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A few years ago, Amazon has introduced Amazon Hub Locker system for more convenient and secure deliveries to customers. In certain public stores such as Whole Foods in this problem, they installed lockers for their Prime deliveries. When customers request to have their Prime orders delivered to Hub Locker, then the products are delivered to a Whole Foods store. When there are available lockers, then the orders will directly be stored in lockers and customers are notified with secure codes, so that they can find their deliveries while they are shopping. When there are no available lockers, then the products will be kept in some space at the store. As soon as there is an available locker, they will be moved to one of the lockers.

Consider a Whole Foods store where 50 lockers are installed, and suppose that the Hub Locker requests (i.e., orders to be delivered in Hub Locker) arrive with a Poisson process with a rate ? = 120 per day. Suppose that when a customer is notified that their orders are stored in Hub Locker, then the time it takes to pick up their orders is exponentially distributed with mean 8 hours.

1. Explain that the Hub Locker system can be modeled as an M/M/s queue. What is the “server,” and how are service times modeled? Specify the parameters ?, µ, and s that can be used for analyzing this system via Q.xls.

2. What is the average delay, defined as the average time between when a product is arrived to the Whole Foods store and when it is moved to one of the lockers?

3. What is the average number of orders that are arrived to the store but not stored in the lockers?

4. What is the probability that an order will not experience the delay, i.e., it will directly be stored to one of the lockers?

5. Suppose that the service becomes more popular and the rate ? is increased to 135 per day. If you would like to meet a service requirement that the proportion of orders which experience the delay is less than 10%, how many additional lockers are needed?