21.1: RELIABILITY OF 4-COMPONENT AND 6- COMPONENT SERIES SYSTEMS
If four identical components each with reliability Ri = 0.98 are con- nected in series, the system reliability is
Rs 4 4 = 0.98
= 0.922 (21.4)
If two more identical components are added in series, the system relia- bility becomes
Rs 2 6 = 0.922 × 0.98
= 0.886 (21.5)
The system with the higher number of components in series is therefore seen to be much less reliable.
Note that as a consequence of the basic laws of the arithmetic operation of multiplication, system reliability for a series conﬁguration is independent of the order in which the components are arranged.
Now consider the system depicted in the bottom panel of Fig 21.1 where the components are arranged in parallel, and again, the reliability of component Ci is Ri. Observe that in this case, if one component fails, the entire system does not necessarily fail. In the simplest case, the system fails when all n components fail. In the special “k-of-n” case, the system will function if at least k of the n components function. Let us consider the simpler case ﬁrst.
In the case when the system fails only if all n components fail, then Rs
is the probability that at least one component functions, which is equiva- lent to 1 − P (no component functions). Now, let Fi be the “unreliability” of
component i, the probability that the component does not function; then by deﬁnition,
Fi = 1 − Ri (21.6)
If Fs is the system “unreliability,” i.e., the probability that no component in the system functions, then by independence,
Fs = n Fi (21.7)
and since Rs = 1 − Fs, we obtain
Rs = 1 − n(1 − Ri) (21.8)
For parallel systems, therefore, we have the product law of unreliabilities ex- pressed in Eq (21.7), from which Eq (21.8) follows. Speciﬁc cases of Eq (21.8) can be informative, as the next example illustrates.