Upon arrival to the hotel, you join the line to check in. There is a single clerk at the check-in counter when you arrive. After studying the system for a while, you learn that new guests arrive to the hotel (to check in) once every 8 minutes on average, and you figure that the inter-arrival times have an exponential distribution. You also estimate that the clerk can process an average of 10 guests per hour and that the service time follows a general distribution with a standard deviation of 3 minutes per guest.

1-1. What is the utilization of the system (i.e., what fraction of time is the clerk busy)?

1-2.What is the total average amount of time (in minutes) that guests spend waiting and being served?

1-3. While you are waiting to check in, an expert in statistics arrives at the hotel (there is a conference on Statistics at the hotel that week) and joins the check-in line. You establish a conversation with the expert and, after talking about the weather and the plans for the day, you engage in a discussion about the wait in the check-in line. The expert congratulates you for your observations regarding the arrival and service processes, but quickly observes that the service time actually follows an exponential distribution. The average processing rate is indeed of 10 guests per hour. What is now the average amount of time (in minutes) that guests spend waiting and being served?

1-4. What is the probability that a customer will wait 10 minutes or more in the check-in area?

1-5. After checking in, you take your suitcases and join the single line to take the elevator up to the room. Because at this time of the day all guests carry luggage, only one guest at a time can fit in the elevator with their suitcases. There are two elevators operating at that time (but a single line feeding both elevators), and you overhear the expert in statistics say that, of course, guests arrive to the elevator line once every 8 minutes on average, and that the inter-arrival times have an exponential distribution. The expert also observes that each elevator takes an average of 9 minutes to go up to the room floors and back to the lobby, and that the distribution of ride times on the elevator is exponential. Based on this information, what is the average time (in minutes) a guest has to wait in line before being able to take an elevator?