$1.$ Consider a simple two$-$station line as shown in the figure below. Both machines take $20$ minutes per job and have $\backslash (\backslash$mathrm$\{$SCV$\}=1\backslash ).$ The first machine can always pull in material, and the second machine can always push material to finished goods. Between the two machines there is a buffer that can hold only $10$ jobs.

a$.$ Model the system using an $\backslash (\backslash$mathrm$\{$M$\}/\backslash$mathrm$\{$M$\}/1/\backslash$mathrm$\{$b$\}\backslash )$ queue. $($Note that $\backslash (\backslash$mathrm$\{$b$\}=12\backslash )$ considering the two machines.$)$

i$.$ What is the throughput?

ii$.$ 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$)?$

iii. 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.$)$

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

b$.$ Reduce the buffer to one $($so that $\backslash (\backslash$mathrm$\{$b$\}=3\backslash ))$ and recompute the above measures. What happens to throughput, cycle time, and WIP? Comment on this as a strategy.

c$.$ Set the buffer to one and make the process time at the second machine equal to $10$ minutes. Recompute the above measures. What happens to throughput, cycle time, and WIP? Comment on this as a strategy.