One inspector has been assigned the full$-$time task of inspecting the output from a group of $10$ identical machines. Jobs to be done by any one of the machines arrive according to a Poisson process at a mean rate of $70$ per hour. The time required by a machine to perform each job has an exponential distribution with a mean of $6$ minutes. Thus, whenever all $10$ machines are busy, the jobs are being completed ready for inspection at a mean rate of $100$ per hour. Unfortunately, the inspector is able to inspect them at a mean rate of only $80$ per hour. $($In particular, her inspection time has an Erlang distribution with a mean of $0\mathrm{.}75$ minute and a shape parameter k$=25.)$ This inspection rate has resulted in a substantial average amount of in$-$process inventory at the inspection station $($i$.$e$.,$ the expected number of jobs waiting to complete inspection is fairly large$),$ in addition to that already found at the group of machines.Management feels that there is too much in$-$process inventory. The cost of this in$-$process inventory is estimated to be $$5$ per hour for each job in process, whether at the machines or at the inspection station. Therefore, management has instructed the production manager to cut down on such inventory. In response, the production manager has made two alternative proposals to reduce the average level of in$-$process inventory.Proposal $1$ is to use slightly less power for the machines $($which would increase their expected time to perform a job to $7$ minutes$),$ so that the inspector can keep up with their output better. This also would reduce the cost for each machine $($operating cost plus capital recovery cost$)$ from $$7\mathrm{.}00$ to $$6\mathrm{.}50$ per hour. $($By contrast, increasing to maximum power would increase this cost to $$7\mathrm{.}50$ per hour while decreasing the expected time to perform a job to $5$ minutes.$)$Proposal $2$ is to substitute a certain younger inspector forthis task. He is somewhat faster $($albeit more variable in this inspection times because of less experience$),$ so he should keep up better. $($His inspection time would have an Erlang distribution with a mean of $0\mathrm{.}72$ minute and a shape parameter k$=2.)$ This inspector is in a job classification that calls for a total compensation $($including benefits$)$ of $$16$ per hour, whereas the current inspector is in a lower job classification where the compensation is $$14$ per hour. $($The probability distribution of inspection time for each of these inspectors is typical of those in the same job classification.$)$The production manager has asked you to $$use the latest then OR techniques to see how mucheach proposal would cut down on in$-$process inventory and then make your recommendations."

$($a$)$ Determine the expected amount of in$-$process inventory $($number of jobs$)$ at the machines and at the inspection station. Then calculate the expected total cost per hour of the in$-$process inventory, the machines, and the inspector.

$($b$)$ Calculate L$,$Ls$,$Lq and W$,$Wq$,$Ws values.