Example Using Success Failure Paths in Phase Diagrams

This example appears in the System Analysis Reference book.

Consider a system whose RBDs for Phase 1 and Phase 2 are as shown in the figure below:



In Phase 1, Block A follows Weibull distribution with Beta = 1.5 and Eta = 1000 hours. Block B follows Weibull distribution with Beta = 1.5 and Eta = 1200 hours. No maintenance.

In Phase 2, Block A follows Weibull distribution with Beta = 1.5 and Eta = 900 hours. Block B follows Weibull distribution with Beta = 1.5 and Eta = 1150 hours. No maintenance.

Example 1
Suppose the phase diagram is as shown next. The maintenance phase has a CM with duration 80 hours and PM with duration 20 hours for both Block A and B. Phase 1 has duration 100 hours and Phase 2 has duration 500 hours. If there is no failure in Phase 1, upon finish, it goes to Phase 2. If there is a failure in Phase 1, it would go to Maintenance Phase 3 immediately. If there is no failure in Phase 2, upon finish, it goes back to Phase 1. If there is a failure in Phase 2, it would go to Maintenance Phase 3 immediately.



The Block Up/Down plot is shown below:


 * 1) At the first cycle (0-100 hours for Phase 1 and 200-600 hours for Phase 2), there is no failure, thus after Phase 2, Phase 1 is executed (back to Phase 1 after this cycle).
 * 2) At 616 hours (Phase 1 of the second cycle), Block A fails. The system doesn't enter Maintenance Phase 3 immediately because in Phase 1, Block A and Block B are parallel. Failure of only Block A does not bring the system down.
 * 3) At 700 hours, after Phase 1, the system goes into Phase 2. In Phase 2, the failure of Block A (passed from Phase 1) brings the system down because in Phase 2, Block A is in series with Block B. Thus the system goes into Maintenance Phase 3 immediately. Block B is not down, thus it gets PM with duration 20 hours. Block A gets CM with duration 100 hours.
 * 4) At 1669 hours, Block A fails in Phase 2, which brings down the system and the system goes into Maintenance Phase 3 immediately. Block A gets CM and Block B gets PM.



Example 2
Now suppose that the phase diagram is as follows, where everything is the same as in Example 1 except that the path from Phase 2 to Maintenance Phase 3 changes from a failure path to a success path. This means that whether or not there is a failure in Phase 2, the system would go to the Maintenance Phase 3 after Phase 2. But if there is a failure in Phase 2, the system would go to the Maintenance Phase 3 after Phase 2, not immediately.



The Block Up/Down plot is shown below:


 * 1) At 219 hours, Block A fails in Phase 2. This failure brings the system down. However, since the path from Phase 2 to Maintenance Phase 3 is a success path, the system doesn't go into Maintenance Phase 3 immediately. Instead, it goes into Maintenance Phase 3 after Phase 2 ends. Block A gets CM and Block B gets PM in the maintenance phase.
 * 2) At 1934 hours, Block B fails in Phase 1. This failure doesn't bring the system down because in Phase 1, the system is in parallel structure. After Phase 1, the system goes into Phase 2 and the failure of Block B brings they system down in Phase 2. However, the system doesn't go into Maintenance Phase 3 immediately because the path between them is a success path. The system goes into Maintenance Phase 3 after Phase 2 ends, and Block B gets CM and Block A gets PM there.