A fast-food franchise is opening a new premise and is considering how best to design their drive-thru service. The drive-thru will operate 24 hours per day. They expect that cars will arrive at the drive- thru at a rate of 18 per hour (throughout the day and night) and that the time between arrivals will be exponentially distributed. The two drive-thru designs under consideration are as follows: A. A single driveway with one service window at which an employee will take the order, process the payment and fill the order. The service time for this design is exponentially distributed with a mean of 3 minutes. Cars are served in the order in which they arrive. B. A single driveway with three service windows. At the first window an employee will take the order, at the second window another employee will take the payment, and at the third window another employee will fill the order. The service times at each window are exponentially distributed with means of 0.5, 1, and 2 minutes respectively. Cars must complete service at each window before moving to the next window. Cars are served in the order in which they arrive. 1. Consider design A. (a) Describe this queue using Kendall-Lee notation. Justify your answer. Total for Question 1: 15 marks (3 marks) (b) Does the steady state distribution for this queueing model exist? Justify your answer. (c) On average, how many cars are in the drive-thru (including those waiting and being served)? Compute using the relevant formula. (3 marks) (3 marks) (3 marks) (d) On average, how long does each car spend in the drive-thru? Compute using the relevant formula. (e) In the long run, what is the probability that there are 10 cars in the drive-thru (including those (3 marks) waiting and being served)? 2. Consider design B. Total for Question 2: 23 marks (3 marks) (4 marks) (8 marks) (a) Describe each stage of this queue using Kendall-Lee notation. Justify your answer. (b) Does the steady state distribution for this queueing model exist? Justify your answer. (c) On average, how many cars are at each stage of the drive-thru and in the drive-thru as a whole (including cars which are waiting and being served)? Compute using the relevant formula. (d) On average, how long does each car spend at each stage of the drive-thru and in the drive-thru (8 marks) as a whole (including cars which are waiting and being served)? Compute using the relevant formula.