Consider a single-server queuing system for which the interarrival times are exponentially distributed. A customer who arrives and finds the server busy joins the end of a single queue. Service times of customers at the server are also exponentially distributed random variables. On completing service for a customer, the server chooses a customer from the queue (if any) in a FIFO manner.

a. Simulate customer arrivals assuming that the mean interarrival time equals the mean service time (e.g., consider that both of these mean values are equal to 1 minute). Create a plot of number of customers in the queue (y-axis) versus simulation time (x-axis). Is the system stable? (Hint: Run the simulation long enough [e.g., 10,000 minutes] to be able to determine whether or not the process is stable.)

b. Consider now that the mean interarrival time is 1 minute, and the mean service time is 0.7 minute. Simulate customer arrivals for 5000 minutes and calculate:

(i) the average waiting time in the queue, (ii) the maximum waiting time in the queue, (iii) the maximum queue length, (iv) the proportion of customers having a delay time in excess of 1 minute, and (v) the expected use of the server.