$($Text Case $13\mathrm{.}2$ Freedom Express, p$\mathrm{.}373)$ Details are in the text, here is the summary. FreeEx a small commute airline is analyzing the fuel consumption occurred by the congestion delay while waiting in the stack $($hanging around in the sky waiting for the landing turn$).$ Fuel cost is $$2\mathrm{.}80/$gallon and average consumption rate is $20$ gallons per minute. The mean arrival rate is $20$ planes per hour, or $1$ every $3$ minutes in the airport. During clear weather the control tower can give a permission to land $1$ plane per minute, or $60$ planes per hour. When the weather is bad, the landing rate is controlled to $30$ per hour because of the safety.

$($a$)$ How many gallons of fuel on average should FreeEx to consume for landing in the clear weather, and in the bad weather, respectively? $($Fuel consumption is considered only during the airport landing process. For example, if there is an aircraft in the stack, the arriving aircraft has to wait until it completes the landing of the aircraft arriving earlier.$)$

$($b$)$ When the plane runs short of the fuel while in the stack, it is given priority to land out of order by communicating with airport control tower. But this advantage is allowed to be used at most $1$ time in $20.$ To ensure this regulation of $95\%$ rule, the planes may carry the reserve fuel $($gallons over and above expected average usage, found in $($a$))$ in case of long waiting. Find the extra reserve fuel required in the clear weather and in the bad weather, respectively? $($Hint : Find the number of airplanes in stack covering at least of $95\%$ in $)$

$($c$)$ In the bad weather, FreeEx has the option to land in another airport nearby by the instruction of control tower. The nearby airport also has the landing rate of $30$ per hour in the bad weather. The combined arrival rate for both airport is $40$ per hour. If it is landed in the nearby airport, FreeEx must pay $$200$ to charter a bus transporting passengers to the original destination airport. Assume that if the option is used, FreeEx$$s aircraft will be diverted to another airport about one$-$half the time. Should FreeEx exercise the option of permitting its aircraft to land at the nearby airport during bad weather?

$($Hint : for not taking the option, it is M$/$M$/1$ system as in $($a$)$ and $($b$).$ If the option is used, the system is M$/$M$/2$ as in the example of two baristas, Class note Service $12.$ Page $28,$ Find the waiting time for each case to compute the cost of consuming fuel. For the case of option, the extra cost for the charter has to be considered$)$