Schips Department Store operates a fleet of $11$ trucks. The trucks arrive at random times throughout the day at the store's truck dock to be loaded with new deliveries or to have incoming shipments from the regional warehouse unloaded. Each truck returns to the truck dock for service two times per $8-$hour day. The service rate is $6$ trucks per hour. Using the Poisson arrivals and exponential service times model with a finite calling population of $11$ trucks, determine the following operating characteristics. $($Round your answers to four decimal places. Report time in hours.$)$

$($a$)$

The probability that no trucks are at the truck dock.

$($b$)$

The average number of trucks waiting for loading$/$unloading$.$

$($c$)$

The average number of trucks in the truck dock area.

$($d$)$

The average waiting time before loading$/$unloading begins.

h

$($e$)$

The average waiting time in the system.

h

$($f$)$

What is the hourly cost of operation if the cost is $$53$ per hour for each truck and $$32$ per hour for the truck dock? $($Round your answer to the nearest cent.$)$

$

$($g$)$

Consider a two$-$server truck dock operation where the second server could be operated for an additional $$32$ per hour. How much would the average number of trucks waiting for loading$/$unloading have to be reduced to make the two$-$server truck dock economically feasible? $($Report time in hours and round to four decimal places.$)$

L $<=$

h

$($h$)$

Should the company consider expanding to the two$-$server truck dock? Explain. $($Report time in hours and round to four decimal places. Report total cost in dollars and round to the nearest cent.$)$

L $=$

h

TC $=$ $