$5\mathrm{.}9.$ Consider again the serial flow factory of Problem $5\mathrm{.}5.$ Management expects

there to be a slow increase in demand $($i$.$e$.,$ arrival rates$)$ over the next few years.

$($a$)$ In order to help management plan for the future, find the system performance

measures $($CTsys and WIPsys$)$ for arrival rates of $0\mathrm{.}51/$hr$,0\mathrm{.}53/$hr$,$ and $0\mathrm{.}55/$hr$.$

$($b$)$ The engineering department is considering some machine changes that will reduce the processing time variance for the bottleneck machine $($i$.$e$.,$ the machine with the largest cycle time$).$ Assuming the arrival rate increases to $0\mathrm{.}51/$hr$,$ what percentage reduction in the C$2$ for the bottleneck machine is necessary so that the average system cycle time remains the same as it for the original system with an arrival rate of $0\mathrm{.}5/$hr$?$

$($c$)$ It turns out that reducing the processing time variance is not possible; however, it is possible to reduce the mean service time while the coefficient of variation remains according to the original system. Assuming that the arrival rate increases to $0\mathrm{.}55/$hr$,$what mean service rate is necessary for the bottleneck machine so that the average system cycle time remains the same as for the original system with an arrival rate of $0\mathrm{.}5/$hr$?$

$($d$)$ If the mean service time of the bottleneck machine is reduced enough, the bottleneck will $$move$$ to a different machine. With an arrival rate of $0\mathrm{.}55/$hr$,$ what is

the mean service time of the current bottleneck machine that is required so that two workstations become $$tied$$ for the bottleneck location?