10.4.1.3 Model C - Series System with Repairable/Replaceable Units
10.4.1.3 Model C - Series System
with Repairable/Replaceable Units
When a series system consists of N units (with independent unit
availabilities) separately repairable or replaceable whenever the system fails
because of any one unit failing, the steady state availability is given
Furthermore, if each "1"> is much less than 1, which is usually the case for most
practical systems, Eq. (10.29) can be approximated by:
Caution is necessary in computing ai , since Eq. (10.30) applies
to the availability of the whole system. Thus, when the units are replaceable
as line replaceable units or system replaceable units, the MTTRi is
the mean time required to replace the unit with a good one at the system
maintenance level and is not the mean
repair time of the failed removed unit. On the other hand, if failed units are
not replaced but are repaired at the system level, MTTRi is the
mean-time-torepair of the unit, which becomes also the downtime for the
system. Thus, when computing the As of the
units and the availability As
of the system, all MTTRs must be those
repair times that the system experiences as its own downtime. The
MTTRi of the ith unit is thus the system mean
repair time when the ith
If we compare Eq. (10.30) with Eq.
(10.20) in Model A we find that they are identical. The system maintenance
time ratio (MTR) is:
But the serial systemís MTTR as shown in Section 4 is given
||MTTR = Sli (MTTRi)/Sli
while its MTBF is
MTBF = (Sli)-1
(MTTRi) = Sai
In other words, the system MTR is the sum of the unit MTRs. The
MTR is actually the average system downtime per system operating hour.
Conceptually, it is very similar to the maintenance ratio (MR) defined as
maintenance man-hours expended per system operating hour. The difference is
that in the MTR one looks only at system downtime in terms of clock hours of
system repair, whereas in the MR one looks at all maintenance man-hours
expended at all maintenance levels to support system operation.
Eq. (10.30) can be still further simplified if "1">
In that case
or the system availability is equal to 1 - (the sum of the unit
Let us work some examples.
Figure 10.4-4 represents a serial system consisting of 5
statistically independent subsystems, each with the indicated MTBF and MTTR.
Find the steady state availability of the system.
Note that for the system, we cannot use any of the simplifying
assumptions since, for example, subsystems 3 and 4 have MTRs of 0.2 and 0.1,
respectively, which are not << than 1.
Also "1"> which is not < 0.1
FIGURE 10.4-4: BLOCK
DIAGRAM OF A SERIES SYSTEM
Therefore, we must use the basic relationship, Eq. (10.27).
Now let us look at a similar series system, consisting of 5
statistically independent subsystems having the following MTBFs and MTTRs, as
shown in the table below.
In this case, each
ai is << than 1 and "1"> ,
so that we can use the simplified Eq.
Of course, the power and speed of modern hand-held calculators
and personal computers tend to negate the benefits of the simplifying