SCS receives on average $1$ data package every $1/50$ seconds, with a standard deviation of $1/50$ seconds, and processes them using its single powerful computing unit, which can process data packages in $1/80$ seconds per package $($with a standard deviation of $1/160$ seconds$).$ Packages will wait indefinitely until they are processed.



$1)$ 

What is the average waiting time of a package, in seconds? $($Please format your answer as $0\mathrm{.}00000)$

Hint: For calculating the average waiting time, ensure you're using the correct formula that incorporates both the arrival rate and the service rate. Remember to account for the variability in arrival and service times as indicated by their standard deviations.

$2)$ 

Suppose utilization increases $10\%$ and everything else remains constant. What is the percent increase in wait time? $($Please format your answer as $00\mathrm{.}00)$

Hint: When calculating the percent increase in wait time due to increased utilization, ensure you understand how changes in utilization affect waiting times in a queue. A small increase in utilization can lead to a disproportionately large increase in waiting time, especially when the system is already highly utilized.