Please solve this stochastic modeling problem and show all work.

. Somehow you nd yourself in your favorite dining hall. But you're not there to eat. You realize that you work there, even though you don't remember how or when you got that job. Strange. Anyway, all of a sudden, students are lining up to get to your station. 0k, you're there, so you need to get the job done! But instead of serving food, when a student walks up to you and asks for something to eat (for example, chicken wings, pizza, or a side of broccoli), rather than serve them the food, you realize you're supposed to use face paint to paint a picture of the food on their forehead. Weird. But, you're there and somehow the paint is there, so you go with it! Suppose that the time between arrivals to the line for your station is exponentially distributed with a rate of 10 per hour. Suppose that the line has innite capacity and that you paint faces on a rstcome-rst-serve basis. Finally, suppose that the time that it takes you to paint each face is exponentially distributed with rate it. (a) (2 points) Assuming that students continue to arrive indenitely, what kind of model can you use to analyze the long-run behavior of the line at your station? (b) (6 points) Your boss wants you to work fast enough such that the long-run ex- pected number of students in your system (including any student you are serving along with any waiting in line) is at most 5. What is the minimum value for p, (in terms of students per hour) that you need in order to achieve this goal? (c) (6 points) Now suppose that your boss wants you to focus on a different perfor- mance measure. In particular, your boss now wants you to work fast enough such that the long-run expected time that a student needs to wait in line (before reach- ing you to start the face painting) is at most 6 minutes. What is the minimum value for p (in terms of students per hour) that you need in order to achieve this