Using the Power Low to Analyze Complex Repairable Systems
Confidence Bounds Examples for Repairable Systems
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Using the Power Low to Analyze Complex Repairable Systems Confidence Bounds Examples for Repairable Systems |
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The parameter β must be positive, thus lnβ is approximately treated as being normally distributed.
(8)
All variance can be calculated using the Fisher Information Matrix.
Λ is the natural log-likelihood function.
(9)
(10)
(11)
Calculate the conditional maximum likelihood estimate of 
:
The Crow 2-sided (1 -
α)
100-percent confidence bounds on β
are: 
(12)
The parameter λ must be positive, thus lnλ is approximately treated as being normally distributed. These bounds are based on:
The approximate confidence bounds on λ
are given as:
where 
.
The variance calculation is the same as Eqns. (9), (10) and (11).
Time Terminated
The confidence bounds on λ for time terminated data are calculated using:
Failure Terminated
The confidence bounds on λ for failure terminated data are calculated using:
Since the growth rate is equal to 1 - β, the confidence bounds are:
(13)
If Fisher Matrix confidence bounds are used then βL and βU are obtained from Eqn. (8). If Crow bounds are used then βL and βU are obtained from Eqn. (12).
The cumulative MTBF, mc(t), must be positive, thus lnmc(t) is approximately treated as being normally distributed.
The approximate confidence bounds on the cumulative MTBF are then estimated
from:
where: 
The
variance calculation is the same as Eqns. (9), (10) and (11). 
To calculate the Crow confidence bounds on cumulative MTBF, first calculate the Crow cumulative failure intensity confidence bounds:
Then
(14)
The instantaneous MTBF, mi(t), must be positive, thus lnmi(t) is approximately treated as being normally distributed.
The approximate confidence bounds on the instantaneous MTBF are then
estimated from: 
where:
The variance calculation is the same as (9), (10) and (11). 
Failure Terminated Data
To calculate the bounds for failure terminated data, consider the following
equation: 
(15)
Find the values p1
and p2 by finding the
solution c to G(n2/c|n) = ζ for 
and 
, respectively. If
using the biased parameters, 
and 
, then the upper and lower confidence
bounds are:
where 
.
If using the unbiased parameters, 
and 
, then the
upper and lower confidence bounds are:
where 
.
Time Terminated Data
To calculate the bounds for time terminated data, consider the following
equation where I1(.) is the modified Bessel function of order one:
(16)
Find the values Π1
and Π2 by
finding the solution x
to 
and 
in the cases corresponding to the lower and upper bounds, respectively.
Calculate 
for each case. If using the biased parameters, 
and 
, then the
upper and lower confidence bounds are:
where 
.
If using the unbiased parameters, 
and 
, then the
upper and lower confidence bounds are:
where 
.
The cumulative failure intensity, λc(t) must be positive, thus lnλc(t) is approximately treated as being normally distributed.
The
approximate confidence bounds on the cumulative failure intensity are
then estimated using: 
where:
and:
The variance calculation is the same as Eqns. (9), (10) and (11): 
The Crow cumulative failure intensity confidence bounds are given by:
The instantaneous failure intensity, λi(t), must be positive, thus lnλi(t) is approximately treated as being normally distributed.
The approximate confidence bounds on the instantaneous failure intensity
are then estimated from: 
where λi(t) = λβtβ-1
and: 
The variance calculation is the same as Eqns. (9), (10) and (11): 
The Crow instantaneous failure intensity confidence bounds are given as:
(17)
The time, T, must
be positive, thus lnT
is approximately treated as being normally distributed. 
The confidence bounds on the time
are given by: 
where:
The
variance calculation is the same as Eqns. (9), (10) and (11). 
Step 1: Calculate:
Step 2: Estimate the number of failures:
Step 3: Obtain the confidence bounds on time given the cumulative failure
intensity by solving for t1
and tu
in the following equations: 
The time, T, must
be positive, thus lnT
is approximately treated as being normally distributed. 
The confidence bounds on the time
are given by: 
where:
The
variance calculation is the same as Eqns. (9), (10) and (11).
Step 1: Calculate the confidence bounds on the instantaneous MTBF as presented in Section 5.5.2.
Step 2: Calculate the bounds on time as follows.
Failure Terminated Data
So the lower an upper bounds on time are:
Time Terminated Data
So the lower and upper bounds on time are:
The time, T, must
be positive, thus lnT
is approximately treated as being normally distributed. 
The confidence bounds on the time
are given by: 
where:
The
variance calculation is the same as Eqns. (9), (10) and (11): 
Step 1: Calculate:
Step 2: Estimate the number of failures:
Step 3: Obtain the confidence bounds on time given the cumulative failure
intensity by solving for t1
and tu
in the following equations: 
These bounds are based on: 
The confidence bounds on the time are given by:
where: 
The
variance calculation is the same as Eqns. (9), (10) and (11). 
Step 1: Calculate 
.
Step 2: Use the equations from 13.1.7.9 to calculate the bounds on time given the instantaneous failure intensity.
These bounds are based on: 
The confidence bounds on reliability are given by: 
The variance calculation is the same as Eqns. (9), (10) and (11). 
Failure Terminated Data
With failure terminated data, the 100(1 - α)%
confidence interval for the current reliability at time t in a specified mission time d
is: 
where: 
ρ1
and ρ2 can be obtained
from Eqn. (15).
Time Terminated Data
With time terminated data, the 100(1 - α)%
confidence interval for the current reliability at time t in a specified mission time τ
is: 
where: 
ρ1
and ρ2 can be obtained
from Eqn. (16).
The time, t,
must be positive, thus lnt
is approximately treated as being normally distributed. 
The confidence bounds on time are calculated by using:
where:
is calculated
numerically from:
The variance calculations are done by:
Failure Terminated Data
Step 1: Calculate 
.
Step 2: Let 
and solve for t1
numerically using 
.
Step 3: Let 
and solve for t2
numerically using 
.
Step 4: If t1 < t2, then tlower = t1 and tupper = t2. If t1 > t2, then tlower = t2 and tupper = t1.
Time Terminated Data
Step 1: Calculate 
.
Step 2: Let 
and solve for t1
numerically using 
.
Step 3: Let 
and solve for t2
numerically using 
.
Step 4: If t1 < t2, then tlower = t1 and tupper = t2. If t1 > t2, then tlower = t2 and tupper = t1.
The mission time, d,
must be positive, thus ln(d) is approximately treated as
being normally distributed. 
The confidence bounds on mission time are given by using:
where:
Calculate 
from:
The variance calculations are done by:
Failure Terminated Data
Step 1: Calculate 
.
Step 2: Let 
and solve for d1
such that:
(18)
Step 3: Let 
and solve for d2
such that:
(19)
Step 4: If d1 < d2, then dlower = d1 and dupper = d2. If d1 > d2, then dlower = d2 and dupper = d1.
Time Terminated Data
Step 1: Calculate 
.
Step 2: Let 
and solve for d1
using Eqn. (18).
Step 3: Let 
and solve for d2
using Eqn. (19).
Step 4: If d1 < d2, then dlower = d1 and dupper = d2. If d1 > d2, then dlower = d2 and dupper = d1.
The cumulative number of failures, N(t), must be positive, thus ln(N(t))
is approximately treated as being normally distributed. 
where: 
The variance calculation is the same as Eqns. (9), (10) and (11). 
where
λi(T)L
and λi(T)U
can be obtained from Eqn. (17).
