MTTF, MTBF, Mean Time Between
Replacements and MTBF with Scheduled Replacements
Seasoned reliability engineers
know that there is a great deal of discussion and confusion
regarding the terms MTTF and MTBF. We certainly hope that the
addition of two more terms in the title won't scare
readers away! In fact, the purpose of this article is to clear
up the confusion by defining these terms and using examples to
illustrate their differences and discuss the applications of
each term. ReliaSoft's
Weibull++,
RGA and
BlockSim
software packages will be used for illustration.
MTTF: Mean time to
failure describes the expected time to failure for a
nonrepairable system.
For example, assume you tested
3 identical systems starting from time 0 until all of them
failed. The first system failed at 10 hours, the second failed
at 12 hours and the third failed at 13 hours. The MTTF is the
average of the three failure times, which is 11.6667 hours.
If these three failures are
random samples from a population and the failure times of this
population follow a distribution with a probability density
function (pdf) of
, then
the population MTTF can be mathematically calculated by:

(1) 
Assuming the failure times
follow a Weibull distribution, we can use Weibull++ to
estimate the parameters for the distribution and calculate the
population MTTF. The analysis settings and estimated parameters
are:
Table 1:
Results from Weibull++
Distribution 
Weibull2P 
Analysis 
RRX 
CBMethod 
FM 
Ranking 
MED 
Beta 
7.2393 
Eta 
12.3559 
Rho 
0.9904 
LKValue 
5.2592 
Fail/Susp 
3/0 
The Mean Life (MTTF) can be
calculated in the Quick Calculation Pad (QCP):
Figure 1: MTTF Calculated in Weibull++
Figure 1 also gives the
twosided 90% confidence bounds of the estimated MTTF. The units
of the calculated MTTF and its bound are the same as the units
of time for the data (which happen to be hours in this example).
MTBDE: Mean Time between Downing Event, describes the expected time between two consecutive
downing events for a repairable system.
For example, assume you are
testing a system that can be repaired when there is a failure. The failures causes the system to go down.
The first failure happens at 10 hours and it takes 5 hours to
fix. The second failure is at 27 hours and the repair duration
is 3 hours. Then after working for 13 hours, the system fails at
43 hours. The repair lasts for 7 hours and the system is
restored at 50 hours. This failure and repair process can be
illustrated using the following graph.
Figure 2: Failure and Repair Process for a Repairable System
without Scheduled Replacements
The MTBDE =
x (T1 + T2) = 16.5 hours, if you use only the
observations of complete cycles. You can add one more cycle by
combining x_{0} and y_{3}. Then
the MTBDE =
⅓
x (T1 + T2 + x_{0} + y_{3})
hours.
If all the uptime durations
x_{i} are independent and identically distributed
(i.i.d) and all the repair durations y_{i} are
i.i.d, then:
MTBDE = MTBF
+ MTTR (Mean Time to Repair) 
(2) 
Eqn. (2) shows that the MTBDE is
the sum of the average uptime and the average downtime (MTTR).
The definition of MTBF is given next.
MTBF: Mean Time between Failures. This average time
excludes the time spent waiting for repair, being repaired, being requalified, and other
downing events such as inspections and preventive maintenance and so on; it is intended to
measure only the time a system is available and operating.
For the above example, it will be:
MTBF
= ⅓
(x_{0} + x_{1} + x_{2}) = 11.6667
The above equation assumes that all the downing events are caused by failures.
The duration of the downing events are the duration of repairs.
Again, this calculation assumes the uptime
durations x_{i} are i.i.d. However, for a repairable
system, the i.i.d assumption for x_{i} is rarely
true unless the system can be treated as brand new after each
repair or the distribution of x_{i} is
exponential. When the i.i.d assumption is not true (for example,
for a nonhomogenous Poisson process [NHPP]), MTBF is a function
of time. Often, the repair duration is relatively short compared
to the time between failures and can be ignored. ReliaSoft's
RGA software package can be used to calculate MTBF for a
repairable system when the repair durations are ignored. For
example, a typical MTBF vs. Time plot in RGA
will be:
Figure 3: MTBF vs. Time Plot
for a Repairable System
The points on the
plot are the observed cumulative MTBFs. These values are
calculated by the following equation:

