$(14$ points total$)1.$ Suppose a grocery store has collected data on customer arrivals to the checkout area and on the service times to check out during a busy period of the week. On average customers arrive to the checkout area at a rate of $45$ per hour. On average, each server in the checkout area can serve $($check out$)12$ customers per hour. Assume Poisson arrivals and exponential service times. $(3$ pts$.)1\mathrm{.}1.$ What is the minimum number of servers required? a$)2.$ b$)3.$ c$)4.$ d$)$ none of the above. $(3$ pts$.)1\mathrm{.}2.$ If there are $5$ identical servers, then the average waiting time in line for a customer is in which range? a$)$ less than $0\mathrm{.}031$ minutes. b$)0\mathrm{.}0311$ minute. c$)13\mathrm{.}75$ minutes. d$)3\mathrm{.}768$ minutes. e$)$ more than $8$ minutes. $(3$ pts$.)1\mathrm{.}3.$ If there are $5$ identical servers, then the average waiting time in the system for a customer is in which range? a$)$ less than $0\mathrm{.}031$ minutes. b$)0\mathrm{.}0311$ minute. c$)13\mathrm{.}75$ minutes. d$)3\mathrm{.}768$ minutes. e$)$ more than $8$ minutes. $(5$ pts$.)1\mathrm{.}4.$ Suppose each server in the checkout area is identical in their rate of service, and each is paid $$18$ per hour. Also suppose that a customer$$s waiting time is valued at $$12$ per hour. How many servers should there be to minimize the total cost $$ and what is the average total hourly cost to provide the service $($including cost for the servers and cost for the customer$$s waiting time$)?$ Number of servers $=\_\_\_4\_\_\_\_\_$ Average total hourly cost $=\_\_$$$84\_\_\_\_\_\_\_\_\_\_\_\_\_$