Question 3. Queuing theory A takeaway restaurant has found that customers arrive for service at an average rate of 10 per hour. At present the service rate is 14 customers on average per hour. There is currently only one server. Assume Poisson arrivals and exponential service times, and the M/M/1 system a. What is the average waiting time in this system, the average number of customers in the queue and the average number of customers in the system? [6 Marks] b. The 1/u ratio plays an important role in queuing models. Provide an intuitive explanation of this ratio and the constraint placed on it to formally solve queuing models. [4 marks] C. If the cost of waiting is 25 per hour for each customer in the system and the cost of a server is 30 per hour, would it be worth adding another server? Assume the same service rate of 14 customers per hour for a new server and the M/M/s system. [5 marks] d. Suppose waiting costs now fall to 20. Set out one factor that might cause a fall in waiting costs. Would your recommendation in 3c still hold? (5 marks]