Two tasks, indexed $1$ and $2$ must be processed by two CPUs Enher CPU takes time to process any task that is distributed as a uniform random variable over the interval $(0\mathrm{.}1)$ seconds. All processing times are independent of each other You can assign tasks in the following two ways Under assignment A$,$ task $1$ is assigned to CPU $1$ and task $2$ is assigned to CPU $2$ at the same time Tasks begin processing as soon as they are assigned The processing of all tasks is considered complete when both task $1$ and $2$ have been processed

Under assignment B$.$ task assignment takes place in two stages. First, tack $1$ is assigned to both the CPUs. As soon as one of the CPUs finishes processing task $1,$ both CPUs are reset and task $2$ is assigned to both of them. The processing of both tasks is considered to finish as soon as one of the two CPUs finishes processing task $2$

We are interested in the time taken to process a task, which is the tume during which a task is being actively processed by any CP$1)$ We are also interested in the total ume taken to process both the taska which is the time that clapses herween the instant when processing starts and the instant when processing of both tasks has finished. Answer the following questions

$($a$)$ Derive the distribution of the time taken to process task $1$ and the distribunon of the tume taken to process task $2$ under assignment A Also derive the distribuuon of the total time it takes to process both the tasks

$($b$)$Derive the distribution of the tune taken to process task $1$ and the distribution of the time taken to process task $2$ under assignment B

$($C$)$ Consider the total time taken to process hotch the tasks. Assume that this total can be well approximated by a suitably chosen Gaussian random variable Derive the probability that assignment A takes a longer total ume than assignment B