Shock model - Proof testing in case of hidden function

• In case of a hidden function a functional test (proof test) is often appropriate to reveal a hidden failure
• This means that a periodic function test is the natural maintenance strategy
• We need to assess the PFD (Probability of Failure on Demand) which expresses the portion of time the system is not able to perform the required function

• For a k oo N system the PFD is given by: $ \color{blue}{ \mathrm{PFD}(\tau) \approx \binom{N}{N-k+1} \frac{(\lambda \tau)^{N-k+1}}{N-k+2}+\beta \lambda \tau /2 } $
  • β is the rate of common cause failures
  • Where a k oo N system is a system comprising N (identical) units where the system is functioning if and only if k or more of the units are functioning
  • In case of a single component the folloing more familiar formula may be used: PFD = λτ/2
• The cost equation to minimiize: C(τ) = C_FT/τ + N⋅C_R(λ - λ²τ/2) + C_H⋅PFD(τ)⋅f_D
  • C_FT is the cost of performing a functional test of all the N units
  • C_R is the cost of repairing one of the N units in case of revealing a failure during the functional test
  • C_H the ecconomical cost of an hazardous event
  • f_D the rate of demands of the hidden funciton, typically the rate of gas leakages, fires etc