$13\mathrm{.}11$ Response time distribution for M$/$M$/14$

In this problem, you are asked to derive the distribution of response time for an M$/$M$/1$ queue with arrival rate $\backslash$lambda and service rate $\backslash$mu $.$ To do this, think about the response time experienced by an arrival, $$job x$.$ Think about the number of jobs that job x sees in the system, and the work associated with each of these jobs. Then express job x$$s response time in terms of these quantities.

$($a$)$ At the time when job x arrives, what is the service requirement $($job size$)$ for each job in the queue? What is the remaining service requirement for the job in service, if there is one?

$($b$)$ Let N denote the total number of jobs in the system that job x sees when it arrives. What is P $\{$N $=$ n$\}?$ Use PASTA.

$($c$)$ Consider a new distribution N$$ where N$$ is the number of jobs in the system seen by job x$,$ plus itself. What is P $\{$N $=$ n$\}?$

$($d$)$ The distribution N$$ has a name. What is the name of the distribution of N$$ and what is the appropriate parameter?

$($e$)$ If Si denotes the service requirement of the ith job in the M$/$M$/1,$ we can express the response time of job x as a sum involving some of the random variables above. Write this sum.

$($f$)$ Fully specify the distribution of response time of job x along with its parameter$($s$).[$Hint: you will need to utilize a result from the exercises in Chapter $11.]$