Components Reliability

L05 Components Reliability

Quantitative Risk Analysis L05Fall 2013

Components Reliability

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Performance Assessment

• ability to realize continuously the intended function (design intent).

• is a random variable that can be estimated probabilistically.

Performance

EfficiencyEffectiveness ofrealizing intendedfunctionComparison of input vs output

CapabilityProbability toattain its functionunder all conditionsReliability varies with time

AvailabilityFraction of time system is operationalReliabilityMaintenance

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Reliability, Availability

• Reliability: Given a system demand, reliability is the ability to start and continue to operate when a unit is online. Includes random failure.

• Availability is the ability of a unit to respond to system demand, so it includes reliability (random failure) and offline intervals for maintenance or repair of the unit.

• Returning the unit to service following failure or maintenance, thereby increasing availability, can be critically important to continued operation of the system.

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System Reliability

• Deterministic view of system: Understand how and why a system fails. How to design and test a system for high reliability.

• Probabilistic view of system: Predict system reliability and system failure: Calculate Pr of system function at T= t and P(failure at T>t)

R(t) = 1– P(fail T ≤ t) = 1–F(t) = P(fail T > t)

• R(t) = P(T > t|c1, c2, …) Conditions affect component and system reliability

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Time to Failure

• Calendar time• Operational time• Number of kilometers driven• Number of cycles for a working item• Number of times a switch is operated• Number of rotations of a bearing• …

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Time Variable

• In many applications we will assume that the time to failure T is a continuous random variable.

• Discrete variables may be approximated by continuous variables.

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Probability Density Function

• A probability density function (pdf), f(t), is a function of probability in terms of a variable, often time,T.

• The f(t) is the probability within dt or dP(t) = f(t)dt.

• The f(t) = dP/dt or slope of probability with respect to the variable t.

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Probability Density Function

• Using f(t) = dP/dt

• Within t1 and t2, the total (cumulative) probability is

• Total P = 1

P(t2) P(t1) d Pdu

dut1

t2 f (u)dut1

t2

f (t)dt0

1

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Distribution Function for T

• For a probability density function (pdf) f(t), the cumulative distribution function (cdf) for T is

• F(t) = cumulative probability of item failure within the interval (0, t)

F(t) P(T t) f(u)du, t 00

t

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Probability Density Function for T

• The probability density function, f(t), for T is expressed in terms of F(t) as

• Therefore, when Δt is small,

f(t) ddt

F(T) limt 0

F(t t) F(t)t

limt 0

P(t T t t)t

P(t T t t) f(t) t at t = 0

Note that this is an unconditional probability, which is independent of history.

slope

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PDF, f(t), and CDF, F(t)

• F(t): cumulative P, 0 to t• f(t): rate of P accumulation

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F(t) and f(t) for T

• P(t1 < T ≤ t2) = F(t2) − F(t1) = the area under the pdf curve between t1 and t2.

F(t2) F(t1) f (t)dtt1

t2

P(t2) P(t1)

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Reliability Function, CDF

• The reliability function (survivor function) is defined by

• R(t) = probability the item will not fail in (0, t)= probability the item will survive to T = t= probability the item will fail for time T > t

R(t) 1– F(t) P(T t)

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Compare R(t) with F(t)

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Conditional Failure Rate Function, h(t)

• The conditional probability that a non-repairable (replaceable) component will fail in (t, t + ∆t) given that the component is functioning at time t is

P(t T t t)P(t T t t |T t)

P(T t)

given

F(t t) F(t)R(t)

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Conditional Failure Rate Function, h(t)

• The conditional pdf h(t) exhibits changes in the conditional probability of failure of a component over its lifetime. Therefore, it is sometimes referred to as the hazard rate.

a conditional probability density function

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for a component that is working at time t and is dependent on history. For identical items, it is the relative proportion of total items working at t but failing in ∆t.

Comparison of h(t) and f(t)

P(t T t t T t) h(t)t

When ∆t is small, conditional probability

The unconditional probability that a new component at t = 0 will fail in (t, t + ∆t) is

P(t T t t) f(t) t

Note that at t = 0, R(t=0) = 1, so h(t=0) = f(t=0).

independent of history

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Probability Function Comparison

• The failure rate probability density function h(t) represents the life distribution of an item and the sensitivity to failure following time t with reliability R(t).

