Hence the reliability or probability of
no failure is
R = 1  Q = 1  q_{1}
q_{2}
For example, assume that A_{1} has a reliability
r_{1} of 0.9 and A_{2} a reliability r_{2} of 0.8.
Then their unreliabilities q_{1} and q_{2} would be
q_{1} = 1  r_{1} = 0.1
q_{2} = 1  r_{2} = 0.2
and the probability of system failure would be
Q = (0.1)(0.2) = 0.02
Hence the system reliability would be
R = 1  Q = 0.98
which is a higher reliability than either of the elements acting
singly. Parallel redundancy is therefore a design tool for increasing system
reliability when other approaches have failed. It should be pointed out that
while redundancy reduces mission failures, it increases logistics
failures.
In general, with n elements in parallel, the overall probability
of failure at time t is

Q(t) = q1(t) • q2(t) • . . . • q (t)_{n} 
(7.9) 
and the probability of operating without failure is

R(t) = 1  Q(t) = 1  q_{1}(t)
q_{2}(t) . . . q_{m}(t) 
(7.10) 
which, because q_{i}(t) = 1  r_{i}(t) for each
component, can also be given as

R(t) = 1  [ 1  r_{1}(t) ] [ 1
 r_{2}(t) ] . . . [ 1  r_{m}(t) ] 
(7.11) 
When each of the component reliabilities is equal, the above
equations reduce to

R(t) = 1  [q(t)]m = 1  [ 1  r(t)
]^{m} 
(7.13) 
Figure 7.54 summarizes the characteristics of simple parallel
active redundancy.