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Reliability and
Maintenance
Management
Failure
Distributions
and Reliability
Measures
Dr. Hazem Smadi
Reliability…. overview
• Reliability is the probability that a component or system will perform
a required function for a given period of time when used under stated
operating conditions.
• Notes:
- Reliability is a “time” oriented quality characteristic.
- Reliability is concerned with the life of a system from
success/failure point of view.
- Reliability is a probability which is a function of time.
- The random variable used to measure reliability is the “time”-tofailure random variable, T.
• How to measure reliability?
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Reliability Function
• Reliability is the probability that a system will function over some
time period t.
• Let T = “time”-to-failure random variable, reliability at time t is the
probability that T is greater than or equal to t:
R(t) = P(T ≥ t), t ≥ 0
• Properties: 0  R(t)  1; R(0) = 1; 𝐥𝐢𝐦 𝑹 𝒕 = 𝟎,
𝒕→∞
monotonically non-increasing
• Two interpretations:
R(t) is the probability that an individual item is functioning at time t.
R(t) is the expected fraction of the population that is functioning at
time t for a large population of items.
• Other names:
Survival Function -- Biostatistics
Complementary CDF
CDF and PDF
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R(t), F(t), and f(t)
Hazard Function
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Hazard Function
Hazard Function
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Plots of R(t), F(t), f(t), h(t) for Normal Distribution
Relationships Among R(t), F(t), f(t), and h(t)
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Relationships Among R(t), F(t), f(t), and h(t)
Bathtub Curve
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Bathtub Curve
Bathtub Curve
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Summary Statistics of Reliability
Expected Life
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Median Life and Bα Life
Mode and Variance
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Conditional Reliability
𝑅 𝑡 𝑇0 = 𝑃 𝑇 > 𝑇0 + 𝑡 𝑇 > 𝑇0 } =
𝑅 (𝑇0 + 𝑡)
𝑅 (𝑇0 )
End of Chapter 2
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