$(20$ marks$)$ Consumers arrive at takeaway window of a fast$-$food restaurant at a rate of $10$ customers every one hour per day. The average time spent for service is $5$ minutes per customers. Currently one takeaway window is open for service. The arrivals are approximated by the Poisson distribution and the service time is exponentially distributed. The fast$-$food restaurant is open $18$ hours per day.

a$.$ What is the average time a customer is waiting in the line of fast$-$food restaurant?

b$.$ What is the total cost, if one staff is employed to serve the customers?

Due to the customer complaints about long time waiting, the fast$-$food restaurant is considering a second window to reduce the customer waiting time. The salary and benefits for each member of staff is to be $8$$ per hour.

c$.$ What is the average time a customer waits in the fast$-$food restaurant if two staff are employed?

d$.$ Should a second staff be added, if the average waiting cost is $20$$ per hour?

$($Please$,$ approach and model the problem in a logical way and technically correct fashion, show the details of your calculations, describe a solution procedure, and write your comments on the findings$).$