Problem $3[30$ pts$]$

A system consists of three periodic real$-$time tasks: $(3,1),(5,2),$ and $(8,3).$ Each pair represents the period and execution time of a task, i$.$e$.,(p_i,e_i).$ We also assume implicit deadlines $($i$.$e$.,$ AAi,$d_i=p_i)$ and synchronous releases $($i$.$e$.,$ all tasks have their first job arrival at time $0).$

$($a$)$ What is the total utilization? $[5$ pts$]$

$($b$)$ Construct a preemptive EDF schedule for this system in the interval $(0,18).$ Label any missed deadlines. In this and the next part, assume that each job that misses its deadline is still executed in its entirety. $[10$ pts$]$

$($c$)$ Construct a preemptive RM schedule for this system in the interval $(0,18).$ Label any missed deadlines. $[10$ pts$]$

$($d$)$ Suppose we want to reduce the execution time of the task with period $3$ to make the task set schedulable $($i$.$e$.,$ all deadlines are guaranteed to meet$)$ with the preemptive EDF algorithm. What is the minimum amount of reduction necessary? $[5$ pts$]$