An engineer has been set the objective of improving the process at a machine served

by a FIFO queue, with the following objectives:

$1.$ Ensure that the percentage of time that there are more than two jobs in the

system $($machine $+$ queue$)$ is less than $25\%$

$2.$ Ensure that no more than $5\%$ of jobs will experience more than $5$ minutes of

queuing time while waiting for service on the machine.

She has studied the existing system and determined that the jobs arrive in the queue

at a rate of $20$ jobs per hour. She has also shown that the time between arrivals is

exponentially distributed. She has determined a series of improvements to the existing

process that will result in an exponentially distributed service time of $1\mathrm{.}5$ minutes.

Answer the following questions:

a$)$ Determine the expected number of customers in the queue and in the system.

$(5$ Marks$)$

b$)$ Determine the expected queuing time and system time for a job. $(4$ Marks$)$

c$)$ Show by means of calculations whether the proposed new process service time

will be sufficient to meet the objectives set out above. $(6$ Marks$)$