Weary travelers arrive at Will Rogers International Airport, pick up their luggage, stumble to their cars, and proceed to the parking lot attendant to pay for their parking. Traveler interarrival times are exponentially distributed, as are the service times of the attendant. On average, travelers arrive every 25 seconds. The attendant can process three travelers per minute.
-Use the information in Scenario C.2.What is the average number of customers in the system?

Question

Quizplus · Accepted Answer

We can use Little's Law to find the average number of customers in the system. Little's Law states that the average number of customers in a system is equal to the average arrival rate multiplied by the average time a customer spends in the system (also known as the average system time).
The arrival rate is 1/25 customers per second, or 0.04 customers per second.
The service rate is 3 customers per minute, or 0.05 customers per second.
Using the formula: L = λW
L = (0.04)(W)
We need to find the average time a customer spends in the system, W. To do this, we can use the formula for the average waiting time in a queue, which is Wq = Lq/λ. Since our queue is the same as the system in this case (there is only one server), we can use Wq as our estimate for W.
Wq = Lq/λ
Lq = λ^2/(μ(μ-λ)) where μ is the mean service rate.
Lq = (0.04)^2/(0.05(0.05-0.04))
Lq = 0.0016/0.0005
Lq = 3.2
Wq = Lq/λ
Wq = 3.2/0.04
Wq = 80 seconds
Now we can use Little's Law to find L, the average number of customers in the system:
L = λW
L = (0.04)(80)
L = 3.2
Therefore, the average number of customers in the system is greater than 3.0 but fewer than or equal to 5.0.

Weary Travelers Arrive at Will Rogers International Airport, Pick Up