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Question 1 - Customer calls are handled by one employee at a small business' call center. The time to answer each call is exponentially distributed, with a mean of two minutes per call. Requests for calls come in at a mean rate of 0.4 per minute, which can be assumed to be Poisson distributed. If the employee is helping a customer, others who call will wait on hold until the employee becomes free. All calls are served in first come first served order.

Determine the following:

a. What is the average number of customers who are on hold?

b. What is the probability of two or more customers in total in the system?

Question 2 - Machines at a production facility break down at the rate of three per hour, and the breakdowns are distributed in a Poisson manner. Downtime of the machines costs the company $20 per hour per machine, and each maintenance worker receives $15 per hour. One worker can service machines at an average rate of five per hour distributed exponentially.

What is the optimal maintenance crew size for servicing the machines in order to minimize the total cost?