The switch, “S,” detects a failure of
the operative element and instantaneously switches from the failed element to
a standby element.
The switch may fail in two ways: (1) the
switch may fail to operate when required, Q_{1} and (2) the
switch may operate without command (i.e., prematurely),
Q_{2}. Q_{1} and
Q_{2} can be represented as (1P_{1}) and (1P_{2}) as the probability of failure (Q) plus the
probability of success (P) equals one.

P_{S} = P(mission success with A
working) P_{A}
+
P(mission success with A failed) (1  P_{A}) P_{S} =
P_{2} P_{A} + (1  P_{2} ) P_{B}
P_{A} + P_{1} P_{B} (1 
P_{A}) P_{S} = P_{A} P_{B} (1 
P_{1}  P_{2}) + P_{A} P_{2} +
P_{B} P_{1} 
(6.28)

The equivalent series reliability mathematical model for this
system is: