1. Consider a simple queueing system with 20 servers. Customers have an average inter-arrival time (the average time between each arrival) of 1 customer every 0.6 minutes with a standard deviation of arrival of 2 minutes. The average service processing time is 11 minutes exponentially distributed. Recall that exponential distributions have a coefficient of variation equal to 1. a) Compute the utilization rate as well as Ii, Ip, and Ti (Tp). How much time in total does a typical flow unit spend in the system? b) The service provider has made some changes. After investing in technology, the average processing time is cut in half to 5.5 minutes, still exponentially distributed. As a result, the number of servers has been reduced to 10. Recalculate your answers. c) Compare your answers from a) and b). Which system would you prefer as a customer? Why?
2. A queueing system has customers arriving at the rate of 31 per hour, exponentially distributed. Servers can process customers at the rate of 2 per hour, exponentially distributed. There are multiple servers. a) What is the minimum number of servers required? b) Suppose that servers are paid a wage of $10/hour. Further, suppose that customers are annoyed at having to wait for service. We value customers waiting time at the rate of $20 per hour. In other words, we treat each hour that each customer waits as a cost of $20. What is the total hourly cost (wages plus annoyance) if we use the minimum number of servers? c) What is the number of servers to use that minimizes the total hourly cost?