Customers at an amusement park arrive at the rate of 10 customers per hour.The entry booth is staffed by one employee.The mean service time at the booth to provide service to each customer is 5 minutes.The arrival rate follows a Poisson distribution,and the service time at the booth follows a negative exponential distribution.Determine the average number of customers waiting in line.
A)3.23 customers
B)4.17 customers
C)8 customers
D)7.65 customers

Question

Quizplus · Accepted Answer

We can use the M/M/1 queueing model to solve this problem, where M indicates the Poisson arrival distribution and 1 indicates one server.

The arrival rate λ = 10 customers per hour
The mean service rate μ = 1/5 customers per minute = 12 customers per hour (since there are 60 minutes in an hour)

The utilization factor ρ = λ/μ = 10/12 = 5/6

The average number of customers in the system L = ρ/(1-ρ) = (5/6)/(1-5/6) = 5 customers
(The "system" includes both the customers in the queue and the one being served at the booth)

The average number of customers waiting in line (in the queue) is Lq = L - ρ = 5 - 5/6 = 4.17 customers.

Therefore, the best choice is B, 4.17 customers.

Customers at an Amusement Park Arrive at the Rate of 10