$(3$ pts$)$ A Poisson flow line has three stations with mean effective process times $(3,2,5).$ Assume a closed system.

$($a$)(1$ pt$)$ Consider $w=W_0$ and $w=2W_0.$ Compare the best CT and TH versus the CT and TH for the Poisson line. $[$Hint: the best CT and TH are obtained in a deterministic world.$]$

$($b$)(0\mathrm{.}5$ pts$)$ What is the minimum amount of WIP for the Poisson line that would result in a TH that is at least $95\%$ of the best TH$?$ What is the associated CT for this WIP level in the Poisson line?

$($c$)(0\mathrm{.}5$ pts$)$ Consider a reduction of $1$ in the bottleneck's mean effective process time. What are the new CT and TH for $w=W_0$ and $w=2W_0?$

$($d$)(0\mathrm{.}5$ pts$)$ Now consider a reduction of $0\mathrm{.}5$ in the mean effective process time at each nonbottleneck station. The bottleneck's process time remains at its original level. What are the new CT and TH for $w=W_0$ and $w=2W_0?$

$($e$)(0\mathrm{.}5$ pts$)$ If there is an improvement from the actions in $($d$),$ explain why it is surprising given the behavior of deterministic lines and why it is important.