### Example

The following RBD illustrates the relationship between the primary modes. The subdiagram blocks A, B and C contain the sub-modes, and the node represented by 2/3 indicates the *k*-out-of-*n* configuration (in this case, 2 out of 3 paths must occur for the component to fail).

The objective of the analysis is to obtain the following:

- The reliability of the component after 1 year (8,760 hrs).
- The B10 life of the component.
- The MTTF (mean-time-to-failure) of the component
- The rank of the modes in order of importance at 1 year.
- The reliability, B10 life and MTTF of the component if mode B were eliminated.

### Analysis

### Step 1: Mode A

There are five independent (i.e., if one mode occurs, the rest are not more likely to occur) sub-modes associated with mode A: events S1, S2, T1, T2 and Y. Assume that:

- Events S1 and S2 each have a constant rate of occurrence with a probability of occurrence of 1 in 10,000 and 1 in 20,000, respectively, in a single year (8,760 hours).
- Events T1 and T2 are more likely to occur in an older component than a newer product (i.e., they have an increasing rate of occurrence) and have a probability of occurrence of 1 in 10,000 and 1 in 20,000, respectively, in a single year and 1 in 1,000 and 1 in 3,000, respectively, after two years.
- Event Y also has a constant rate of occurrence with a probability of occurrence of 1 in 1,000 in a single year.

There are three possible ways for mode A to manifest itself:

- Events S1 and S1 both occur.
- Event T1 or T2 occurs.
- Event Y and either event S1 or event S2 occur (i.e., events Y and S1 or events Y and S2).

The following RBD illustrates the conditions for mode A.

The RBD includes a starting block (NF) and an end node (2/2). The starting block is set to a reliability equal to 1 or 100% so that it cannot fail and, therefore, will not affect the results of the analysis. The end node indicates that both paths leading into the node must work in order for mode A to occur.

Based on the given probabilities, compute the distribution parameters for each block. For events S1, S2 and Y, you can use an exponential distribution because a constant rate of occurrence was assumed. Figures 3 and 4 show how you can use the Quick Parameter Experimenter (QPE) in BlockSim to compute the mean time of event S1. The mean time for events S2 and Y can be computed in a similar manner.

Events T1 and T2 need to be modeled using a life distribution that does not have a constant failure rate. The following picture shows the computed parameters of event T1 using a 2-parameter Weibull distribution. The parameters for T2 can be computed in a similar manner.

### Step 2: Mode B

There are three **dependent** sub-modes associated with mode B: events BA, BB and BC. Two out of the three events must occur for mode B to occur. Events BA, BB and BC have an exponential distribution with a mean of 50,000 hours. The events are dependent (i.e., if BA, BB or BC occurs, the remaining events are more likely to occur). Specifically, when one event occurs, the MTTF of the remaining events is cut in half. This is basically a load sharing configuration. The reliability function for each block will change depending on the other events. Therefore, the reliability of each block is not only dependent on time, but also on the stress (load) that the block sees.

The following picture shows the RBD of mode B. The blocks representing the sub-modes are inside a load sharing container.

The following picture shows the Block Properties window of the load sharing container. The Number of paths required field is set to 2, indicating that 2 out of the 3 contained events must occur for mode B to occur.

To describe the dependency between the events, BlockSim uses a multiplier called the *weight proportionality factor*, which determines how the load will be shared. For example, if a container holds two blocks and one block has a factor of 3 while the other has a factor of 9, then the first block will receive 25% of the load (3/12) and the second block will receive 75% of the load (9/12).

In this case, all three contained blocks have a weight proportionality factor equal to 1, indicating that they will share the load evenly (33.33% of the load each) when all are operating. If one fails, then the other two will take over the load. The weight proportionality factor needs to be set up in each block in the load sharing container, as shown in the following example.

### Step 3: Mode C

There are two sequential sub-modes associated with mode C: events CA and CB. Both events must occur for mode C to occur. Event CB will occur only if event CA has occurred. If event CA has not occurred, then event CB will not occur. Both events, CA and CB, occur based on a Weibull distribution. For event CA, beta = 2 and eta = 30,000 hours. For event CB, beta = 2 and eta = 10,000 hours.

This scenario is similar to standby redundancy. Basically, if CA occurs then CB gets initiated. The following picture shows the RBD of mode C. The blocks representing the sub-modes are inside a standby container.

### Step 4: Modes D, E and F

Modes D, E and F can all be represented using the exponential distribution. The failure distribution properties for modes D, E and F are presented next.

- D: MTTF = 200,000 hours
- E: MTTF = 175,000 hours
- F: MTTF = 500,000 hours

### Step 5: Component

The last step is to set up the RBD of the component in BlockSim (as shown in Figure 1), and then calculate the answers to the questions posed earlier.

- Using the Analytical QCP, the reliability of the component at 1 year (8,760 hours) is estimated to be 97.3517%
- Using the Analytical QCP, the B10 life of the component is equal to 14,715.5509 hours.
- Using the Analytical QCP, the mean life of the component is equal to 31,685.8921 hours.
- The reliability importance plot displays the ranking of the modes after 1 year, as shown next.

- Assuming that mode B is removed, the results are:
- R = 98.7007%
- B10 = 16,867.4551 hours
- MTTF = 34,321,2133 hours