1. (Single Queue, Single Server System Simulation) The following data show the arrival times and service times (in minutes) of 10 customers. Performance measures are 1) average number of customers in queue, 2) average delay, and 3) server utilization. The system starts its operation at 9am.

Arrival Times: 1, 6, 9, 15, 19, 21, 30, 34, 39, 50 Service Times: 4, 20, 5, 10, 5, 9, 2, 8, 4, 5

That is, customer 1 arrives at 9:01am and will require 4 minutes for his/her service to be completed once he/she enters the service. At the beginning of the simulation, there’s no customer in the system and the server is idle. There’s no limit on the queue size. The simulation will terminate when the 10th delay is observed.

(a) What are the system state(s) that need(s) to be introduced to calculate the three performance measures?

(b) What are the events?

(c) Draw the event timeline diagram.

(d) Draw the system state(s) graph(s) (i.e., a graph with a system state (y axis) vs simulation time (x axis))

(e) Find departure times of 10 customers.

(f) Calculate the three performance measures. Show the steps in detail. To do the part (c)–(f), create a table, similar to the one used in class, that shows simulation clock, next arrival time, next departure time, customer delay, number of delays, number of customers in queue, and server status. (Note that you are expected to do hand simulation and provide the handwritten solution.)

2. (MM1 Computer Simulation) Now we assume that the inter-arrival times and service times follow an exponential distribution with mean of 2 minutes and 1.5 minutes, respectively. Simulation will terminate when 10th delay is observed. Modify the MM1 VBA code provided in the course to run the simulation 20 times. Find the average and standard deviation of the three performance measures. Explain in words how you get the solution. (e.g., introduction of new variables, new loops introduced to run simulation 20 times, how to get across-simulation averages, etc)

3. (Single Queue, Two Server System Hand Simulation) Now we assume that there are two servers, instead of one. The arriving customers will enter the service when any of the two servers is idle. If both servers are idle, the arriving customer will go to server 1. The customers will form a single queue and the the first waiting customer in queue will enter service when any of the servers becomes idle. Use the data in Problem 1, consider four performance measures: 1) average number of customers in queue, 2) average delay, 3) server 1 utilization, and 4) server 2 utilization, and repeat part (a)–(f). Note that you are required to do hand simulation.