Answer Happy

14. A supermarket has 7 collection boxes. Customers reach them following a Poisson process with an average rate of 100 customers/hour. The staff that attends each collection box needs on average 5 seconds to process each unit of purchase (each element that the customer wants to buy) In addition, it is estimated that the collection time of a client, either in cash or by card, is 1.5 minutes. It is estimated that customers are divided into two groups: - Group A: customers who buy baskets of up to x units, - Group B: users who carry in their carts more than x units. Group A accounts for 10% of total customers. In this group, the average number of units in purchases is 9 units/customer, while in group B it amounts to 30 units/customer. With a view to reducing the average waiting time in line, the store manager is considering applying one of the following two solutions: Add a collection box equivalent to the existing ones. - Add a quick collection box. The fast box would dispatch all group A customers and none from group B. a) Calculate the average waiting time in line in the initial situation, with 7 collection boxes. (b) Which of the two proposed solutions would you recommend to the person in charge? Justify the answer with the appropriate calculations.