How do you get c = 4 servers?

6. Queuing Models (25 minutes) (21 points) If you need to make any assumptions, please state them clearly! The successful coffee shop chain Moonbucks is interested in evaluating a new project, developed by the consulting company Analytics Edge. It consists of a cell phone app that would allow their customers to use it to place an order, pay, and pick it up at their preferred time in one of the 150 locations in Toronto without waiting in line! In order to evaluate this project, Moonbuck considers a representative store in Toronto. All the average measures that are valid for this representative store can be multiplied by 150 to get a good estimate of the impact that a full implementation in the city of Toronto would have. At the representative store, the inter-arrival times for customers are 30 seconds on average, with a standard deviation of 1 minute; while the service times are of 1 minute and 20 seconds on average, with a standard deviation of 2 minutes. Analytics Edge has presented convincing evidence, collected from small scale exper- iments, that suggests that the app would reduce variability in both the customers' arrival process and in the service process. Moreover, they claim that the implemen- tation of the app could reduce the standard deviation of the inter-arrival times by 50%, while at the same time reducing the standard deviation of the service times by one third. Analytics Edge has offered to lead the successful implementation of this project, in- cluding the development and implementation of the app, training the staff in the new processes required to serve the customers using the app, and an advertising campaign in the subway, street cars, buses and the radio. For this whole project, Analytics Edge would charge Moonbucks CAD$1.5 million, and they claim that Moonbuck would al- ready be making a profit out of the project by the end of the first year after starting the implementation. Moonbuck has hired you to verify claim. Further information that can be useful to evaluate Analytics Edge's claim is that Moonbucks estimates that each minute that their customers wait in line on average in each store costs them CAD$10,000 annually, in terms of loss of goodwill. For this reason, the stores operate with the appropriate staff that ensures that the average wait time at the queue in their stores is no more than 5 minutes. On the other hand, each employee costs Moonbuck CAD$40,000 annually, therefore they keep the number of servers at each store at the minimum level such that this goal is met. (b) (6 points) What are the average system utilization, individual server utilization, queue lenght, and waiting time at the representative store before the app is Page 13 of 16 M 270 L0101/L0201/L0301 Midterm Examination 2016 Winter Term introduced? How many servers are working at the store? 4/3 Solution: 1= * = 2 customers/minute H = 173 = .75 customers/minute Ca= 1 = 2 Cg = = 1.5 c=4 servers T= /4u = 2/3 is both the system and individual server utilization 1,= 2(c+1) x 04+= 2.6 customers T, = 1/1 = 1.3 minutes from Little's Law Note that c=3 servers gives T, = 10.08 > 5 minutes, which is not acceptable to Moonbucks. 1-7