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The soda machine outside the lecture hall sees a steady stream of customers throughout the day. While the machine takes 4 seconds to deliver a can of carbonated sugar water once the button is pressed, the student customers introduce variation into the delivery process by pondering their selection for a while. Thus, we can safely say that the service time is governed by a negative exponential probability distribution that averages 10 seconds. Over the course of a twelve hour lecture day, students arrive at the rate of 240 per hour, Poisson distributed of course.
a. On average, how many students are standing directly in front of the soda machine?
b. Over the course of the twelve hour day, how many hours is the soda machine not happily dispensing product?
c. Nathan is cutting it close today. At 10:00:00 AM, the professor will close the door to the classroom and ignore any requests to enter the room once his scintillating lecture has begun. Nathan approaches the soda machine at 9:59:20. Will he make it to class on time?
d. At the end of an average twelve hour day, how many cans of soda will the vending machine have dispensed?
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