$($This problem is fairly involved and could be considered a small project.$)$ Consider a simple two$-$station line as shown in Figure $8\mathrm{.}8.$ Both machines take $20$ minutes per job and have $SCV=1.$ The first machine can always pull in material, and the second machine can always push material to finished goods. Between the two machines is a buffer that can hold only $10$ jobs $($see Sections $8\mathrm{.}7\mathrm{.}1$ and $8\mathrm{.}7\mathrm{.}2).$

$($a$)$ Model the system using an $\frac{M}{M}\frac{?}{?}\frac{1}{b}$ queue. $($Note that $b=12$ 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 $b=3)$ 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.

$($d$)$ Keep the buffer at one, make the process times for both stations equal to $20$ minutes $($as in the original case$),$ but set the process CVs to $0\mathrm{.}25(SCV=0\mathrm{.}0625).$

i$.$ What is the throughput?

ii$.$ Compute an upper bound on the WIP in the system.

iii. Compute an $($approximate$)$ upper bound on the total cycle time. Is this upper bound an acceptable cycle time?

iv$.$ Comment on reducing variability as a strategy.