PART$1$: Saul Goodman loves a quick bite at Los Pollos Hermanos. Saul knows from Gustavo that Los Pollos receives a new customer every$3$minutes on average$($coefficient of variation$=1)$and on average it takes$2$ minutes$($standard deviation of this time is$2$mins$)$for the food to be prepared and served. The counter can serve at most one person at a time and the first arriving customer is served first.

A$)$What is the inter$-$arrival time in minutes?

B$)$What is the capacity of Los Pollos Hermanos? You can express your answer in either customers per minute or customers per hour.

C$)$What is the utilization of Los Pollos Hermanos?

D$)$What is the coefficient of variation of the service process of Los Pollos Hermanos?

E$)$ How long on average, in minutes, does Saul have to wait before getting served?

PART$2$: Gustavo runs the store for eight hours every day. An average order earns $$4$in revenue and costs $$1$in raw materials. The labor costs are $$160$ per day. Assume all customers that come within the$8$hour window are served. What are Gustavo$$s average daily earnings$($revenues$$ cost$)$in dollars?$\{$Hint: Remember all arriving customers are served.$]$Gustavo puts in measure to reduce the standard deviation of the service$($or processing$)$time$.$ Now the coefficient of variation of service time is$0\mathrm{.}5.$

F$)$ When he is not serving a customer, Gustavo$$cooks his books$.$ How many minutes each day, on average, can Gustavo devote to this?$[$Hint: Think idle time which is$(1-$utilization$)...]$

For G$-$H: Gustavo puts in measure to reduce the standard deviation of the service $($or processing$)$ time. Now the coefficient of variation of service time is $0\mathrm{.}5.$

G$)$ What is the new standard deviation of service time?

H$)$ How will the utilization change?

I$)$How will the time in queue change?

J$)$Word has spread about the reduction in standard deviation, and customers are flocking to Los Pollos Hermanos. On average,$25$ customers now arrive per hour. What is the average inter$-$arrival time in minutes?