, h(t) rises as R(t) drops

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Failure Rate Relations

The explicit form of h(t) is needed to calculate reliability, R(t), of a component during operational life.

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Cumulative Failure Rate

• The cumulative, conditional failure distribution or cumulative hazard function H(t) is

• For a given time interval (0, t), the average failure rate is

R(t) e h(x)dx

0

t

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Cumulative Failure, Reliability

• As shown, the reliability cdf is related to h(t):

• If h(t) is constant = λ, R(t) is the exponential cdf:

• F(t) is the cdf of failure, which is complementary to R(t):

R(t) e h(x)dx

0

t

R(t) et

F(t) 1 et

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Cumulative Failure, Exponential

• H(t), cdf of failure for constant h(t) = λ:

• The exponential function has no memory:

• So a used item that is working is just as good as a new item! This approximation must be tested!

P(t T t t | T t) F(t t) F(t)R(t)

et e(tt)

et 1 et, assumes constant

H(t) h(x)dx0

t t

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Life Distribution of a Component

• Divide (0, t) into intervals of length ∆t• n(i): number of items that fail in interval i• m(i): number of items working in start of i

• h(t)Δt represents the fraction of total systems that fail in interval i with length Δt

• = probability that n(i) fail in i given m(i) are working at start of i.

from the classical definition of Pr, n/N

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Bathtub Curve

h(t)

h(t)

test data to estimate h(t)

λ(t) λ(t)

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Mean Time To Failure

• Mean time to failure, (MTTF) illustrates the expected time during which the item will perform it’s function successfully – Expected Life

• For

0 0( ) ( ) ( )

MTTF E T T t f t dt t R t dt

(by partial integration)

limt t f (t) 0

0( )

MTTF R t dt

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R(t) Calculations

• R(t), F(t), cumulative distribution functions, cdf:

P(t T t t)P(t T t t T t)

P(T t)

1 2

1 2R(t ) R(t )P(t T t T t)

R(t)

R(t) R(t t)R(t)

F(t t) F(t)R(t)

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Conditional Reliability Distribution

• An item new at t = 0 and still working at t. The probability that it survives an added time x is the conditional reliability (survivor) function, R(x|t)

R(x | t) P(T x t | T t) P(T x t)P(T t)

R(x t)

R(t)

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Reliability Function Relationships

F(t) f(t) R(t) h(t)

F(t) f(u)du0

t

1R(t) 1 exp h(u)du0

t

f(t) ddt

F(t) ddt

R(t) h(t) exp h(u)du0

t

R(t) 1F(t) f(u)dut

exp h(u)du0

t

h(t) dF(t) / dt1F(t)

f(t)

f(u)dut

ddt

lnR(t)

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Example 5.1

• A device time to failure follows the exponential distribution. If the device has survived up to time t, determine its residual MTTF.

• Given– exponential distribution– useful functions

– conditional reliability

– MTTF

( ) exp( ) f u uexp( ) exp( ) t dt t

( )( | )( )

R tR t

R t

0( )

MTTF R t dt

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Mean Residual Life (MRL)

• Mean residual life (MRL) or conditional time to failure = μ(t):

• What is the remaining life in a component put into operation at t = 0 and still working at time = t ?

The R(x|t) is the Pr that the item working at age t survives an additional time x = Δt.

R(x | t) P(T t x | T t) P(T t x)P(T t)

R(t x)

R(t)

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Mean Residual Life (MRL)

• The mean residual life (MRL) is

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Example 5.1, solution

• Mean residual life (MRL) for a component that follows the exponential reliability model:

• Note that if λ is constant, a used item is just as reliable as a new item. The exponential R(t) has “no memory.”

independent of t

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Other Measures of a Life Distribution

• Median life: P(X ≤ xm) = R(tm) = 0.50

Item will fail before tm with a 0.5 probability and will fail after tm with a 0.5 probability.

