You have been asked to analyze the queue for registrations at a large Las Vegas hotel.$$ Guests arrive to register at the rate of $3$ people every $12$ minutes $($Poisson distributed$).$ It takes each host an average of $6$ minutes to register a guest $($following a negative exponential distribution$),$ any number of which can be hired by the hotel at the cost of $$15$ per hour.$$ The hotel recognizes a waiting cost of $$55$ per hour per guest in line. The hotel has decided that guests' average wait in line is not more than $6\mathrm{.}5$ minutes.

A single step waiting line is set up$,$ but can accommodate multiple servers.$$

a$.$ Based on the goal $($average wait in line is less than $6\mathrm{.}5$ minutes$)$ stated above, what is the minimum number of hosts that could be hired for this system?$[$ Select $]["3","4","1","2","5"]$ hosts

b$.$ Based on wait cost, what is the optimal number of servers needed to minimize total costs?$[$ Select $]["2","5","4","1","3"]$ hosts

c$.$ What is total $($providing service plus time in queue costs$)$ daily cost at the optimal number of servers if the registration desk is open for $3$ hours? $[$ Select $]["$$$58\mathrm{.}02","$$$62\mathrm{.}46","$$$203\mathrm{.}31","$$$174\mathrm{.}08","$$$342\mathrm{.}89"]$

d$.$ At the optimal number of servers, what is the average time to check in an attendee? $[$ Select $]["1\mathrm{.}11$ minutes", $"6\mathrm{.}94$ minutes", $"8\mathrm{.}12$ minutes", $"\mathrm{.}94$ of a minute", $"\mathrm{.}55$ of a minute"$]$