Consider a single server queuing system. The system starts at time t=0. The arrival time ofcustomers is: 0.8, 1.4, 2.7, 3.2, 3.8, 8.0, 8.6, 9.0, 9.2, and 9.8. The departure times are: 2.2,4.0, 5.0, 6.2, and 10.0. Time is in minutes. The first in first out queuing discipline isfollowed. Simulate this system for six clients and estimate:i) The average delay in the waiting line (3 Marks)ii) The average number of clients in the waiting line at any time t (5 Marks)iii) The expected utilization of server. (2 Marks)

To solve this problem, we need to simulate the queuing system step by step.

i) The average delay in the waiting line:

First, we need to calculate the delay for each customer. The delay is the difference between the arrival time and the departure time.

For the first customer, the delay is 2.2 - 0.8 = 1.4 minutes. For the second customer, the delay is 4.0 - 1.4 = 2.6 minutes. For the third customer, the delay is 5.0 - 2.7 = 2.3 minutes. For the fourth customer, the delay is 6.2 - 3.2 = 3.0 minutes. For the fifth customer, the delay is 10.0 - 3.8 = 6.2 minutes.

The average delay is the sum of these delays divided by the number of customers, which is 5.

So, the average delay is (1.4 + 2.6 + 2.3 + 3.0 + 6.2) / 5 = 3.1 minutes.

ii) The average number of clients in the waiting line at any time t:

To calculate this, we need to find the total time that there were customers in the line and divide it by the total time.

The total time that there were customers in the line is the sum of the delays, which is 15.5 minutes.

The total time is the last departure time, which is 10.0 minutes.

So, the average number of clients in the line is 15.5 / 10.0 = 1.55.

iii) The expected utilization of server:

The utilization of the server is the total time that the server was busy divided by the total time.

The server was busy whenever there was a customer in the line, which is the same as the total time that there were customers in the line, 15.5 minutes.

So, the utilization of the server is 15.5 / 10.0 = 1.55, or 155%.

This is greater than 100% because the server was busy for more time than the total time, which means that there were customers in the line waiting to be served even after the last customer arrived.