Reliability Allocation Using
Lambda Predict When developing a new product
or improving an existing one, engineers are often faced with the
task of designing a system that must meet a certain set of
reliability specifications. This involves a balancing act in
order to determine how to allocate reliability among the
subsystems/components in the system. In this article we will
introduce several different reliability allocation methods,
which are included in ReliaSoft's
Lambda Predict
software.
Reliability allocation involves
solving the following inequality:
where:
 R_{i}
is the reliability allocated to the i^{th} subsystem/component.
 f
is the functional relationship between the subsystem/component
and the system.
 R_{s}
is the required system reliability.
Several algorithms for
reliability allocation have been developed [1]:
 Equal apportionment
 AGREE apportionment
 ARINC apportionment
 Feasibility of Objectives apportionment
 Repairable Systems apportionment
In the next section, we will use an example to
explain these algorithms.
Example
Consider a
power circuit with several subsystems, or function blocks: power supply block,
transformer block, switch block and load block. Figure 1 shows the system tree
and the failure rate of the current design for each block.
Figure 1: Function Blocks of a Power Circuit
The system is required to have
a reliability of 0.9 at the time of 8760 hours. To achieve this, we need to
evaluate the current design and specify the reliability for each
function block in order to meet this goal.
Equal Apportionment
The simplest apportionment technique is to distribute the
reliability uniformly among all components. This method is
called Equal apportionment. Equal apportionment assumes a series
of n subsystems, all in series and having an exponential failure
distribution. Each subsystem is assigned the same reliability.
The mathematical model can be expressed as:
or:
where:
 R_{s }is the system reliability goal.

R′_{i}
is the reliability allocated to the i^{th} subsystem.
 i is the subsystem index.
 n is the total number of subsystems.
Using Lambda Predict, we get
the following results:
Figure 2: Equal Apportionment
Figure 2 shows that in order to
meet the system reliability goal of 0.9 at the operating time of
8760 hours, the reliability allocated to each block should be
0.9740. Thus, the allocated failure rate should be 3.0069 FITS.
Here, the exponential failure distribution is assumed for
each function block. It is clear that all four blocks have
exceeded the expected failure rate, as shown in the Current
Failure Rate column. A revised design is required
for each function block in order to meet the allocated
reliability.
AGREE Apportionment
The AGREE apportionment method, designed by the Advisory Group on
Reliability of Electronic Equipment, determines a minimum
acceptable mean life for each subsystem in order to fulfill a
minimum acceptable system mean life. The AGREE method assumes that all
subsystems are in series and have an exponential failure distribution. This method takes into account both the complexity
and the importance of each subsystem. The mathematical model is:
and
where:
 R_{s}(t)
is the system reliability.

R′_{i}(t_{i})
is the allocated reliability for the i^{th} subsystem.
 t
is the system operating time.
 t_{i}
is the operating time of the i^{th} subsystem.
 i is the subsystem index.
 n is the total number of subsystems.
 w_{i}
is the importance factor for the i^{th} subsystem.
 n_{i}
is the number of subelements for the i^{th} subsystem.
 n is the
total number of subelements, which is given by
.
Using Lambda Predict, we get
the following results:
Figure 3: AGREE Apportionment
Figure 3 shows that in the
current design, the failure rates for the power supply block and the
transformer block have exceeded the allocated failure rates. New
designs with lower failure rates for the power supply and the transformer
are necessary.
ARINC Apportionment
The
ARINC apportionment method was designed by ARINC Research Corporation, a subsidiary of Aeronautical Radio, Inc. The method assumes that all subsystems are in series
and have an exponential failure distribution. From the present allocation
of the subsystems, allocation improved system failure rates are
derived based on weighting factors. The mathematical expression
is:
where:
 n is the total number of subsystems.
 λ_{i}
is the present failure rate of the i^{th} subsystem.
 λ_{S}
is the required system failure rate.
 λ′_{i}
is the failure rate allocated to the i^{th} subsystem.
Figure 4 shows the ARINC apportionment
for this example. None of the four blocks meet the expected
failure rates. Based on this method, a revised design should be
considered.
Figure 4: ARINC Apportionment
Feasibility of Objectives
Feasibility of Objectives apportionment is based on numerical
ratings of the designs state of the art, the system complexity,
the mission operating time and the environment for each item to
which the product reliability will be allocated, assuming that
all subsystems are in series and have an exponential failure
distribution. Ratings are
assigned based on the engineer's experience and judgment. Ratings for each
factor range from a low of 1 to a high of 10. These four
criteria ratings are multiplied together to get an overall
weighting and are normalized so that the product sum is 1. The
mathematical model can be described as:
and
where:
 T is the operating duration.
 λ_{S}
is the system failure rate.
 λ_{i}
is the allocated subsystem i failure rate.
 C_{i}
is the percent weighting factors of the i^{th} subsystem.
 W_{i}
is the composite rating for the i^{th} subsystem.
 N is the total number of subsystems.
 r_{ik} is the
k^{th} rating result for the i^{th} subsystem.
Figure 5 shows the Feasibility of Objectives
apportionment for this example. None of the four blocks
meet the expected failure rates. Based on this method, a new design should be
considered.
Figure 5: Feasibility of Objectives Apportionment
Repairable System Apportionment
Another reliability allocation
method, called Repairable Systems apportionment, is designed for
repairable systems. Since this circuit is usually not
repairable, we will not apply this method to the example
discussed above. The algorithm is, however, briefly discussed
next.
Repairable Systems apportionment allocates subsystem failure
rates to allow the system to meet an availability objective for
a repairable system. This technique assumes all subsystems to be
in series, with exponential failure distributions and constant repair rates.
By determining the ratio of the allocated failure rate to the repair
rate for each subsystem based on a steadystate availability
calculation, the failure rate allocated to each subsystem can be
determined.
The math expression of this
method is:
where:
 A_{s} is
the required system availability.
 A_{i}
is the allocated availability for the i^{th} subsystem.
 n is the total number of subsystems.
 θ_{i}
is the ratio of allocated failure rate to the repair rate for the
i^{th}
subsystem.
 u_{i}
is the repair rate for the i^{th} subsystem.
 λ_{i}
is the allocated failure rate for the i^{th} subsystem.
Conclusions
In this article, different reliability allocation techniques
were
discussed. The simplest technique is Equal apportionment, which
distributes system reliability equally among all the subsystems.
The AGREE, ARINC and Feasibility of Objectives techniques take
additional weighting
factors into consideration during allocation. Repairable Systems
apportionment allocates failure rates for subsystems through
the ratio of the allocated failure rate to the repair rate for each
subsystem. Through reliability allocation, reliability
parameters are assigned to different system elements; in this
way, the whole system reaches the established reliability
target. To obtain good results, it is important to choose an
appropriate apportionment method based on the system reliability
requirement and the system properties. For more complicated
cases (e.g. if the distribution is not exponential and
the cost is a factor that is considered),
BlockSim can be
employed. For more on using BlockSim for allocation, see
the
Hot Topics article in the Reliability HotWire Issue 6.
References
1. ReliaSoft Corporation, Lambda Predict
Users Guide, Tucson, AZ: ReliaSoft Publishing, 2007.
