6.4.5.2 Mathematical Models for
Reliability Prediction
For the simplest case of equipment or system configurations
consisting of N independent elements or subsystems in series, the reliability
equation is:
where:
R_{s} is the
equipment or system reliability
R_{i}
is the reliability of each of the elements or
subsystems
For the case where time is a
factor
where:
R_{s}(t) 
= 
The probability that the system will not fail before
time t. (In this case a “system” is
considered to be any device consisting of n elements, none of which
can fail without system failure). 



R_{i}(t) 
= 
The probability that the i^{th}
element of the system will not fail before time
t. 
Finally, if one assumes that each of the R_{i}(t) 's is
exponentially distributed with constant failure rate of l_{i}, then

Ri(t) = exp ( l_{i}t
) 
(6.38) 
Then,
Also,
where:
l_{s} = system failure rate
l_{i} = failure rate of each of the independent elements of
the system
And,
Eqs. (6.38), (6.39), and
(6.41) are the basic equations used in the reliability prediction of
electronic equipment/systems.
The use of the exponential
distribution of time to failure for complex systems is usually justified
because of the many forces that can act upon the system and produce failure.
For example, different deterioration mechanisms, different part hazardrate
functions, and varying environmental conditions often result in, effectively,
random system failures.
Another justification for assuming
the exponential distribution in longlife complex systems is the so called “approach to a stable state,” wherein the
system hazard rate is effectively constant regardless of the failure pattern
of individual parts. This state results from the mixing of part ages when
failed elements in the system are replaced or repaired. Over a period of time,
the system hazard rate oscillates, but this cyclic movement diminishes in time
and approaches a stable state with a constant hazard
rate.