A fast-food restaurant makes hamburgers on a grill. At any given time, only four hamburgers can fit on the grill. If there is no room on the grill, the customers are asked to order a different item that does not require the grill. Assume that the time between hamburger orders and the cook time of hamburgers are both exponentially distributed. Furthermore suppose that, on average, one customer asks for a hamburger every 10 minutes, and it takes an average of 7 minutes to cook a hamburger.
(a) Construct the rate diagram for this CTMC. Make sure to clearly define your states.
(b) Develop the balance equations and solve these equations to find the limiting probabilities.
(c) What is the average number of hamburgers on the grill?
1(d) Assume that the restaurant makes a revenue of 7 dollars per hamburger sold (price paid by customer minus cost of ingredients), and it is open for 8 hours per day. If the fixed cost (light, water, etc.) of keeping the restaurant open is 75 dollars per day and the restaurant has a two, equally paid employees, how much should the owner pay to her employees per hour (assuming the employee works for 8 hours per day) to ensure that the restaurant makes an average net profit of at least 100 dollars per day?
(e) Suppose that if a customer cannot order a hamburger, they become angry and leave the restaurant without ordering anything else. The owner of the fast food chain has said that he wants at least 95 percent of his customers to leave happy (assuming that everyone that eats a burger leaves happy). Is this goal being met? Write down the percentage of customers who leave happy.