The overall system in the figure below includes three systems: Power Supply System, System 1, and System 2. System 1 and System 2 are arranged in series, and they both need the Power Supply System to operate. The internal devices of System 1 (Device 1A and Device 1B) are arranged in parallel. The internal subsystems of System 2 (Subsystem 2.1 and Subsystem 2.2) are also arranged in parallel. At the device level, Device 2A and Device 2B of Subsystem 2.1 are arranged in series, whereas Device 2C and Device 2D of Subsystem 2.2 are arranged in parallel.
In order to continue operating, the system needs the following conditions to hold:
• The Power Supply is operational, and
• System 1 is operational; System 1 operates if Device 1A or Device 1B, or both, are operational, and
• System 2 is operational; System 2 operates if Subsystem 2.1 or Subsystem 2.2 are operational (i.e., Subsystem 2.1 operates if Device 2A and Device 2B are operational; Subsystem 2.2 operates if Device 2C or Device 2D, or both, are operational).
The time study period is one year, and the individual probabilities of systems/devices being in failed states are as follows:
• Power Supply: 0.005
• Device 1A: 0.025
• Device 1B: 0.025
• Device 2A: 0.001
• Device 2B: 0.001
• Device 2C: 0.025
• Device 2D: 0.025
1) Build a Markov transition model for the system. The Markov solution should include all possible states and transitions from good (success) states to failed states. The transition failure rates are considered the same as probabilities of systems/devices being in failed states. Calculate the probabilities of being in all possible states and the final probabilities of system success and failure.
2) Furthermore, it was decided that the devices, the sub-systems, and the systems can be repaired during operation. Add the repair transitions to the already developed Markov model, considering that there could be more than one device repair operations performed in the same time interval. No calculations are required.