8. Assessing Product Reliability
8.1. Introduction
8.1.8. How can you evaluate reliability from the "bottom-up" (component failure mode to system failure rate)?

8.1.8.4.

R out of N model

An r out of n model is a system that survives when at least r of its components are working (any r)

An "r out of n" system contains both the series system model and the parallel system model as special cases. The system has n components that operate or fail independently of one another and as long as at least r of these components (any r) survive, the system survives. System failure occurs when the (n-r+1)th component failure occurs. 

When r = n, the r out of n model reduces to the series model. When r = 1, the r out of n model becomes the parallel model. 

We treat here the simple case where all the components are identical. 

Formulas and assumptions for r out of n model (identical components):

1. All components have the identical reliability function R(t).
2. All components operate independently of one another (as far as failure is concerned).
3. The system can survive any (n-r) of the components failing. The system fails at the instant of the (n-r+1)th component failure.

Formula for an r out of n system where the components are identical

System reliability is given by adding the probability of exactly r components surviving to time t to the probability of exactly (r+1) components surviving, and so on up to the probability of all components surviving to time t. These are binomial probabilities (with p = R(t)), so the system reliability is given by: 

Note: If we relax the assumption that all the components are identical, then R_s(t) would be the sum of probabilities evaluated for all possible terms that could be formed by picking at least r survivors and the corresponding failures. The probability for each term is evaluated as a product of R(t)'s and F(t)'s. For example, for n = 4 and r = 2, the system reliability would be (abbreviating the notation for R(t) and F(t) by using only R and F)

R_s = R₁R₂F₃F₄ + R₁R₃F₂F₄ + R₁R₄F₂F₃ + R₂R₃F₁F₄
        + R₂R₄F₁F₃ + R₃R₄F₁F₂ + R₁R₂R₃F₄ + R₁R₃R₄F₂
        + R₁R₂R₄F₃ + R₂R₃R₄F₁ + R₁R₂R₃R₄