1-R)
R3
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Reliability calculation a more complicated
example
R Rc Rs2 (1-Rc) Rs1
System
Assuming C is faulty
S1
B
E
Assuming C is fault free
A
F
D
S2
Rs1 can be calculated using parallel series
formulae
Needs further reduction
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Rs2 RE Rs3 (1-RE) Rs4
S2
Assuming E is faulty
S4
Assuming E is fault free
D
A
F
S3
S3
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Maintainability

• Maintainability of a system is the probability of
  isolating and repairing a fault in the system
  within a given time.
• Maintainability is given by
• M(t) 1 exp(-µt)
• Where µ is the repair rate
• And t is the permissible time constraint for the
  maintenance action
• µ 1/(Mean Time To Repair) 1/MTTR
• M(t) 1 exp(-t/MTTR)

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Availability

• Availability of a system is the probability that
  the system will be functioning according to
  expectations at any time during its scheduled
  working period.
• Availability System up-time / (System up-time
  System down-time)
• System down-time No. of failures MTTR
• System down-time System up-time ? MTTR
• Therefore,
• Availability System up-time / (System up-time
  (System up-time ? MTTR)
• 1 / (1 (? MTTR)
• Availability MTBF / (MTBF MTTR)