(3) 
where:
The curve in Figure 3 is the
estimated MTBF by the Crow AMSAA model for repairable systems.
Mean Time Between
Replacements: This metric is usually used for
nonrepairable components or subsystems in a repairable system.
For example, a light bulb in a machine is replaced after every
Tp hours of operation or replaced at failure. The mean time
between replacements metric describes the average time between
two consecutive replacements under these conditions.
If the replacement time is
short and can be ignored, there is a closed form solution for
mean time between replacements. The expected time between two
adjacent replacements is given by:

(4) 
The first term in the above
equation is for the case when the replacement occurs at the
scheduled interval T_{p}. The second term is for
the case when the replacement occurs at the first failure time
x (0 < x
< T_{p} ).
For example, if the light bulb
has a Weibull distribution with
β
= 1.5, η
= 5000 and T_{p} = 3000, the mean time between
replacements is 2515, calculated by Eqn. (4). You also can use
ReliaSoft's BlockSim to estimate this value through
simulation. Since the replacement duration is ignored in Eqn.
(4), it is set to a small number, such as 0.0001, in the
simulation. The simulation settings are shown next.
Figure 4: Simulation Settings in BlockSim
The simulation results are:
Figure 5: Simulation Results in BlockSim
From the results, we can see
that the total number of events (replacements) is 39.274. The
simulation time is 100,000. Therefore, the mean time between
replacements is:
Mean Time
between Replacements
= 100,000/39.274 = 2546
This simulation result is close
to the analytical solution, 2515.
MTBF with Scheduled
Replacements (MTBF_SR): This metric is used in the same
situations as mean time between replacements, but describes
something different. Usually, it is used for nonrepairable
components or subsystems in a repairable system. For
example, a light bulb in a system is replaced every Tp hours of
working or replaced at failures. This metric describes the
average time between two consecutive failures under these
conditions.
For example, a failure and
replacement process is given in Figure 6.
Figure 6: Failure and Replacement Process for a System with
Scheduled Replacements
In Figure 6, T1 is the time to
the first failure, T2 is the duration between failure 1 and 2
and T3 is the duration between failure 2 and 3. The MTBF_SR is
the average of these three values.
The MTBF with Scheduled
Replacements metric also has a closed form solution if the
replacement time is small enough that it can be ignored. The
formula is:

(5) 
For the example used in the
previous section, the MTBF with scheduled replacements is 6766,
calculated from Eqn. (5). From the simulation results shown in
Figure 5, we know that the number of failures is 14.695. So the
mean time between failures with scheduled replacements can be
calculated as:
MTBF with
Scheduled Replacements = 100,000/14.695 = 6805
This result is close to the
analytical solution, 6766. If you increase the number of
simulations and use a larger simulation end time, you will get a
result that is even closer to the analytical solution.
When there are multiple
replaceable subsystems with different scheduled replacement
intervals, it is not easy to find a closed form solution for
MTBF_SR and mean time between replacements for the whole system.
Using simulation is a better choice. MTBF_SR and mean time
between replacements can be used to evaluate whether or not the
scheduled replacement intervals are good. With other
information, such as the logistic delays, crew costs and part
costs, you can find an optimum replacement interval.
Conclusion
In this article, four commonlyused terms in reliability
engineering are discussed. Examples show how they are used for
different purposes. MTTF is usually used for nonrepairable
systems. MTBF, the most wellknown term, is usually used for
repairable systems and is also widely used for the case where
the failure distribution is exponential. Mean time between
replacements and MTBF with scheduled replacements are applied to
repairable systems with scheduled preventive maintenance. Mean
time between replacements can be used to find the optimum
maintenance interval to minimize the cost per unit time. For
details, please read
http://reliawiki.org/index.php/Introduction_to_Repairable_Systems#Preventive_Maintenance_2. 