Consider a simple two$-$station line as shown in Figure. The first machine takes $20$ minutes per job and has $\backslash (\backslash$mathrm$\{$SCV$\}=1\backslash ).$ The first machine can always pull in material, and the second machine can always push material to finished goods. The process time at the second machine is equal to $10$ minutes with $\backslash (\backslash$mathrm$\{$SCV$\}=\backslash )1.$ Between the two machines is a buffer that can hold only $1$ job $($see Sections $8\mathrm{.}7\mathrm{.}1$ and $8\mathrm{.}7\mathrm{.}2).$

$1.$ Model the system using an $\backslash ($ M $/$ M $/1/$ b $\backslash )$ queue.

Note that $\backslash ($ b$=3\backslash )$ considering the two machines. Compute the below measures. Comment on this as a strategy.

$1.$ What is the throughput and utilization of the line and the first machine?

$2.$ What is the partial WIP $($i$.$e$.,$ WIP waiting at the first machine or at the second machine, but not in process at the first machine$)?$

$3.$ What is the total cycle time for the line $($not including time in raw material$)?($Hint: Use Little's law with the partial WIP and the throughput and then add the process time at the first machine.$)$

$4.$ What is the total WIP in the line? $($Hint: Use Little's law with the total cycle time and the throughput.$)$

$5.$ What is the utilization of the second machine?