Issue 98, April 2009

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 sub-elements for the i^th subsystem.
• n is the total number of sub-elements, 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 steady-state 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.

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