• Mode: f(tmode) = max f(t); tmode is the most likely failure time (with highest probability)

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Other Measures of a Life Distribution

f(tmode) = max f(t)

f(tm) = 0.50

Compare this pdf, skewed to the right, with the Gaussian pdf.

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Reliability, Methods

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Reliability Assessment

• Methods for components and systems based on analytical models and empirical data.

• Used to estimate reliability and failure rates, R(t), f(t), h(t), H(t), which are used for:– Failure probabilities for reliability and risk metrics– Risk decisions and risk management– Economic decisions

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Selection of Reliability Models

• Model types:– Physical models representing detailed failure mechanisms:

most useful but often not available– Selection and fitting of standard engineering distributions

(parametric models)– Non-parametric method in which the model structure is

determined by the data (not based in advance on a particular population distribution)

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Test Data to Support Reliability Models

• Usefulness of periodic test data– Selection of component models for behavior– Reliability model parameter updating

• Nonparametric method: Estimation of reliability from evaluation of sample data

• Parametric method: Represent reliability using hypothesized models with parameters obtained from the sample data with tests of models and calculation of parameter confidence intervals.

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Graphical Nonparametric Procedures

• A failure test to completion of n components yield time to failure for each item: t1 ≤ t2 ≤ … ≤ tn.

• Failure data also can be grouped into equal time-to-failure (TTF) increments, ∆t

• Based on the definition of reliability as the probability of success estimated by the relative frequency of occurrence, n/N, a nonparametric estimate of a reliability cdf at a specific time is

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R(ti ) NS(ti )

N

units working at t i

all units

nN


Graphical Nonparametric Procedures

• Nonparametric estimate of reliability cdf at specific time ti :

• The failure pdf and the conditional probability of failure pdf are estimated by:

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R(ti ) NS(ti )

N

NS(ti) = # of working units, successN = # of all unitsti = upper endpoint of each ∆t

f (ti ) Nf (ti )N t Nf (ti) = # failed units in (ti , ti+∆t)

h(ti ) f (ti )R(ti )

Nf (ti )

NS(ti )t NS(ti) = NS(ti-1) – Nf(ti)

Average conditional failure rate within the interval (ti , ti+∆t)

previous current

success fail


Time to Failure (TTF) Data for a Component

• Test data are observed during the three stages of a component life: infant mortality, random failure (useful life region), and wear out.

• Given collected data, calculate the empirical failure rate pdf, reliability cdf, and the conditional failure pdf.

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Time, t, in hours

λ ~ constant

λ ~ decreasing

λ ~ increasing

Time to Failure (TTF) Data for a Component

• TTF Data for an Electronic Component During 3 Stages

• H/W– replicate the

calculations and– draw the histogram– submit excel file

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TTF Parameter Data

• TTF parameters or failure on demand distribution can be based on:– Field data– Reliability tests within realistic conditions and credible upset

conditions that are be encountered in service

• Test types:– with replacement of the failed items– without replacement of failed items

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TTF Test Samples

• Sample types for TTF (time-between failures) tests:– Complete samples in which all items fail during the

observation period and all failure times are known. Usually not practical, especially for highly reliable items that would require extremely long test periods

– Time terminated, yielding Type I right-censored data (data incomplete for failures at longer time): n items observed over a preset time period. (# failures = random variable)

– Failure terminated, yielding Type II right-censored data (data incomplete for longer time failures: observation of n items terminated following a preset # of failures. (time of test = random variable)

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Type I Life Test (Time T Fixed) With Replacement

• With n non-repairable items observed with replacement, and terminated after test time t0.

• Accumulated item time T, total time of items on test(failed and not failed) = T = nt0.

• The exponential MLE (maximum likelihood estimate) for λ is , where r failures (random variable) were observed in total item time T = nt0. (T = unit time on test; t0 = observed test time.)

• The MTTF = T/r

• Number of observed units on test, n’ = n + r

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r T

r Treplaced


Type I Life Test (Time T Fixed) Without Replacement

• With n nonrepairable items observed without replacement, terminated after time t0 during which r failures (random variable) occurred.

