A Superstore has a self$-$checkout system that involves a single queue and up

to $8$ self$-$checkout stations that can be opened. The cost of service at one of these stations

is only $$0\mathrm{.}42/$min$.,$ while the cost of waiting in line is $$2\mathrm{.}50/$min$.$ If the arrival rate to the

system is $1\mathrm{.}667/$min$.,$ and the average service time per customer is $2\mathrm{.}0$ min., answer the

following questions, showing all work and clearly indicating your final answers.

i$.$ What is the minimum number of stations that should be open at all times

in the system $($i$.$e$.$ to keep the utilization $<1\mathrm{.}0)?$

ii$.$ If there are typically $5$ stations operational at once, what is the avg. server

utilization, avg. # in the queue, avg. queue waiting time, avg. time in the

system, and the total cost of the system per minute.

iii. Consider the cases of $4,5,6,7$ or $8$ stations operational. What is the

optimal number of open stations to minimize total cost?