10.4.1.1 Model A  Single Unit
System (Point Availability)
Consider first a single unit system or a strictly serial system
that has a reliability, R(t); its availability, A(t), that it will be in an
“up” state (i.e., will be operating) at time, t, when it started in an “up”
condition at t = 0 is given by:
where:
l is the failure rate and m
is the repair rate
If it started in a “down” state at
t = 0
This assumes that the probability density functions for failures
and repairs are exponentially distributed and given by, respectively:
We may write Equation 10.13
also in terms of the reciprocal values of the failure and repair rates, i.e.,
in terms of the MTBF and the MTTR, remembering, however, that both
timetofailure and timetorepair must be exponentially distributed for the
equation to hold.


(10.17) 
When we study this equation
we see that as t increases the second term on the right diminishes and that
availability in the limit becomes a constant, i.e.,
We call this the steadystate availability or inherent uptime
ratio of a serial system. It is equivalent to the intrinsic availability,
Ai, discussed in Section
5.
Figure 10.42 shows plots of A(t), instantaneous availability,
and A_{i} or A_{s} (steady state
availability) for a single system having a failure rate, (l), of 0.01 failures/hour and
a repair rate (m), of 1 repair/hour.
Note that the transient term decays rather rapidly; it was shown
in Section
5 that the transient term becomes negligible for
An important point to be
made is that Eq. (10.18) holds regardless of the probability distribution of
timetofailure and timetorepair.
Looking again at Eq. (10.18), we may divide the numerator and
denominator by the MTBF and write the steady state availability as
follows:
where:
a 
= 
MTTR/MTBF, the maintenance time ratio (MTR), or
alternatively,

a 
= 
l/m which the reader may recognize from queuing
theory as the “utilization” factor. Thus, the availability, A, does
not depend upon the actual values of MTBF or MTTR or their reciprocals
but only on their
ratio. 
Since there is a whole range of
MTBF (1/l) and
MTTR (1/m) values
which can satisfy a given availability requirement, the system designer has
the option of trading off MTBF and MTTR to achieve the required system
availability within technological and cost constraints. This will be discussed
later.
Another observation to be made from
Eq. (10.20) is that if a, which is equal to MTTR/MTBF, or l/m, is less than 0.10, then
A_{i} can be approximated by 1  MTTR/MTBF, or 1 
l/m.
Thus far we have discussed inherent
or intrinsic availability which is the fundamental parameter used in
equipment/system design. However, it does not include preventive maintenance
time, logistic delay time, and administrative time. In order to take these
factors into account, we need several additional definitions of
availability.
For example, achieved availability,
A_{a}, includes preventive
maintenance and is given by the formula:
where M is the mean active
corrective and preventive maintenance time and MTBM is the mean interval
between corrective and preventive maintenance actions equal to the reciprocal
of the frequency at which these actions occur, which is the sum of the
frequency or rate (l) at which corrective maintenance actions occur and the
frequency or rate (f) at which preventive maintenance actions
occur.
Therefore,
MTBM = 1/(l + f)
Operational availability, A_{o}, includes, in addition to A_{a}, logistic time, waiting time, and administrative time,
so that the total mean downtime MDT becomes:
MDT = M + Mean Logistic Time +
Mean Administrative Time
and adds to the uptime the ready time, RT, i.e.,
It is important to realize that RT is the system average ready
time (available but not operating) in a complete operational cycle, the cycle
being MTBM + MDT + RT.
Example 3: Illustration of Availability Calculations
The following example is provided to clarify the concepts in the
subsection. A ground radar system was found to have the following R&M
parameters. Determine A_{i},
A_{a}, and
A_{o}:
MTBF = 100 hours
MTTR = 0.5 hour
Mean active preventive maintenance time = 0.25 hours
Mean logistic time = 0.3 hour
Mean administrative time = 0.4 hours
MTBM = 75 hours for either corrective or preventive
maintenance actions
Mean ready time = 20 hours
Intrinsic or Inherent Availability = A_{i}
"1">
Achieved Availability = A_{a}
"1">
Operational Availability = A_{o}