$8.(10$ points$)$

In a RTOS system where $$Priority Ceiling$$ is implemented and used, there are only three tasks A$,$ B$,$ and C$,$

whose priorities are in the order of A$>$B$>$C$.$ For each task, it takes $1$ time unit to request a resource $($no

matter successfully or not$).$ If the resource is locked by other tasks, this requesting task needs to wait $($be

blocked$)$ until the resource is unlocked. It takes $1$ time unit to unlock a resource if the resource is hold by

the task. For each task, it takes $3$ time units to finish executing its critical section.

Assume that at time unit $1,$ task C is created and scheduled to run immediately, task C requests for Resource

X$,$ and X is also unlocked at time unit $1.$ Task B is created at time $3,$ and Task A is created at time $5.$ Tasks B

and A need to acquire Resource X before executing their critical sections.

$(1)$ You may assume that Resource X is the only protected resource in the system, and tasks A$,$ B and C

only compete for this single resource. What is the ceiling of Resource X$?$

$(2)$ Please complete the following analysis at each time unit. You need to specify: $(1)$ if a task is holding

resource X$,(2)$ if it is blocked or running, and $(3)$ the task$$s priority level.