• Accumulated item time T, total time on test =

• The exponential ML estimates for λ and MTTF are:

MTTF = T/r

• Number of observed units, n’ = n (no replacements)

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r T

T ti (n r )t0i1

r = r failed time + (n r ) survived time


Type II Life Test (r Failures Fixed) With Replacement

• With n nonrepairable items observed with replacement, and terminated after time tr (random variable) when the rth failure has occurred.

• Accumulated item time T, total time on test (failed and not failed) = T = ntr.

• The exponential MLE (maximum likelihood estimate) for λ is , where r failures were observed in total unit time T = ntr.

• The MTTF = T/r• Number of observed units, n’ = n + r – 1 (last unit is not

replaced)

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r T

r T


Type II Life Test (r Failures Fixed) Without Replacement

• With n nonrepairable items observed without replacement, terminated after time tr (random variable) during which r failures occurred.

• Accumulated item time T, total time on test =

• The exponential ML estimates for λ and MTTF are:

MTTF = T/r

• Number of observed units, n’ = n (no replacement)

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r T

T ti (n r )tri1

r = r failed time + (n r ) survived time


Life Tests, Summary

• Type I Life Test (t0 fixed) with replacement: n’ = n + r

T = nt0 ; λ = r/T ; MTTF = T/r

• Type I Life Test (t0 fixed) without replacement: n’ = n

T = Σ ti + (n – r)t0 ; λ = r/T ; MTTF = T/r

• Type II Life Test (r fixed) with replacement: n’ = n + r – 1

T = ntr ; λ = r/T ; MTTF = T/r

• Type II Life Test (r fixed) without replacement: n’ = n

T = Σ ti + (n – r)tr ; λ = r/T ; MTTF = T/r

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Time Terminated

Failure Terminated


Example 5.2

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Sample size = 12

Find: Type of Data and T, λ ?



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Time terminated, r is a random variable

= 51 years, time of test = t0

Sample size = 12

Observed r = 8 failuresT tii1

r (n r ) t0 208 4(51) 412yr λ = r/T =8/412 =0.019/yr

TAMUQ

Sticky Note

Add till 48 to get ti


Example 5.3

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Sample size = 12

Find: Type of Data and T, λ ?


Model Reliability Data

• Estimate H(t) and h(t) using an empirical reliability (survivor) function, S(t), for i failures from sample of size n

• Type I, time terminated: test interval is (0, t0), t0 fixed, to observe r failures (r is random variable).

• Type II, failure terminated: test interval is (0, tr), tr random variable, with time as long as needed to observe r failures

• If the data are complete without censoring, r = n

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S(t)

1 0 t t1

n in

ti t ti1 i 1,2,...,n 1

0 tn t

(n – i) items working in ti-ti+1


Uses of Empirical Data

• Estimating parameters of a reliability model. R(t) = exp[–H(t)]

• Evaluate the conditional functions h(t) and H(t) (conditional failure rate functions)

H(t) = –ln[R(t)]; h(t) = dH(t)/dt

• H(t) can be represented by –ln[Sn(t)] from component reliability data.

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Empirical Estimate of TTF CDF

• The empirical estimate, Fn(t), of the TTF cdf (time to failure cumulative distribution function), F(t), is the complement of the empirical estimate of the Survivor function or Reliability cdf, R(t).

• The empirical estimate of the cumulative failure cdf, Fn(t), is therefore found from the empirical estimate, Sn(t), of the Reliability cdf:

Fn(t) = 1 – Sn(t)

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Example 5.4

• failure times (years) for 19 identical units

26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 50, 56

• Time to failure is a random variable, because the 19 units exhibit random failure times even though all units were produced from the same design and process

57Ayyub, RAEE



• T = Σti = 686 yrλ = r/T

= 19/686 = 0.028/yr

58Ayyub, RAEE

1

0

Sn(t)

Complete data: All 19 units observedto failure


Complete DataSn(t) ~ a – λt

Based on test, λ = 0.028/yr Question: Is a near linear dependence of Sn(t) on λ expected?


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end of test

Slide 18Example 5.3 (cont’d)

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Longest TTF observed

End of test

Sn(t) ~ 1 – λt

Based on test, λ = 0.019/yr

Example 5.3 (cont’d)

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Components Reliability

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