For a single server, single queue system with an arrival rate of 6 customers per hour and a service rate of 15 customers per hour, what is the probability of exactly one customer in the system (to two decimal places)?
A) 0.90
B) 0.40
C) 0.24
D) Cannot be determined from the information given

Question

Quizplus · Accepted Answer

This is a M/M/1 queuing system where the arrival rate (λ) is 6 customers per hour and the service rate (μ) is 15 customers per hour. 
The utilization factor (ρ), which is the probability that the server is busy, can be calculated as:
ρ = λ/μ = 6/15 = 0.4
The probability of exactly one customer in the system can be calculated as:
P(1 customer in the system) = P(1 customer in the queue) × P(server is idle) + P(1 customer being served) 
= ((1-ρ) × ρ^1) + (1-ρ) 
= ((1-0.4) × 0.4^1) + (1-0.4) 
= 0.24
Therefore, the probability of exactly one customer in the system is 0.24, so the best choice is C.

For a Single Server, Single Queue System with an Arrival