7.5.4 __Redundancy in Time
Dependent Situations__

The reliability of elements used in redundant configurations is
usually time dependent. If the relation between element reliability and time
is known, inclusion of the time factor does not change the basic notation and
approach to redundancy computation outlined above. As an example, assume two
active independent elements in parallel. System reliability is given by:

R
= p_{a} + p_{b} -
p_{a} p_{b}

This equation is applicable for one time interval. To express
reliability over a segment of time, the reliability of each element must be
expressed as a function of time.

Hence,

R(t)
= p_{a}(t) + p_{b}(t) -
p_{a}(t) p_{b}(t)

where:

R(t)
= system reliability for time t, t >
0

and

p_{a}(t) , p_{b}(t) = element
reliabilities for time t

The failure pattern of most components is described by the
exponential distribution, i.e.:

R(t) = e^{-lt} = e^{-t /
q}

where l is the constant failure rate; t is the time interval
over which reliability, R, is measured; and q
is the
mean-time-between-failure.

For two elements in series with
constant failure rates la and lb, using the product rule of reliability
gives:

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The system reliability, R(t), function
is also exponential. With redundant elements present in the system, however,
the system reliability function is not itself exponential. This is illustrated
by two operative parallel elements whose failure rates are constant.
From:

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which is not of the simple
exponential form e^{-lt}. Element failure rates cannot, therefore, be combined
in the usual manner to obtain the system failure rate if considerable
redundancy is inherent in the design.

Although a single failure rate
cannot be used for redundant systems, the mean-time-to-failure of such systems
can be evaluated. The mean life of a redundant "pair" whose failure rates are
l_{a} and l_{b}, respectively, can be
determined from:

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If the failure rates of both elements are equal, then,

R(t)
= 2e^{-lt}
- e^{-2lt}

and

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For three independent
elements in parallel, the reliability function is:

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and

If

l_{a} =
l_{b} = l_{c} =
l

then

R(t)
= 3e^{-lt}
- 3e^{-2lt}
+ e^{-3lt}

and

In general, for n active
parallel elements, each element having the same constant failure rate,
l,

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and

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