Reliability Demonstration Test Planning

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Overview

A Reliability Demonstration Test (RDT) provides statistical evidence that a product meets its reliability requirement. The rdt() function computes either:

the required test time given a fixed sample size, or
the required sample size given a fixed test duration.

The calculation uses the Weibull distribution; setting beta = 1 (the default) gives the exponential special case.

Test Time

For a zero-failure test plan (f = 0), the required cumulative test time \(T\) for \(n\) units satisfying reliability \(R\) at mission time \(t_m\) with confidence \(C\) is:

\[T = t_m \left(\frac{-\ln C}{-\ln R}\right)^{1/\beta}\]

When allowable failures \(f > 0\), the chi-squared quantile replaces \(-\ln C\):

\[T = \frac{t_m}{(-\ln R)^{1/\beta}} \cdot \left(\frac{\chi^2_{1-C,\,2(f+1)}}{2n}\right)^{1/\beta}\]

Solving for Test Time

Suppose you need to demonstrate 90% reliability at 500 hours with 90% confidence, and you have 20 units available for testing:

plan <- rdt(
  target       = 0.90,   # 90% reliability
  mission_time = 500,    # hours
  conf_level   = 0.90,   # 90% confidence
  n            = 20      # 20 test units
)
print(plan)
#> Reliability Demonstration Test (RDT) Plan
#> -----------------------------------------
#> Distribution:  Exponential 
#> Weibull Shape Parameter (Beta):  1 
#> Allowed Failures (f):  0 
#> Target Reliability:  0.9 
#> Mission Time:  500 
#> Input Sample Size (n):  20 
#> Required Test Time (T):  546.36

Each unit must be tested for Required_Test_Time hours (or failure, whichever comes first) with zero failures allowed.

Solving for Sample Size

Alternatively, fix the test duration at 800 hours per unit:

plan2 <- rdt(
  target       = 0.90,
  mission_time = 500,
  conf_level   = 0.90,
  test_time    = 800
)
print(plan2)
#> Reliability Demonstration Test (RDT) Plan
#> -----------------------------------------
#> Distribution:  Exponential 
#> Weibull Shape Parameter (Beta):  1 
#> Allowed Failures (f):  0 
#> Target Reliability:  0.9 
#> Mission Time:  500 
#> Input Test Time (T):  800 
#> Required Sample Size (n):  14

Effect of Weibull Shape

For products with wear-out (beta > 1) or infant mortality (beta < 1), the Weibull shape parameter changes the required test time. Here we compare three scenarios:

betas <- c(0.8, 1.0, 1.5)
for (b in betas) {
  p <- rdt(target = 0.90, mission_time = 500, conf_level = 0.90, n = 20, beta = b)
  cat(sprintf("beta = %.1f  ->  required test time = %.1f hours\n",
              b, p$Required_Test_Time))
}
#> beta = 0.8  ->  required test time = 558.6 hours
#> beta = 1.0  ->  required test time = 546.4 hours
#> beta = 1.5  ->  required test time = 530.4 hours

Higher beta (wear-out) demands shorter tests because failures concentrate near end-of-life; lower beta (infant mortality) demands longer tests.

Allowing Failures

A zero-failure plan is conservative. Allowing a small number of failures can substantially reduce the test burden:

for (f in 0:3) {
  p <- rdt(target = 0.90, mission_time = 500, conf_level = 0.90, n = 20, f = f)
  cat(sprintf("f = %d  ->  required test time = %.1f hours\n",
              f, p$Required_Test_Time))
}
#> f = 0  ->  required test time = 546.4 hours
#> f = 1  ->  required test time = 923.0 hours
#> f = 2  ->  required test time = 1262.9 hours
#> f = 3  ->  required test time = 1585.2 hours

Summary

Parameter	Role
`target`	Required reliability R(t_m)
`mission_time`	The time at which reliability is evaluated
`conf_level`	Statistical confidence in the demonstration
`beta`	Weibull shape (1 = exponential)
`f`	Allowable failures (0 = most conservative)
`n`	Sample size → solves for test time
`test_time`	Test duration per unit → solves for sample size

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