A queuing system with a Poisson arrival rate and exponential service time has three servers, an average arrival rate of 7.5 customers per hour, and an average service time of 20 minutes per customer.
Answer the following questions. Explain ALL formulas and calculations used in your response.
a. What is the minimum number of queues required to avoid an overloaded system?
b. How many queues are required to ensure the utilization is less than 50%
c. If the system had a single queue, what would be required to ensure that the queue length is not at risk of approaching infinity?

1. Clarke’s Shoe Factory has an annual demand of 5,000 for a particular shoe style. One of their suppliers made an irresistible offer on shoelaces for the popular shoe. Shoelaces usually cost 10 cents each; however, if Clarke’s buy 10,000 laces in each order for the upcoming year, they will pay only 5 cents each; and if they buy 50,000 in each order, you will pay only 3 cents each. It costs $15 to place an order. Carrying cost for each shoelace is 20% of purchase price.
a. Showing working, determine which of these two quantities represents the most optimal order quantity?
b. Explain the response you gave justifying ALL formula selections and calculations.

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