10.4.1.2 __Model B -
Average or Interval Availability__

What we discussed in the previous section is the concept of
point availability which is the probability that the system is “up” and operating at
any point in time. Often, however, one may be interested in knowing what
percent or fraction of a time interval (a,b) a system can be expected to
operate. For example, we may want to determine the availability for some
mission time. This is called the interval or average availability,
A_{AV} , of a system and is given by the time average of the
availability function A(t) averaged over the interval (a,b):

For instance, if we want to know the fraction of time a system
such as shown in Figure 10.4-2 will be operating counting from t = 0 to any
time, T, we substitute A(t) of Eq. (10.13) into Eq. (10.23) and perform the
integration. The result is:

Figure 10.4-3 shows the
relationship of A(t) to A_{AV}(t) for the exponential case. Note that
in the limit in the steady state we again get the availability A of Eq.
(10.18), i.e.,

But in the transient state of the process, as shown in the
figure for an interval (0, T), before equilibrium is reached A_{AV}(t)
is in the exponential case larger than A(t) for an interval (0, t). This is
not true for all distributions, since A(t) and A_{AV}(t) may be
subject to very large fluctuations in the transient state.

From Eq. (10.24) we may also get the average or expected “on”
time in an interval (0, t) by multiplying A_{AV}(t) and t, the length
of the time interval of interest. Ref. [8], pp.
74-83, contains an excellent mathematical
treatment of the pointwise and interval availability and related
concepts.

Unavailability (U) is simply one
minus availability (1-A).