1. A cafeteria has one cashier. Customers carrying their food wait in a queue to pay the cashier. Arrivals of customers to the queue form a Poisson process with a rate of 56 customers per hour, and service times at the cashier are independent and exponentially distributed. The average time in system (queueing plus service) is 4 minutes. There is a counter right before the cashier for customers to place their tray of food while they wait in the queue. When the counter space is full, additional customers have to carry their tray while they wait in the queue. Calculate the service rate (mu) of this system in customers per hour.
2. Using the same arrival stream as in the prevous question, suppose that the lunch counter cashier has a service rate of 74 customers per hour. What is the average time (in minutes) spent waiting in the queue at the lunch counter? Express your answer to two decimal places.
3. The counter at the cashier is 8 feet long. Assume a cashier service rate of 76 customers per hour and assume that each tray measures 2 feet in length. What is the probability that an arriving customer will find that the counter is full and she has to just hold her tray for a bit? (The counter length includes both the person paying for a meal and those waiting in line) Enter your answer as a decimal (not a percentage) to three decimal places.