# 21.1: RELIABILITY OF 4-COMPONENT AND 6- COMPONENT SERIES SYSTEMS If four identical components each w

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.

Parallel  Systems

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,

n

Fs = n Fi                                                      (21.7)

i=1

and since  Rs  = 1 − Fs, we obtain

n

Rs  = 1 − n(1 − Ri)                                               (21.8)

i=1

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.