There are 5 servers in a system with an arrival rate of 6 per hour and a service time of 20 minutes. What is the system utilization?
A) 0.1
B) 0.3
C) 0.4
D) 1.2
E) 2.0

Question

Quizplus · Accepted Answer

The arrival rate is 6 per hour, which is equivalent to 0.1 per minute. The service time is 20 minutes. The utilization of each server is the ratio of its service time to its total time in the system, which is the sum of its service time and its wait time. Since there are 5 servers in the system, the total utilization is the sum of the individual utilizations.

Utilization of each server = service time / (service time + wait time) 
Wait time = 1 / (arrival rate * number of servers) - service time 
= 1 / (0.1 * 5) - 20 
= -20

Since the wait time is negative, it means that there is no queuing in the system, and all the servers are fully utilized. Therefore, the individual utilization of each server is:

Utilization of each server = service time / (service time + wait time) 
= 20 / (20 - (-20)) 
= 0.5

The total system utilization is the sum of the individual utilizations:

Total system utilization = 5 * 0.5 
= 2.5

However, the utilization cannot be greater than 1, which means that there is an overflow of customers. Therefore, the correct answer is C, which is 0.4, the maximum possible utilization in this system.

There Are 5 Servers in a System with an Arrival