An operating system can fail either fully or partially, and from the partially operating state it fails completely. The fully operational system undergoes periodic PM$.$ More specifically, the system can be in either of four states: operational, partially operational, down for PM$,$ or failed. The system is repaired from partially and fully failed states to the normal operating state. Use Markov model to derive expressions for system availability, probability of system down for PM$,$ and probability of system failure. Explicitly mention all the assumptions that are associated with your model.