Question: 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. Upon 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 min). 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 min] to be able to determine whether or not the process is stable.)

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

(v) the expected utilization of the server.