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# Phase Analysis with Complex Diagrams

Reliability Phase Diagrams or RPDs, can be thought of as an extension of the Reliability Block Diagram (RBD) approach. RPDs graphically describe the sequence of different operational and/or maintenance phases experienced by a system. This means that, unlike an RBD which is limited to systems with fixed configurations, RPDs can be used to represent complex systems that may change over time. The change may be in the reliability configurations (i.e. the RBD) and/or other properties, such as the availability of resources or throughput properties. In this article, we will use BlockSim to illustrate the use of reliability phase diagrams to study a complex system.

In complex systems, each stage during a mission can be represented by a phase block. The properties of the phase block are inherited from an RBD corresponding to the system's reliability configuration in that phase, along with any associated resources of the system during that time. A reliability phase diagram is then a series of such phase blocks connected in chronological order. For more information on reliability phase diagrams, see .

## Example

Consider a process where a robot manipulator is used. Because the system operates in a remote location and stopping the operation is costly, individual components that fail but do not cause a system failure are not repaired immediately. The failed components are repaired only during the scheduled yearly maintenance of the entire operation. System failures, however, are given immediate attention, and these repairs are costly, time-consuming and incur additional warranty costs per hour of downtime. The robot manipulator is warranted for 5 years. The goal of the analysis is to determine the warranty costs incurred by the system. In order to accomplish this, three metrics are of interest: the expected number of failures of the system, the system downtime associated with unplanned system failures and the expected number of component replacements during the yearly maintenance.

### The Reliability Phase Diagram

Representing the robot system with a single RBD would result in limitations. For example, a corrective repair of a component that is due to a system failure needs to be handled differently than a corrective repair performed during the yearly maintenance because they incur different costs. Therefore, the following RPD will be used. Figure 1 shows the properties of each phase.  Figure 1: Phase properties for the robot manipulator system RPD.

By using two phases to represent the system, we segregate the results obtained from different stages (normal vs. yearly maintenance) as well as any input variations. The Normal Operation phase represents the system while in operation. In this phase, blocks that fail but do not bring down the system are not repaired upon failure but remain down until the scheduled yearly maintenance. However, the failure of a block that causes a system failure will result in the repair of that block as well as a repair of all other failed components. The Normal Operation phase inherits its properties from an RBD representing the system (Figure 2). Meanwhile, the Scheduled Yearly Maintenance phase represents the portion of the mission time when the system is brought down each year so that maintenance actions can be performed on some or all of its components. In Blocksim, a maintenance phase block is defined by, and linked to, a maintenance template. This template can be thought of as a list of the specific components (blocks) that are designated to undergo inspection, repair or replacement actions during the maintenance phase. The template and subtemplates for this example are shown in Figure 3.

Note that BlockSim 7 recognizes that a component is present in more than one phase by checking the name of the component. In order to keep the integrity of this name convention, the maintenance template must follow the same structure as the original system RBD. In other words, if the robot system is represented by one main diagram with five subdiagrams, the maintenance template must be defined as one main template with five subtemplates.

 Robot Manipulator Failure Modes Jam Subdiagram  Brake Subdiagram Watchdog Subdiagram  Comm Subdiagram Mechanical Failure Subdiagram  Figure 2: Robot Manipulator System diagram and subdiagrams.

 Robot Manipulator Template Jam Subtemplate  Break Subtemplate Watchdog Subtemplate  Comm Subtemplate Mechanical Failure Subtemplate  Figure 3: Scheduled Yearly Maintenance template and subtemplates.

### The Analysis

Table 1 gives the failure and repair inputs of the system.

 Block Name Failure DistributionNormal Operation(in days) Repair DistributionNormal Operation(in days) Repair DistributionYearly Maintenance(in days) JF WBL(Beta=4, Eta=6000) EXP(Mean=12) EXP(Mean=3) KRF WBL(Beta=2, Eta=7000) EXP(Mean=12) XP(Mean=4) COM EXP(Mean=5000) EXP(Mean=16) EXP(Mean=4) EStop EXP(Mean=4000) EXP(Mean=14) EXP(Mean=2) MSF WBL(Beta=3, Eta=10000) EXP(Mean=13) EXP(Mean=4) RCF EXP(Mean=5000) EXP(Mean=14) EXP(Mean=5) Primary Brake Cold Standby, Active distribution:WBL(Beta=2, Eta=1000) EXP(Mean=10) EXP(Mean=5) Secondary Brake Cold Standby, Active distribution:WBL(Beta=2, Eta=1000) EXP(Mean=10) EXP(Mean=5) Processor I Load sharing, WBL-IPL:Beta=1.5, K=2.38E-4, n=1.32) EXP(Mean=9) EXP(Mean=2) Processor II Load sharing, WBL-IPL:(Beta=1.5, K=2.38E-4, n=1.32) EXP(Mean=9) EXP(Mean=2) Sensor EXP(Mean=5000) EXP(Mean=9) EXP(Mean=2) Primary PN EXP(Mean=5000) EXP(Mean=12) EXP(Mean=3) Secondary PN EXP(Mean=5000) EXP(Mean=12) EXP(Mean=3) Third PN EXP(Mean=5000) EXP(Mean=12) EXP(Mean=3) Fourth PN EXP(Mean=5000) EXP(Mean=12) EXP(Mean=3) Other EXP(Mean=100000) EXP(Mean=15) EXP(Mean=3)

Table 1: Failure and repair properties.

We can now run a simulation for the mission time of interest (5 years or 1825 days). Figure 4: Simulation settings.

BlockSim 7 provides multiple results at the overall mission level, as well as at the phase level. The results of interest are highlighted in Figures 5 and 6. Figure 5: The expected number of failures and the component downtime
associated with system failures. Figure 6: The expected number of component replacements during yearly maintenance.

## Conclusion

This article presented an example of how to use reliability phase diagrams for modeling a system with multiple levels of complexity, including subdiagrams and container constructs, along with the use of a maintenance phase. We also demonstrated how RPD analysis can obtain specific metrics of interest that can be used to perform a warranty analysis.

## References

 ReliaSoft Corporation, System Analysis: Reliability, Availability and Optimization Reference. ReliaSoft Publishing, 2007. http://reliawiki.org/index.php/Reliability_Phase_Diagrams_(RPDs) 