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)?

## 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 the $$r$$ 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: $$R_S(t) = \sum_{i=r}^n \left( \begin{array}{c} n \\ i \end{array} \right) \left[ R(t) \right]^i \left[ 1 - R(t) \right] ^{n-i} \,\, .$$ 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 $$F$$) $$\begin{eqnarray} R_S & = & R_1 R_2 F_3 F_4 + R_1 R_3 F_2 F_4 + R_1 R_4 F_2 F_3 + R_2 R_3 F_1 F_4 \\ & + & R_2 R_4 F_1 F_3 + R_3 R_4 F_1 F_2 + R_1 R_2 R_3 F_4 + R_1 R_3 R_4 F_2 \\ & + & R_1 R_2 R_4 F_3 + R_2 R_3 R_4 F_1 + R_1 R_2 R_3 R_4 \,\, . \end{eqnarray}$$