Consider a production line with three single$-$machine stations in series. Each has processing times with mean $2$ hours and standard deviation of $2$ hours. $($Note that this makes it identical to the line represented in the practical worst case of Chapter $7.)$

$($a$)$ Suppose we run this line as a push system and release jobs into it at a rate of $0\mathrm{.}45$ per hour with arrival variability given by $c_a=1.$ What is the average WIP in the line?

$($b$)$ Compute the throughput of this line if it is run as a CONWIP line with a WIP level equal to your answer in $($a$).$ Is the throughput higher or lower than $0\mathrm{.}45?$ Explain this result.